The

Power

of

Local

Optimization

Approximation

Algorithms

for

Maximum-leaf

Spanning

Tree

-LRB-

DRAFT

-RRB-

Hsueh-I

Lu

and

R.

Ravi

Department

of

Computer

Science

Brown

University

Providence

Rhode

Island

January

The

Power

of

Local

Optimization

Approximation

Algorithms

for

Maximum-leaf

Spanning

Tree

-LRB-

DRAFT

-RRB-

Hsueh-I

Lu

hil@cs.brown.edu

Department

of

Computer

Science

Brown

University

R.

Ravi

ravi+@andrew.cmu.edu

Graduate

School

of

Industrial

Administration

Carnegie

Mellon

University

November

Abstract

Given

an

undirected

graph

G

finding

a

spanning

tree

of

G

with

maximum

number

of

leaves

is

NP-complete

We

use

the

simple

technique

of

local

optimization

to

provide

the

first

approximation

algorithms

for

this

problem

Our

algorithms

run

in

polynomial

time

to

produce

locally

optimal

solutions

We

prove

that

locally

optimal

solutions

to

this

problem

are

globally

nearoptimal

In

particular

we

prove

that

two

such

algorithms

have

performance

ratios

of

and

The

latter

algorithm

employs

more

powerful

local-improvement

steps

than

the

former

and

hence

has

higher

running

time

This

may

indicate

an

interesting

trade-off

between

the

performance

ratios

and

the

running

times

of

the

series

of

algorithms

we

describe

Introduction

Given

an

undirected

graph

G

-LRB-

V

E

-RRB-

the

Maximum

Leaf

Spanning

Tree

problem

is

to

find

a

spanning

tree

of

G

with

maximum

number

of

leaves

This

problem

finds

applications

in

communication

networks

circuit

layouts

and

in

other

graph-theoretic

problems

-LSB-

-RSB-

An

interesting

application

is

mentioned

in

-LSB-

-RSB-

In

-LSB-

-RSB-

E.

W.

Dijkstra

studied

the

problem

of

self-stabilizing

a

set

of

processors

in

the

presence

of

distributed

control

and

proposed

a

solution

based

on

mutual

exclusion

A

variant

of

this

solution

-LSB-

-RSB-

seeks

a

spanning

tree

of

a

graph

such

that

the

product

of

the

degrees

of

all

the

nodes

in

the

tree

is

minimum

We

can

use

a

spanning

tree

with

maximum

number

of

leaves

as

a

heuristic

solution

-LRB-

with

no

guarantees

-RRB-

for

the

latter

problem

Maximum

Leaf

Spanning

Tree

is

NP-complete

-LSB-

-RSB-

In

this

paper

we

use

the

simple

technique

of

local

optimization

to

provide

the

first

approximation

algorithms

for

this

problem

These

algorithms

perform

a

series

of

local-improvement

steps

until

a

local

optimum

is

reached

and

output

a

locally

optimal

solution

This

notion

of

applying

local-improvement

heuristics

to

hard

optimization

problems

was

around

-LSB-

-RSB-

long

before

the

invention

of

NP-completeness

-LSB-

-RSB-

It

has

been

applied

heuristically

to

solve

a

variety

of

NP-hard

problems

in

combinatorial

optimization

-LSB-

-RSB-

A

breakthrough

in

this

area

was

the

work

of

F?urer

and

Raghavachari

-LSB-

-RSB-

who

applied

this

technique

The

preliminary

version

of

this

paper

appeared

in

Proceedings

Thirtieth

Annual

Allerton

Conference

on

Communication

Control

and

Computing

-LRB-

October

-RRB-

pp.

to

the

Minimum-Degree

Spanning

Tree

problem

They

showed

a

version

of

a

local

optimization

algorithm

that

runs

in

polynomial

time

to

produce

near-optimal

solutions

Our

work

is

inspired

by

that

of

F?urer

and

Raghavachari

We

demonstrate

that

local

optimization

can

be

used

to

design

polynomial-time

approximation

algorithms

for

the

Maximum

Leaf

Spanning

Tree

problem

Our

algorithms

output

spanning

trees

with

as

many

leaves

as

at

least

a

constant

fraction

of

the

maximum

Our

algorithms

perform

local

changes

that

increase

the

number

of

leaves

in

the

resulting

spanning

tree

For

the

problem

at

hand

there

is

a

natural

notion

of

such

a

local

change

namely

swapping

a

tree

edge

for

a

non-tree

edge

This

notion

can

be

extended

to

include

swapping

many

tree

edges

for

an

equal

number

of

non-tree

edges

in

a

single

local

change

operation

Our

algorithms

perform

such

local

changes

that

increase

the

number

of

leaves

until

no

improvement

in

the

solution

is

possible

and

output

a

locally

optimal

solution

as

an

approximate

solution

By

varying

the

limit

on

the

number

of

tree

edges

that

can

participate

in

a

single

local-change

operation

we

derive

a

series

of

approximation

algorithms

Increasing

this

limit

corresponds

to

allowing

for

more

powerful

local-improvement

steps

Thus

the

higher

is

this

limit

the

higher

is

the

running

time

of

the

corresponding

algorithm

A

local-change

operation

involving

at

most

k

tree

edges

is

termed

a

k-change

Thus

k-changes

are

costlier

to

implement

than

-LRB-

k

-RRB-

changes

Intuitively

we

expect

that

an

algorithm

using

k-changes

would

perform

better

than

an

algorithm

using

only

-LRB-

k

-RRB-

changes

In

this

paper

we

corroborate

this

intuition

and

prove

that

local-improvement

algorithms

using

and

have

performance

ratios

of

and

respectively

The

following

are

our

main

results

Theorem

There

is

an

O

-LRB-

-RRB-

algorithm

using

for

finding

a

spanning

tree

of

any

undirected

graph

G

on

n

nodes

such

that

the

number

of

leaves

of

this

spanning

tree

is

at

least

as

many

as

a

fifth

of

the

number

of

leaves

of

any

spanning

tree

of

G.

Theorem

There

is

an

O

-LRB-

-RRB-

algorithm

using

for

finding

a

spanning

tree

of

any

undirected

graph

G

with

n

nodes

such

that

the

number

of

leaves

of

this

spanning

tree

is

at

least

as

many

as

a

third

of

the

number

of

leaves

of

any

spanning

tree

of

G

We

prove

that

the

running

time

of

the

approximation

algorithm

using

k-changes

increases

with

k

as

follows

Lemma

The

running

time

of

the

approximation

algorithm

using

k-changes

on

a

graph

with

n

nodes

is

O

-LRB-

-RRB-

The

pattern

exhibited

in

Theorems

and

lead

us

to

conjecture

that

there

is

a

trade-off

between

the

running

times

and

the

performance

ratios

exhibited

in

the

series

of

algorithms

we

have

defined

Conjecture

The

performance

of

the

approximation

algorithm

using

k-changes

is

strictly

better

than

that

of

the

algorithm

using

-LRB-

k

-RRB-

changes

In

the

next

section

we

survey

related

work

We

then

describe

the

approximation

algorithms

after

introducing

relevant

definitions

In

the

following

section

we

illustrate

our

proof

technique

by

providing

a

loose

bound

on

the

performance

guarantee

In

the

subsequent

sections

we

prove

Theorems

and

We

conclude

with

some

remarks

on

related

problems

Related

Work

The

technique

of

local

optimization

has

been

applied

to

provide

heuristic

solutions

for

a

variety

of

hard

problems

in

combinatorial

optimization

Chapter

of

-LSB-

-RSB-

surveys

a

few

applications

of

this

technique

Some

notable

examples

are

its

applications

to

the

graph-partitioning

problem

-LSB-

-RSB-

and

the

Traveling

Salesperson

problems

-LSB-

-RSB-

A

number

of

complexity

results

relating

to

the

paradigm

of

local

optimality

and

the

time

complexity

of

computing

a

locally

optimal

solution

are

presented

in

-LSB-

-RSB-

In

this

paper

we

are

interested

in

the

application

of

the

local

search

techniques

to

design

efficient

approximation

algorithms

namely

those

that

run

in

polynomial

time

and

provide

provably

good

solutions

with

values

close

to

that

of

the

optimum

F?urer

and

Raghavachari

-LSB-

-RSB-

showed

such

an

application

of

local

search

to

the

problem

of

computing

a

spanning

tree

of

a

given

graph

whose

maximum

degree

is

minimum

This

is

termed

the

Minimum

Degree

Spanning

Tree

problem

and

is

NP-complete

-LSB-

-RSB-

F?urer

and

Raghavachari

show

that

local

optimization

can

be

applied

to

produce

spanning

trees

and

even

Steiner

trees

of

nearly

minimum

degree

-LSB-

-RSB-

The

maximum

degree

of

the

spanning

trees

obtained

by

their

polynomial-time

approximation

algorithms

is

at

most

one

plus

the

maximum

degree

of

the

minimum-degree

spanning

trees

Unless

P

NP

this

performance

guarantee

is

the

best

bound

achievable

in

polynomial

time

The

result

is

generalized

to

work

on

the

Minimum

Degree

Steiner

Tree

problem

with

the

same

performance

guarantee

-LSB-

-RSB-

In

-LSB-

-RSB-

Ravi

Raghavachari

and

Klein

generalized

their

technique

to

obtain

approximation

algorithms

for

a

variety

of

minimum-degree-network-design

problems

In

this

paper

we

add

to

the

list

of

problems

that

are

amenable

to

good

approximate

solutions

by

localsearch

methods

Very

recently

using

the

technique

of

local

optimization

Halld?orsson

-LSB-

-RSB-

gives

the

approximation

algorithms

with

the

currently

best

performance

ratios

for

a

set

of

problems

including

-LRB-

a

-RRB-

k-Set

Cover

-LRB-

b

-RRB-

Independent

Set

Vertex

Cover

k-Set

Packing

k-Dimensional

Matching

and

k-Clique

Packing

for

-LRB-

k

-RRB-

claw

free

graphs

-LRB-

c

-RRB-

Induced

Colorable

Subgraph

and

Vertex

Arboricity

which

are

special

cases

of

Hereditary

Induced

Subgraph

and

etc.

Previous

work

on

finding

spanning

trees

with

many

leaves

have

focused

on

graphs

with

minimum

degree

at

least

k

for

some

fixed

k

For

such

graphs

good

lower

bounds

on

the

number

of

leaves

achievable

in

a

spanning

tree

are

derived

in

-LSB-

-RSB-

These

lower

bounds

are

typically

proved

algorithmically

by

constructing

a

spanning

tree

with

the

desired

number

of

leaves

Thus

these

algorithms

can

be

interpreted

as

approximation

algorithms

for

the

Maximum

Leaf

Spanning

Tree

problem

for

special

classes

of

graphs

in

which

the

minimum

degree

of

a

node

is

at

least

k.

However

to

the

best

of

our

knowledge

no

previous

approximation

algorithms

for

this

problem

are

known

in

the

general

case

that

we

consider

in

this

paper

The

best-known

lower

bound

for

the

number

of

leaves

achievable

in

a

spanning

tree

of

an

n-node

graph

with

minimum

degree

k

is

-LRB-

b

ln

k

k

-RRB-

n

where

b

is

any

constant

exceeding

and

k

is

sufficiently

large

-LSB-

-RSB-

The

best

lower

bounds

for

the

cases

k

and

are

n

and

respectively

-LSB-

-RSB-

There

has

also

been

work

on

polynomial-time

solutions

to

the

problem

of

determining

if

a

given

graph

has

a

spanning

tree

with

at

least

k

leaves

for

fixed

k

The

first

such

algorithm

was

due

to

Fellows

and

Langston

-LSB-

-RSB-

The

running

time

of

their

algorithm

was

improved

by

Bodlaender

-LSB-

-RSB-

Papadimitriou

and

Yannakakis

-LSB-

-RSB-

identified

a

class

of

NP-hard

optimization

problems

interreducible

to

one

another

using

approximation-preserving

reductions

and

thus

took

a

step

towards

classifying

NP-complete

problems

with

respect

to

hardness

of

approximating

them

They

called

this

class

MAX

SNP

It

has

been

shown

that

the

problems

complete

for

this

class

do

not

permit

a

fully

polynomial-time

approximation

scheme

unless

P

NP

-LSB-

-RSB-

Galbiati

Maffioli

and

Morzenti

show

that

Maximum

Leaf

Spanning

Tree

is

MAX

SNP-complete

-LSB-

-RSB-

which

settles

the

open

question

raised

in

the

preliminary

version

of

this

paper

-LSB-

-RSB-

Therefore

there

exists

some

constant

such

that

there

is

no

-LRB-

-RRB-

approximation

for

Maximum

Leaf

Spanning

Tree

unless

P

NP

What

the

is

for

the

Maximum

Leaf

Spanning

Tree

problem

remains

open

Preliminaries

Let

G

be

an

undirected

graph

Let

be

a

subgraph

of

G

We

use

E

-LRB-

-RRB-

to

denote

the

set

of

edges

in

and

use

V

-LRB-

-RRB-

to

denote

the

set

of

nodes

in

A

node

v

is

in

if

v

V

-LRB-

-RRB-

an

edge

e

is

in

if

e

E

-LRB-

-RRB-

We

shall

sometimes

refers

to

an

edge

uw

of

G

as

the

edge

incident

to

u

and

w

We

say

e

is

incident

to

if

an

endpoint

of

e

is

in

The

degree

of

a

node

v

in

is

the

number

of

edges

of

that

are

incident

to

v

We

use

Vi

-LRB-

-RRB-

to

denote

the

set

of

nodes

that

have

degree

i

in

G

and

use

Vi

-LRB-

-RRB-

to

denote

the

set

of

nodes

that

have

degree

at

least

i

in

Clearly

-LRB-

G

-RRB-

V

-LRB-

G

-RRB-

if

is

connected

and

has

at

least

two

nodes

Suppose

G

has

more

than

one

node

Let

T

be

a

spanning

tree

of

G

A

leaf

of

T

is

a

node

in

-LRB-

T

-RRB-

An

internal

node

of

T

is

a

node

in

-LRB-

T

-RRB-

A

high-degree

node

of

T

is

a

node

in

-LRB-

T

-RRB-

A

leaf

of

T

is

special

if

it

is

adjacent

in

T

to

a

high-degree

node

of

T

A

leaf

of

T

is

normal

if

it

is

not

special

Therefore

a

normal

leaf

of

T

is

adjacent

in

T

to

a

node

in

-LRB-

T

-RRB-

unless

G

has

only

two

nodes

We

use

-LRB-

T

-RRB-

to

denote

the

set

of

special

leaves

of

T

and

use

-LRB-

T

-RRB-

to

denote

the

set

of

normal

leaves

of

T

We

use

pathT

-LRB-

u

w

-RRB-

to

denote

the

path

that

connects

u

and

w

in

T

Note

that

u

and

w

are

not

necessarily

distinct

We

define

pathT

-LRB-

u

u

-RRB-

to

be

u

itself

A

of

T

is

a

maximal

path

in

T

that

contains

only

nodes

in

-LRB-

T

-RRB-

Namely

pathT

-LRB-

u

w

-RRB-

is

a

of

T

if

all

nodes

in

pathT

-LRB-

u

w

-RRB-

are

in

-LRB-

T

-RRB-

and

all

nodes

in

-LRB-

T

-RRB-

V

-LRB-

pathT

-LRB-

u

w

-RRB-

-RRB-

are

not

adjacent

to

u

or

w

in

T

A

P

of

T

is

short

if

jV

-LRB-

P

-RRB-

j

otherwise

it

is

long

We

use

PS

-LRB-

T

-RRB-

to

denote

the

set

of

short

of

T

and

user

PL

-LRB-

T

-RRB-

to

denote

the

set

of

long

of

T

We

say

a

node

v

is

enclosed

by

an

edge

uw

in

G

T

if

v

u

v

w

and

v

is

in

pathT

-LRB-

u

w

-RRB-

A

region

of

T

is

a

connected

component

of

the

subgraph

of

T

obtained

from

T

by

removing

all

nodes

in

PL

-LRB-

T

-RRB-

and

all

edges

of

T

that

have

endpoints

in

PL

-LRB-

T

-RRB-

A

region

of

T

is

special

if

it

contains

special

leaves

of

T

A

region

of

T

is

normal

if

it

is

not

special

Suppose

R

and

are

two

distinct

regions

of

T

The

long

of

T

that

connects

R

and

in

T

is

called

the

long

incident

to

R

and

The

number

of

long

incident

to

a

region

of

T

is

the

degree

of

the

region

A

region

is

internal

if

it

has

degree

at

least

two

A

region

is

external

if

it

is

not

internal

We

use

RiS

-LRB-

T

-RRB-

to

denote

the

set

of

special

regions

of

T

that

have

degree

i

and

use

RiN

-LRB-

T

-RRB-

to

denote

the

set

of

normal

regions

of

T

that

have

degree

i.

Let

RS

-LRB-

T

-RRB-

and

RN

-LRB-

T

-RRB-

be

the

sets

of

special

and

normal

regions

of

T

respectively

There

are

some

straightforward

relations

among

these

notations

Contracting

every

in

a

tree

to

a

single

edge

and

bounding

the

number

of

internal

nodes

-LRB-

each

resulting

internal

node

has

degree

at

least

-RRB-

in

this

contracted

tree

by

the

number

of

leaves

we

have

the

following

inequality

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

-RRB-

By

discarding

special

leaves

along

with

their

incidental

edges

from

a

tree

contracting

all

to

single

edges

and

bounding

the

number

of

contracted

edges

in

the

resulting

tree

by

the

number

of

nodes

we

have

the

following

inequality

jPL

-LRB-

T

-RRB-

j

jPS

-LRB-

T

-RRB-

j

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

-RRB-

Suppose

we

regard

T

as

a

tree

of

regions

by

contracting

every

long

of

T

to

a

single

edge

The

number

of

nodes

-LRB-

i.e.

regions

of

T

-RRB-

is

exactly

one

plus

the

number

of

edges

-LRB-

i.e.

long

e

e

e

-LRB-

b

-RRB-

e

-LRB-

a

-RRB-

T

-LRB-

c

-RRB-

Figure

Examples

of

local

improvements

In

-LRB-

a

-RRB-

the

tree

T

is

shown

in

dark

lines

and

the

nontree

edges

are

dashed

The

-LRB-

feg

-RRB-

is

a

for

T

resulting

in

with

one

more

leaf

as

shown

in

-LRB-

b

-RRB-

does

not

admit

any

but

admits

the

-LRB-

-RRB-

to

give

shown

in

-LRB-

c

-RRB-

of

T

-RRB-

Also

the

sum

of

the

degrees

of

nodes

is

exactly

two

times

the

number

of

edges

jPL

-LRB-

T

-RRB-

j

jRS

-LRB-

T

-RRB-

j

jRN

-LRB-

T

-RRB-

j

-LRB-

-RRB-

-LRB-

T

-RRB-

j

X

i

i

-LRB-

jRiS

-LRB-

T

-RRB-

j

jRiN

-LRB-

T

-RRB-

j

-RRB-

-LRB-

-RRB-

Since

each

special

region

of

T

contains

at

least

one

special

special

leaf

of

T

Therefore

we

have

jRS

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

-RRB-

One

can

easily

verify

the

following

lemmas

Lemma

Let

T

be

a

spanning

tree

of

G.

Let

u

v

and

w

be

three

distinct

nodes

of

G

If

v

is

in

pathT

-LRB-

u

w

-RRB-

then

v

is

not

a

leaf

of

T

The

Algorithm

In

this

section

we

formally

define

k-changes

and

describe

the

approximation

algorithms

Edge

Changes

Improvements

and

Locally

Optimal

Trees

Suppose

T

is

a

spanning

tree

of

G.

Let

E

be

a

subset

of

edges

in

T

and

a

subset

of

edges

in

G?T

where

jEj

k

for

some

k

If

T

is

still

a

spanning

tree

of

G

then

we

call

making

edges

in

tree

edges

and

making

edges

in

E

non-tree

edges

a

k-change

denoted

-LRB-

E

-RRB-

for

T

We

use

T

-LRB-

E

-RRB-

to

denote

T

E

A

k-change

-LRB-

E

-RRB-

for

T

is

a

k-improvement

if

-LRB-

T

-LRB-

E

-RRB-

-RRB-

j

-LRB-

T

-RRB-

j

A

k-change

can

be

regarded

as

a

batched

version

of

a

sequence

of

k

A

k-change

is

more

powerful

than

k

In

Figure

we

illustrate

a

spanning

tree

T

that

admits

a

On

applying

the

we

get

a

spanning

tree

that

does

not

admit

any

but

has

a

A

k-locally

optimal

tree

-LRB-

k-LOT

-RRB-

of

a

given

graph

G

is

a

spanning

tree

of

G

that

does

not

admit

any

k-improvement

By

definition

an

k-LOT

is

also

a

j-LOT

for

any

j

k

Since

a

spanning

tree

has

exactly

n

edges

a

maximum-leaf

spanning

tree

is

just

an

-LRB-

n

-RRB-

LOT

The

Algorithm

and

Termination

The

idea

is

simply

using

k-LOTs

to

approximate

the

maximum-leaf

spanning

trees

for

small

values

of

k

The

approximation

algorithm

is

as

follows

e

e

e

e

e

eviolate

violate

Figure

The

dark

edges

represent

tree

edges

Each

dashed

edge

is

a

non-tree

edge

violating

one

of

the

properties

and

Note

that

the

-LRB-

feg

-RRB-

in

each

graph

is

a

Procedure

LocallyOptimalTree

-LRB-

G

k

-RRB-

Let

T

be

a

spanning

tree

of

G

While

there

exists

a

k-improvement

-LRB-

E

-RRB-

for

T

do

Let

T

T

-LRB-

E

-RRB-

Output

T

as

a

k-LOT

of

G

The

algorithm

starts

with

an

arbitrary

spanning

tree

which

can

be

found

in

time

O

-LRB-

-RRB-

where

n

is

the

number

of

nodes

in

G

It

then

keeps

applying

k-improvements

to

the

current

spanning

tree

until

it

becomes

a

k-LOT

Clearly

the

second

step

dominates

the

time

complexity

One

can

determine

in

polynomial

time

whether

a

tree

is

a

k-LOT

since

the

number

of

distinct

k-changes

allowed

for

a

spanning

tree

is

at

most

-LRB-

-RRB-

n

k

n

which

is

O

-LRB-

-RRB-

The

number

of

iterations

required

by

the

while-loop

is

at

most

n

since

each

iteration

increases

the

number

of

leaves

of

the

current

T

by

at

least

one

and

T

has

at

most

n

leaves

Therefore

it

takes

time

O

-LRB-

-RRB-

to

obtain

a

k-LOT

This

proves

Lemma

Our

major

concern

is

the

performance

guarantee

of

this

approximation

algorithm

In

other

words

how

much

less

can

the

number

of

leaves

of

a

k-LOT

be

than

that

of

the

optimum

In

the

following

sections

we

prove

that

the

performance

ratios

of

and

are

and

respectively

However

we

conjecture

that

have

a

better

performance

ratio

than

what

we

prove

Properties

of

Locally

Optimal

Trees

Instead

of

jumping

into

proving

the

performance

guarantees

of

and

we

start

with

showing

some

straightforward

properties

of

locally

optimal

trees

At

the

end

of

the

section

we

illustrate

our

proof

strategy

by

proving

a

loose

bound

on

the

performance

ratio

of

Let

G

be

an

undirected

graph

Let

T

be

a

of

G.

By

definition

a

has

the

following

properties

They

are

illustrated

in

Figure

Let

uw

be

an

edge

in

G

T

where

u

is

an

internal

node

of

T

Then

pathT

-LRB-

u

w

-RRB-

does

not

contain

two

consecutive

nodes

in

-LRB-

T

-RRB-

Let

u

be

a

normal

leaf

of

T

and

let

w

be

an

internal

node

of

T

Then

there

is

no

edge

uw

in

G

T

Let

uw

be

an

edge

in

G

T

where

u

and

w

are

both

internal

in

T

Then

pathT

-LRB-

u

w

-RRB-

does

not

contain

any

node

in

-LRB-

T

-RRB-

The

following

is

critical

to

showing

a

constant

performance

ratio

of

v

v

u

w

v

Figure

Lemma

Let

T

be

a

of

an

undirected

graph

G.

Let

T

be

a

spanning

tree

of

G

Then

every

of

T

contains

at

most

four

leaves

of

T

Proof

Assume

for

a

contradiction

that

a

P

of

T

contains

more

than

four

leaves

of

T

Let

v

and

be

five

of

them

in

order

along

P

Let

be

a

node

not

in

P

Let

u

be

the

node

closest

to

v

in

pathT

-LRB-

v

-RRB-

such

that

u

is

not

in

P

Let

w

be

the

node

closest

to

w

in

pathT

-LRB-

v

u

-RRB-

-LRB-

See

Figure

-RRB-

Clearly

uw

is

an

edge

of

T

T

and

w

is

in

P

By

we

know

that

T

T

does

not

have

any

edge

incident

to

distinct

nodes

in

P

It

follows

that

w

must

be

in

pathT

-LRB-

-RRB-

Therefore

uw

encloses

either

and

or

and

This

contradicts

the

of

T

by

With

the

help

of

the

above

lemma

we

show

that

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

for

any

T

of

G

and

any

spanning

tree

T

of

G.

Note

that

this

relation

holds

even

if

T

is

a

maximum-leaf

spanning

tree

of

G.

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

jPS

-LRB-

T

-RRB-

j

jPL

-LRB-

T

-RRB-

j

-RRB-

j

-LRB-

T

-RRB-

j

by

Lemma

-LRB-

T

-RRB-

j

-LRB-

-LRB-

T

-RRB-

j

j

-LRB-

T

-RRB-

j

-RRB-

j

-LRB-

T

-RRB-

j

by

-LRB-

-RRB-

-LRB-

T

-RRB-

j

by

-LRB-

-RRB-

Performance

Guarantee

of

In

this

section

we

prove

the

following

restatement

of

Theorem

Theorem

Let

be

a

of

an

undirected

graph

Let

be

a

of

and

T

a

spanning

tree

of

Then

-LRB-

T

-RRB-

j

-LRB-

-RRB-

j.

Augmentation

In

order

to

simplify

the

proof

of

Theorem

we

augment

together

with

as

follows

For

every

normal

leaf

of

adjacent

to

a

short

we

subdivide

the

tree

edge

incident

to

the

normal

leaf

by

inserting

a

pseudo-node

of

degree

in

this

edge

Lemma

Suppose

is

an

directed

graph

and

a

of

Let

G

and

T

be

the

augmented

versions

of

and

respectively

Then

T

is

a

of

G.

Figure

Proof

Assume

for

a

contradiction

that

T

is

not

a

of

G

It

follows

that

there

exists

a

for

T

involving

addition

of

an

edge

e

G

T

Since

is

a

of

a

pseudo-node

v

must

be

enclosed

by

e

in

T

Note

that

no

edge

in

G?T

is

incident

to

v.

To

enclose

v

e

must

be

incident

to

the

normal

leaf

of

adjacent

to

v

Since

is

a

and

one

of

the

endpoints

of

e

is

a

normal

leaf

of

by

the

other

endpoint

of

e

should

be

a

leaf

of

as

well

By

the

definition

of

augmentation

e

is

incident

to

two

leaves

of

T

too

However

no

non-tree

edge

incident

to

two

leaves

can

participate

in

a

This

contradicts

the

fact

that

edge

e

yields

a

for

T

Thus

T

is

a

An

example

is

shown

in

Figure

The

tree

edges

are

in

solid

lines

and

the

non-tree

edges

are

in

dashed

lines

The

tree

at

left

is

a

with

two

external

short

The

picture

at

right

is

the

augmented

graph

and

tree

where

the

gray

nodes

are

the

inserted

pseudo-nodes

One

can

easily

verify

that

the

augmented

tree

is

still

a

Moreover

this

example

shows

that

the

augmentation

might

destroy

the

of

a

It

is

not

hard

to

see

that

the

tree

at

left

is

a

and

the

tree

at

right

is

not

a

We

say

T

is

an

augmented

of

G

if

T

and

G

is

augmented

from

and

where

is

a

of

By

definition

of

augmentation

an

augmented

does

not

have

any

external

short

Therefore

every

normal

of

leaf

of

an

augmented

is

a

single-node

external

region

Therefore

every

augmented

T

has

the

following

property

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

jRN

-LRB-

T

-RRB-

j

-LRB-

-RRB-

The

following

lemma

ensures

that

if

we

can

prove

that

the

performance

ratio

of

augmented

is

at

most

five

then

Theorem

is

proved

Lemma

Let

be

a

of

Let

G

and

T

be

the

augmented

versions

of

and

respectively

Then

-LRB-

T

-RRB-

j

-LRB-

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

where

T

is

a

maximum-leaf

spanning

tree

of

and

T

is

a

maximum-leaf

spanning

tree

of

G.

Proof

The

equality

is

trivial

since

the

augmentation

does

not

change

the

number

of

leaves

in

The

first

inequality

follows

from

the

definition

of

T

The

last

inequality

can

be

proved

by

constructing

a

spanning

tree

of

G

that

has

at

least

-LRB-

T

-RRB-

j

leaves

We

start

with

T

For

each

pseudo-node

v

to

be

inserted

if

the

unique

edge

whose

subdivision

resulted

in

v

is

in

T

we

subdivide

this

edge

and

insert

the

pseudo-node

v

in

T

Otherwise

let

the

neighbors

of

v

in

G

be

and

To

augment

T

to

include

v

we

simply

attach

v

to

In

either

case

we

obtain

a

spanning

tree

of

G

that

has

at

least

-LRB-

T

-RRB-

j

leaves

The

lemma

is

thus

proved

It

follows

from

the

above

lemma

that

-LRB-

T

-RRB-

j

-LRB-

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

Therefore

in

order

to

prove

Theorem

it

suffices

to

prove

the

following

theorem

Theorem

Let

T

be

an

augmented

of

G

where

G

is

an

undirected

graph

Then

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

holds

for

any

spanning

tree

T

of

G.

For

the

rest

of

the

section

we

assume

that

T

is

an

augmented

of

G

as

given

in

Theorem

Invalid

Regions

Let

T

be

a

spanning

tree

of

G

A

region

R

of

T

is

invalid

in

T

if

either

-LRB-

-RRB-

R

is

a

special

region

of

T

and

each

special

leaf

of

T

in

R

is

a

leaf

of

T

or

-LRB-

-RRB-

R

is

an

internal

normal

region

of

T

A

region

of

T

is

valid

in

T

if

it

is

not

invalid

in

T

Theorem

can

be

proved

using

the

following

lemma

Lemma

Let

T

be

an

augmented

of

G.

Let

T

be

a

spanning

tree

of

G

Then

the

number

of

leaves

of

T

that

are

in

long

of

T

is

at

most

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

where

r

is

the

number

of

special

regions

of

T

that

are

invalid

in

T

Proof

of

Theorem

Since

r

is

the

number

of

special

regions

of

T

that

are

invalid

in

T

there

are

exactly

jRS

-LRB-

T

-RRB-

j

r

special

regions

of

T

that

are

valid

in

T

By

definition

of

invalid

special

regions

each

special

region

of

T

that

is

valid

in

T

contains

at

least

one

special

leaf

of

T

that

is

not

a

leaf

of

T

It

follows

that

there

are

at

least

jRS

-LRB-

T

-RRB-

j

r

special

leaves

of

T

that

are

not

leaves

of

T

Therefore

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

jRS

-LRB-

T

-RRB-

j

r

-RRB-

-LRB-

-RRB-

It

follows

that

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

V

-LRB-

PS

-LRB-

T

-RRB-

-RRB-

j

-LRB-

T

-RRB-

V

-LRB-

PL

-LRB-

T

-RRB-

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

jPS

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

V

-LRB-

PL

-LRB-

T

-RRB-

-RRB-

j

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

jPS

-LRB-

T

-RRB-

j

-LRB-

-LRB-

T

-RRB-

j

r

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-RRB-

j

-LRB-

T

-RRB-

j

by

Lemma

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

jPS

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

r

-LRB-

T

-RRB-

j

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

jRS

-LRB-

T

-RRB-

j

r

-RRB-

jPS

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

r

-LRB-

T

-RRB-

j

j

-LRB-

T

-RRB-

j

by

-LRB-

-RRB-

-LRB-

T

-RRB-

j

jPS

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

jPS

-LRB-

T

-RRB-

j

jRS

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

j

-LRB-

T

-RRB-

j

by

-LRB-

-RRB-

jPS

-LRB-

T

-RRB-

j

jRS

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

j

-LRB-

T

-RRB-

j

jPS

-LRB-

T

-RRB-

j

jPL

-LRB-

T

-RRB-

j

jRN

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

j

-LRB-

T

-RRB-

j

by

-LRB-

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

jRN

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

j

-LRB-

T

-RRB-

j

by

-LRB-

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

by

-LRB-

-RRB-

-LRB-

T

-RRB-

j

by

-LRB-

-RRB-

For

the

rest

of

the

section

we

prove

Lemma

Charge

of

Regions

and

Deficient

Long

The

key

for

proving

Lemma

is

a

tighter

bound

on

-LRB-

T

-RRB-

V

-LRB-

PL

-LRB-

T

-RRB-

-RRB-

j

than

the

bound

of

-LRB-

T

-RRB-

j

given

in

Lemma

Let

R

be

a

region

of

T

Let

the

charge

of

R

in

T

denoted

chargeT

-LRB-

R

-RRB-

be

defined

by

chargeT

-LRB-

R

-RRB-

if

R

is

a

special

region

of

T

if

R

is

a

normal

leaf

of

T

and

a

leaf

of

T

otherwise

We

have

the

following

refined

version

of

Lemma

Lemma

Let

T

be

an

augmented

of

G.

Let

T

be

a

spanning

tree

of

G.

Let

R

and

be

two

distinct

regions

of

T

Let

P

be

the

long

of

T

incident

to

R

and

Then

P

contains

at

most

chargeT

-LRB-

R

-RRB-

chargeT

-LRB-

-RRB-

leaves

of

T

Let

G

T

T

P

R

and

be

as

in

Lemma

The

long

P

is

deficient

in

T

if

ffl

chargeT

-LRB-

R

-RRB-

chargeT

-LRB-

-RRB-

and

ffl

P

contains

at

most

chargeT

-LRB-

R

-RRB-

chargeT

-LRB-

-RRB-

leaves

of

T

The

following

lemma

relates

the

number

of

invalid

regions

and

that

of

deficient

long

Lemma

Let

T

be

an

augmented

of

G.

Let

T

be

a

spanning

tree

of

G.

Let

k

be

the

number

of

regions

of

T

that

are

invalid

in

T

Then

there

are

at

least

k

distinct

long

of

T

that

are

deficient

in

T

Lemma

can

therefore

be

proved

as

follows

Proof

of

Lemma

Let

d

-LRB-

R

-RRB-

be

the

degree

of

the

region

R

of

T

Then

X

-LRB-

T

-RRB-

-LSB-

RN

-LRB-

T

-RRB-

chargeT

-LRB-

R

-RRB-

d

-LRB-

R

-RRB-

k

the

number

of

regions

of

T

that

are

invalid

in

T

It

follows

from

Lemma

and

Lemma

that

the

number

of

leaves

of

T

that

are

in

the

long

of

T

is

at

most

-LRB-

k

-RRB-

We

show

that

-LRB-

k

-RRB-

is

at

most

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j.

Note

that

chargeT

-LRB-

R

-RRB-

for

every

R

-LRB-

T

-RRB-

-LRB-

T

-RRB-

Therefore

X

-LRB-

T

-RRB-

chargeT

-LRB-

R

-RRB-

d

-LRB-

R

-RRB-

X

-LRB-

T

-RRB-

-LRB-

-LRB-

T

-RRB-

-LRB-

T

-RRB-

-RRB-

chargeT

-LRB-

R

-RRB-

d

-LRB-

R

-RRB-

X

-LRB-

T

-RRB-

-LRB-

R

-RRB-

X

-LRB-

T

-RRB-

d

-LRB-

R

-RRB-

X

-LRB-

T

-RRB-

-LRB-

T

-RRB-

d

-LRB-

R

-RRB-

X

i

i

jRiS

-LRB-

T

-RRB-

j

X

i

i

jRiN

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

It

follows

from

-LRB-

-RRB-

and

-LRB-

-RRB-

that

-LRB-

T

-RRB-

j

X

i

i

jRiN

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

X

i

i

jRiN

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

-RRB-

By

definition

a

region

R

of

T

is

invalid

in

T

if

-LRB-

i

-RRB-

R

is

an

internal

normal

region

of

T

-LRB-

i.e.

R

RN

-LRB-

T

-RRB-

-LRB-

T

-RRB-

-LRB-

T

-RRB-

-RRB-

or

-LRB-

ii

-RRB-

R

is

a

special

region

of

T

that

is

in

valid

in

T

Therefore

-LRB-

k

-RRB-

-LRB-

jRN

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

r

-RRB-

-LRB-

jRN

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

r

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

-RRB-

where

the

inequality

above

is

by

-LRB-

T

-RRB-

Combining

-LRB-

-RRB-

-LRB-

-RRB-

and

-LRB-

-RRB-

we

have

-LRB-

k

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

X

i

i

jRiN

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

The

lemma

is

proved

We

prove

Lemma

and

Lemma

respectively

in

the

following

two

subsections

Proof

of

Lemma

Let

i

chargeT

-LRB-

R

-RRB-

and

j

chargeT

-LRB-

-RRB-

Assume

for

a

contradiction

that

P

contains

more

than

i

j

leaves

of

T

Let

vi

v

be

i

j

of

them

in

order

along

P

from

the

side

of

R

to

that

of

i

j

Since

chargeT

-LRB-

R

-RRB-

chargeT

-LRB-

-RRB-

we

know

R

and

are

both

leaves

of

T

and

normal

leaves

of

T

It

follows

that

T

is

a

path

Since

v

is

a

leaf

of

T

but

internal

in

T

we

know

there

must

be

some

edges

in

T

T

By

and

we

know

that

the

only

possible

edge

in

T

T

is

the

edge

incident

to

R

and

Since

R

and

are

both

leaves

of

T

there

is

no

way

for

v

to

be

connected

to

R

or

in

T

This

contradicts

the

fact

that

T

is

a

spanning

tree

of

G.

j

Let

be

a

node

in

Let

P

pathT

-LRB-

v

-RRB-

Let

u

be

the

node

closest

to

v

in

P

such

that

u

is

not

in

P

Let

w

be

the

node

in

pathT

-LRB-

u

v

-RRB-

such

that

uw

is

an

edge

in

P

Clearly

uw

is

an

edge

of

T

T

and

w

is

in

P

It

follows

from

Lemma

that

of

vi

is

in

P

By

we

know

that

there

can

not

be

any

edges

in

T

T

that

is

incident

to

two

distinct

nodes

in

P

It

follows

that

w

must

be

in

pathT

-LRB-

vj

-RRB-

We

first

show

that

u

is

not

in

Assume

for

a

contradiction

that

u

is

in

Since

w

is

in

pathT

-LRB-

vj

-RRB-

we

know

that

uw

encloses

By

and

we

know

that

u

must

be

a

special

leaf

of

T

Thus

is

a

special

region

of

T

which

implies

that

j

Since

uw

encloses

and

the

of

T

is

contradicted

by

It

can

be

shown

that

u

is

not

in

R.

Assume

for

a

contradiction

that

u

is

in

R.

For

exactly

the

same

reason

as

above

we

know

that

i

Therefore

we

have

i

It

follows

from

the

definition

of

chargeT

-LRB-

R

-RRB-

that

R

which

is

exactly

u

is

a

leaf

of

T

This

contradicts

the

fact

that

u

is

in

P

by

Lemma

since

u

v

and

u

Clearly

u

is

not

in

any

region

of

T

that

is

not

R

or

since

otherwise

uw

would

enclose

all

the

nodes

of

some

long

of

T

and

thus

the

of

T

is

violated

by

By

definition

of

u

we

know

that

u

is

not

in

P

It

follows

from

the

above

discussion

that

u

must

be

in

a

long

of

T

that

is

not

P

Therefore

u

and

w

are

both

internal

nodes

of

T

This

implies

that

i

since

otherwise

uw

must

enclose

either

or

and

thus

the

of

T

is

contradicted

by

However

i

means

that

R

is

an

external

normal

region

which

contains

only

a

single

normal

leaf

of

T

Therefore

uw

is

forced

to

enclose

Therefore

the

of

T

is

contradicted

by

i

The

contradiction

for

this

case

can

be

obtained

in

a

way

similar

to

case

The

lemma

is

thus

proved

Proof

of

Lemma

If

k

the

lemma

trivially

holds

Therefore

we

assume

that

k

Let

Rk

be

the

k

regions

of

T

that

are

invalid

in

T

We

identify

k

distinct

long

denoted

Pk

of

T

that

are

deficient

in

T

Let

R

be

a

region

of

T

-LRB-

not

necessarily

invalid

in

T

-RRB-

Let

adj

-LRB-

R

-RRB-

be

the

maximum

subset

of

V

-LRB-

PL

-LRB-

T

-RRB-

-RRB-

such

that

each

node

in

adj

-LRB-

R

-RRB-

is

an

endpoint

of

an

edge

of

T

that

is

incident

to

a

node

of

R.

Claim

Let

R

be

a

region

of

T

that

is

invalid

in

T

Suppose

u

is

a

high-degree

node

in

R

and

w

is

a

node

not

in

adj

-LRB-

R

-RRB-

-LSB-

R

Then

pathT

-LRB-

u

w

-RRB-

contains

an

edge

of

T

that

is

incident

to

a

node

in

adj

-LRB-

R

-RRB-

and

a

node

not

in

adj

-LRB-

R

-RRB-

-LSB-

R.

Proof

Let

P

pathT

-LRB-

u

w

-RRB-

Clearly

P

contains

an

edge

such

that

adj

-LRB-

R

-RRB-

-LSB-

R

and

adj

-LRB-

R

-RRB-

-LSB-

R

We

show

must

be

in

adj

-LRB-

R

-RRB-

By

definition

of

invalid

regions

R

is

either

an

internal

normal

region

of

T

or

a

special

region

of

T

that

is

invalid

in

T

Since

T

is

augmented

R

does

not

contain

any

normal

leaf

of

T

By

definition

adj

-LRB-

R

-RRB-

contains

only

internal

nodes

of

T

Therefore

is

not

a

normal

leaf

of

T

Since

R

is

invalid

in

T

a

special

leaf

of

T

in

R

if

any

must

be

a

leaf

of

T

It

follows

from

Lemma

that

P

i.e.

pathT

-LRB-

u

w

-RRB-

does

not

contain

any

special

leaf

of

T

in

R

since

u

is

not

a

leaf

of

T

and

w

is

not

in

R.

Thus

is

not

a

special

leaf

of

T

Therefore

is

internal

in

T

since

is

in

P

Assume

for

a

contradiction

that

is

in

R

It

follows

that

must

enclose

a

node

in

adj

-LRB-

R

-RRB-

By

we

know

that

is

a

leaf

of

T

Thus

is

in

a

region

of

T

other

than

R

This

contradicts

the

of

T

by

since

encloses

all

then

nodes

of

some

long

of

T

We

then

know

that

is

not

in

R.

Therefore

is

in

adj

-LRB-

R

-RRB-

By

definition

of

invalid

regions

it

can

be

easily

verified

that

each

Ri

contains

at

least

one

internal

node

of

T

Let

vi

be

an

arbitrary

internal

node

of

T

in

Ri

for

every

i

k.

Let

P

i

pathT

-LRB-

vi

vk

-RRB-

for

every

i

k

Clearly

vk

is

not

in

adj

-LRB-

Ri

-RRB-

-LSB-

Ri

for

every

i

k

By

Claim

we

know

that

P

i

contains

an

edge

incident

to

a

node

in

adj

-LRB-

Ri

-RRB-

and

a

node

not

in

adj

-LRB-

Ri

-RRB-

-LSB-

Ri

Let

uiwi

where

ui

adj

-LRB-

Ri

-RRB-

and

wi

adj

-LRB-

Ri

-RRB-

-LSB-

Ri

be

the

edge

in

P

i

such

that

ui

is

closest

to

vi

in

P

i

Let

Pi

be

the

long

of

T

that

contains

ui

We

show

by

proving

the

following

claim

that

Pk

are

k

distinct

long

of

T

that

are

deficient

in

T

Claim

For

every

i

k

Pi

is

deficient

in

T

Moreover

Pj

Pi

for

every

j

i

k

adj

Ri

-LRB-

-RRB-

uiv

i

u

R

-LRB-

-RRB-

Radj

w

w

i

vk

R

i

P

i

Figure

The

dark

lines

are

the

edges

of

T

The

gray

path

is

P

i

i.e.

pathT

-LRB-

vi

vk

-RRB-

We

show

in

the

proof

that

w

must

be

in

adj

-LRB-

R

-RRB-

-LSB-

R.

Proof

Since

ui

vi

ui

vk

and

ui

is

in

pathT

-LRB-

vi

vk

-RRB-

it

follows

from

Lemma

that

ui

is

not

a

leaf

of

T

Let

R

be

the

other

region

of

T

that

Pi

is

incident

to

Let

w

be

the

node

closest

to

ui

in

pathT

-LRB-

ui

vk

-RRB-

such

that

w

is

not

in

Pi

Let

uw

be

the

edge

in

pathT

-LRB-

ui

w

-RRB-

Clearly

u

is

in

Pi

and

thus

each

node

in

pathT

-LRB-

ui

u

-RRB-

are

not

leaves

of

T

-LRB-

See

Figure

-RRB-

We

know

that

w

can

not

be

in

a

region

of

T

that

is

not

Ri

or

R

since

otherwise

uw

would

enclose

the

nodes

of

some

long

of

T

and

thus

the

of

T

would

be

contradicted

by

We

show

that

w

is

in

adj

-LRB-

R

-RRB-

-LSB-

R.

Assume

for

a

contradiction

that

w

is

not

in

adj

-LRB-

R

-RRB-

-LSB-

R.

ffl

u

ui

Clearly

w

wi

Since

w

is

not

in

Pi

by

definition

of

w

and

wi

is

not

in

adj

-LRB-

Ri

-RRB-

-LSB-

Ri

by

definition

of

wi

we

know

that

w

is

not

in

adj

-LRB-

Ri

-RRB-

-LSB-

Ri

-LSB-

Pi

By

assumption

w

is

not

in

R

It

follows

that

w

must

be

in

some

long

of

T

that

is

not

Pi

Since

w

is

not

in

adj

-LRB-

Ri

-RRB-

by

assumption

and

u

is

in

adj

-LRB-

Ri

-RRB-

by

definition

of

ui

we

know

that

uw

must

enclose

either

a

node

in

adj

-LRB-

Ri

-RRB-

or

a

node

in

Pi

This

contradicts

the

of

T

by

since

u

and

w

are

both

internal

nodes

of

T

ffl

u

ui

Since

u

ui

we

know

that

u

is

not

in

adj

-LRB-

Ri

-RRB-

-LSB-

Ri

Note

that

w

is

not

in

adj

-LRB-

R

-RRB-

by

assumption

It

follows

that

w

can

not

be

in

a

long

of

T

that

is

not

Pi

since

otherwise

uw

would

enclose

either

a

node

in

Pi

or

a

node

in

adj

-LRB-

R

-RRB-

which

contradicts

the

of

T

by

Since

w

is

not

in

Pi

by

definition

of

w

we

know

that

w

must

be

in

some

regions

of

T

However

we

already

know

that

w

can

not

be

in

any

region

of

T

that

is

not

Ri

or

R.

Therefore

w

must

be

in

Ri

by

the

assumption

that

w

is

not

in

adj

-LRB-

R

-RRB-

-LSB-

R

Since

w

is

in

Ri

uw

has

to

enclose

ui

By

and

we

know

that

w

is

a

special

leaf

of

T

and

thus

Ri

is

a

special

region

of

T

Since

Ri

is

invalid

in

T

w

is

a

leaf

of

T

This

contradicts

the

fact

that

w

is

in

pathT

-LRB-

vi

vk

-RRB-

by

Lemma

since

w

vi

and

w

vk

By

definition

of

invalid

regions

we

know

that

chargeT

-LRB-

Ri

-RRB-

We

prove

the

deficiency

of

Pi

by

showing

that

chargeT

-LRB-

R

-RRB-

and

the

number

of

leaves

of

T

in

Pi

is

at

most

chargeT

-LRB-

R

-RRB-

for

the

following

two

cases

ffl

w

adj

-LRB-

R

-RRB-

Since

w

Pi

w

must

be

in

some

long

of

T

that

is

not

Pi

By

we

know

that

u

is

in

adj

-LRB-

R

-RRB-

since

otherwise

the

node

in

Pi

adj

-LRB-

R

-RRB-

would

be

enclosed

by

uw

Since

u

is

in

adj

-LRB-

R

-RRB-

and

u

is

in

Pi

Pi

does

not

contain

any

leaf

of

T

It

suffices

to

show

that

chargeT

-LRB-

R

-RRB-

-LRB-

a

-RRB-

-LRB-

b

-RRB-

Figure

-LRB-

a

-RRB-

An

example

of

-LRB-

b

-RRB-

The

maximum-leaf

spanning

tree

of

the

graph

in

-LRB-

a

-RRB-

Since

w

is

in

adj

-LRB-

R

-RRB-

and

w

is

not

in

Pi

we

know

that

the

degree

of

R

is

at

least

two

Therefore

chargeT

-LRB-

R

-RRB-

since

R

can

not

be

an

external

normal

region

ffl

w

R

Since

w

vi

and

w

vk

it

follows

from

Lemma

that

w

is

not

a

leaf

of

T

We

therefore

know

that

chargeT

-LRB-

R

-RRB-

If

u

is

in

adj

-LRB-

R

-RRB-

then

Pi

does

not

contain

any

leaf

of

T

and

thus

Pi

is

deficient

in

T

If

u

is

not

in

adj

-LRB-

R

-RRB-

then

uw

encloses

a

node

in

adj

-LRB-

R

-RRB-

It

follows

from

and

that

w

is

a

special

leaf

of

T

Therefore

R

is

special

region

of

T

and

thus

chargeT

-LRB-

R

-RRB-

It

suffices

to

show

that

Pi

contains

at

most

one

leaf

of

T

Assume

for

a

contradiction

that

Pi

contains

two

leaves

of

T

Since

we

know

that

every

node

in

pathT

-LRB-

ui

u

-RRB-

is

not

a

leaf

of

T

it

follows

that

both

leaves

of

T

in

Pi

are

enclosed

by

uw

The

of

T

is

contradicted

by

since

u

is

internal

in

T

Now

we

prove

the

second

statement

of

the

lemma

Assume

for

a

contradiction

that

Pj

Pi

for

some

j

i

k

We

know

that

Pi

is

incident

to

Ri

and

Pj

is

incident

to

Rj

Since

Ri

Rj

we

know

that

Pi

must

be

incident

to

Ri

and

Rj

Therefore

R

is

invalid

in

T

since

R

Rj

It

has

been

shown

above

that

w

is

in

adj

-LRB-

R

-RRB-

-LSB-

R.

Therefore

w

ui

w

vk

Since

w

is

in

pathT

-LRB-

ui

vk

-RRB-

it

follows

from

Lemma

that

w

is

internal

in

T

We

know

that

R

does

not

contain

any

normal

leaf

of

T

and

every

special

leaf

of

T

in

R

if

any

is

also

a

leaf

in

T

It

follows

that

w

must

be

an

internal

node

in

T

Thus

uj

the

only

node

in

Pi

adj

-LRB-

R

-RRB-

must

be

u

since

otherwise

uw

would

enclose

uj

and

the

of

T

would

be

contradicted

by

It

follows

that

uj

is

in

Pi

and

thus

uj

is

in

pathT

-LRB-

ui

vk

-RRB-

If

we

switch

the

role

of

Ri

and

Rj

know

that

ui

is

in

pathT

-LRB-

uj

vk

-RRB-

for

exactly

the

same

reason

as

above

However

this

contradicts

the

fact

that

T

is

a

spanning

tree

of

G

It

follows

from

Claim

that

we

have

identified

k

distinct

long

of

T

that

are

deficient

in

T

This

completes

the

proof

of

Lemma

Tightness

of

the

Performance

Ratio

We

conclude

this

section

by

presenting

an

example

of

such

that

the

bound

of

in

Theorem

is

approachable

arbitrarily

closely

It

is

easy

to

verify

that

the

tree

in

Figure

-LRB-

a

-RRB-

is

a

Since

the

structure

is

symmetric

and

repetitive

we

only

need

to

verify

four

different

kinds

of

nontree

edges

as

candidates

for

Since

of

these

three

types

of

non-tree

edges

can

be

involved

in

any

the

tree

is

a

The

maximum-leaf

spanning

tree

for

this

graph

is

shown

in

Figure

-LRB-

b

-RRB-

Clearly

the

longer

this

is

the

closer

to

five

the

ratio

is

Performance

Guarantee

of

In

this

section

we

prove

the

following

restatement

of

Theorem

Theorem

Let

T

be

a

of

G

Then

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

for

any

spanning

tree

T

of

G

If

T

is

a

path

Theorem

is

immediate

from

Lemma

Therefore

without

loss

of

generality

we

assume

that

T

is

not

a

path

for

the

rest

of

the

section

Note

that

we

do

not

use

augmentation

as

we

did

in

the

previous

section

because

the

augmentation

shown

in

the

previous

section

might

turn

a

into

a

tree

that

is

not

a

as

shown

in

Figure

We

call

a

short

of

T

external

if

it

is

adjacent

to

a

-LRB-

normal

-RRB-

leaf

of

T

otherwise

we

call

it

internal

Let

-LRB-

T

-RRB-

be

the

set

of

the

external

short

of

T

Let

-LRB-

T

-RRB-

be

the

set

of

the

internal

short

of

T

One

can

easily

verify

that

-LRB-

T

-RRB-

j

j

-LRB-

T

-RRB-

j

-LRB-

-RRB-

since

every

internal

short

of

T

is

adjacent

in

T

to

two

nodes

in

-LRB-

T

-RRB-

Basic

Strategy

In

order

to

prove

Theorem

it

suffices

to

show

the

following

lemma

Lemma

Let

T

be

a

of

G

Then

-LRB-

T

-RRB-

-LRB-

-LRB-

T

-RRB-

-LRB-

T

-RRB-

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

for

any

spanning

tree

T

of

G

Since

-LRB-

T

-RRB-

-LRB-

T

-RRB-

-LRB-

T

-RRB-

-LRB-

T

-RRB-

it

follows

from

Lemma

that

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

-LRB-

T

-RRB-

-LRB-

T

-RRB-

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

-LRB-

-RRB-

Since

V

-LRB-

G

-RRB-

V

-LRB-

T

-RRB-

-LRB-

T

-RRB-

-LRB-

T

-RRB-

-LRB-

T

-RRB-

we

have

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

j

-LRB-

T

-RRB-

j

by

-LRB-

-RRB-

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

j

-LRB-

T

-RRB-

j

-LRB-

T

-RRB-

j

by

-LRB-

-RRB-

and

-LRB-

-RRB-

Therefore

it

remains

to

prove

Lemma

Second

Connections

Cross

Connections

and

Normal

Connections

There

are

three

kinds

of

edges

in

G

T

that

could

enclose

the

nodes

in

-LRB-

T

-RRB-

-LRB-

T

-RRB-

ffl

An

edge

uw

in

G

T

is

a

second

connection

of

T

if

w

is

in

a

long

P

of

T

and

uw

encloses

exactly

one

node

of

P

It

follows

from

and

that

u

must

be

a

special

leaf

of

T

Moreover

P

must

be

incident

to

the

special

region

of

T

that

contains

u.

Let

KS

-LRB-

T

-RRB-

denote

the

set

of

the

second

connections

of

T

Figure

The

tree

in

dark

lines

is

a

T

The

dashed

lines

are

the

non-tree

edges

where

KX

-LRB-

T

-RRB-

KN

-LRB-

T

-RRB-

KS

-LRB-

T

-RRB-

KN

-LRB-

T

-RRB-

KX

-LRB-

T

-RRB-

and

KX

-LRB-

T

-RRB-

KN

-LRB-

T

-RRB-

ffl

An

edge

in

G

T

is

a

cross

connection

of

T

if

its

endpoints

are

in

different

regions

of

T

It

follows

from

and

that

both

endpoints

of

a

cross

connection

of

T

must

be

leaves

of

T

Let

KX

-LRB-

T

-RRB-

denote

the

set

of

the

cross

connections

of

T

ffl

An

edge

in

G

T

is

a

normal

connection

of

T

if

one

of

its

endpoints

is

in

-LRB-

T

-RRB-

It

follows

from

that

the

other

endpoint

of

a

normal

connection

of

T

must

be

a

leaf

of

T

Note

that

a

normal

connection

of

T

could

be

a

cross

connection

of

T

Let

KN

-LRB-

T

-RRB-

denote

the

set

of

the

normal

connections

of

T

Let

K

-LRB-

T

-RRB-

denote

the

set

of

the

connections

of

T

which

is

defined

as

KS

-LRB-

T

-RRB-

-LSB-

KX

-LRB-

T

-RRB-

-LSB-

KN

-LRB-

T

-RRB-

An

example

is

shown

in

Figure

We

claim

that

even

if

we

consider

only

the

endpoints

of

the

edges

in

K

-LRB-

T

-RRB-

E

-LRB-

T

-RRB-

the

number

of

those

endpoints

that

are

in

-LRB-

T

-RRB-

-LRB-

T

-RRB-

is

already

no

less

than

that

required

in

Lemma

For

the

rest

of

the

section

we

prove

the

following

lemma

from

which

Lemma

is

immediate

Lemma

Let

T

be

a

of

G.

Suppose

T

is

a

spanning

tree

of

G

Then

-LRB-

T

-RRB-

-LRB-

-LRB-

T

-RRB-

-LRB-

T

-RRB-

-RRB-

j

jV

-LRB-

K

-LRB-

T

-RRB-

E

-LRB-

T

-RRB-

-RRB-

-LRB-

-LRB-

T

-RRB-

-LRB-

T

-RRB-

-RRB-

j

A

node

is

green

if

it

is

in

-LRB-

T

-RRB-

-LRB-

-LRB-

T

-RRB-

-LRB-

T

-RRB-

-RRB-

A

node

is

yellow

if

it

is

in

V

-LRB-

K

-LRB-

T

-RRB-

E

-LRB-

T

-RRB-

-RRB-

-LRB-

-LRB-

T

-RRB-

-LRB-

T

-RRB-

-RRB-

In

other

words

a

green

node

is

a

leaf

of

T

that

is

in

a

long

of

T

or

a

external

short

of

T

A

yellow

node

v

is

an

endpoint

of

an

edge

of

T

that

is

also

a

connection

of

T

such

that

v

is

a

leaf

of

T

but

not

a

leaf

of

T

Clearly

a

green

node

is

not

yellow

and

vice

versa

since

every

green

node

is

a

leaf

of

T

and

every

yellow

node

is

not

a

leaf

of

T

We

prove

Lemma

by

showing

that

the

number

of

green

nodes

is

no

more

than

the

number

of

yellow

nodes

A

connection

of

T

is

yellow

if

it

has

a

yellow

endpoint

One

can

easily

verify

that

the

-LRB-

only

-RRB-

yellow

endpoint

of

a

yellow

second

connection

of

T

must

be

a

special

leaf

of

T

since

the

other

endpoint

is

not

a

leaf

of

T

Proof

of

Lemma

The

proof

has

the

following

steps

Step

For

every

green

node

v

in

-LRB-

T

-RRB-

we

show

that

there

exists

a

yellow

normal

connection

uw

of

T

that

encloses

v

such

that

-LRB-

-RRB-

u

is

a

normal

leaf

of

T

adjacent

to

v

in

T

and

-LRB-

-RRB-

w

is

yellow

Let

y-edge

-LRB-

v

-RRB-

uw

and

y-node

-LRB-

v

-RRB-

w

We

mark

v

and

w.

Step

For

every

green

node

v

in

-LRB-

T

-RRB-

we

do

the

following

If

v

is

not

enclosed

by

any

yellow

second

connection

of

T

then

we

leave

v

alone

Otherwise

let

uw

be

a

yellow

second

connection

of

T

that

encloses

v

where

w

is

yellow

Let

y-edge

-LRB-

v

-RRB-

uw

and

y-node

-LRB-

v

-RRB-

w

We

mark

v

and

w.

Step

For

any

two

distinct

unmarked

green

nodes

v

and

we

show

that

y-node

-LRB-

v

-RRB-

y-node

-LRB-

-RRB-

Step

We

show

that

the

number

of

unmarked

green

nodes

is

no

more

than

that

of

the

unmarked

yellow

nodes

By

the

first

two

steps

we

know

every

marked

green

node

has

a

corresponding

marked

yellow

node

The

third

step

ensures

that

the

number

of

the

marked

green

nodes

is

equal

to

that

of

the

marked

yellow

nodes

Lemma

then

follows

from

the

last

step

We

elaborate

these

steps

in

the

following

subsections

Step

Let

v

be

a

green

node

in

-LRB-

T

-RRB-

By

definition

of

-LRB-

T

-RRB-

we

know

v

are

adjacent

to

exactly

two

nodes

say

u

and

in

T

Moreover

one

of

u

and

say

u

must

be

a

normal

leaf

of

T

Let

w

be

the

node

such

that

uw

is

an

edge

in

pathT

-LRB-

u

-RRB-

Let

y-node

-LRB-

v

-RRB-

w

We

show

w

is

yellow

Lemma

Let

v

be

a

green

node

in

-LRB-

T

-RRB-

Let

y-node

-LRB-

v

-RRB-

be

chosen

as

described

above

Then

y-node

-LRB-

v

-RRB-

is

yellow

Proof

We

prove

the

lemma

by

showing

w

-LRB-

T

-RRB-

and

w

-LRB-

T

-RRB-

Assume

for

a

contradiction

that

uw

is

an

edge

of

T

It

follows

that

w

v

since

uv

is

the

only

edge

of

T

that

is

incident

to

u.

Clearly

u

v

and

are

distinct

by

the

choice

of

u

and

It

follows

from

Lemma

that

w

can

not

be

a

leaf

of

T

This

contradicts

the

fact

that

v

is

green

Since

uw

is

not

an

edge

of

T

and

u

is

a

normal

leaf

of

T

we

know

that

uw

is

in

E

-LRB-

T

-RRB-

KN

-LRB-

T

-RRB-

It

follows

from

the

property

of

normal

connections

that

w

must

be

a

leaf

of

T

It

suffices

to

show

that

w

is

not

a

leaf

of

T

Assume

for

a

contradiction

that

w

is

a

leaf

of

T

Since

w

is

in

pathT

-LRB-

u

-RRB-

it

follows

from

Lemma

that

w

Since

each

endpoint

of

a

normal

connection

of

T

is

a

leaf

of

T

we

know

is

a

leaf

of

T

Since

v

is

adjacent

in

T

to

only

u

and

and

u

and

are

both

leaves

of

T

it

follows

that

T

is

equal

to

the

path

This

contradicts

the

assumption

that

T

is

not

a

path

Step

We

do

not

have

to

prove

anything

about

this

step

since

it

is

just

an

instruction

However

we

show

a

lemma

about

the

number

of

unmarked

green

nodes

in

a

long

of

T

when

this

step

is

finished

A

long

of

T

is

green

if

it

contains

unmarked

green

nodes

A

green

long

of

T

is

darkgreen

if

it

contains

more

than

one

green

node

It

follows

from

the

following

lemma

that

every

long

of

T

contains

at

most

two

unmarked

green

nodes

Lemma

Let

and

be

two

distinct

unmarked

green

nodes

in

the

same

long

of

T

Then

is

an

edge

of

T

Proof

Let

P

pathT

-LRB-

-RRB-

Assume

for

a

contraction

that

v

is

a

node

in

P

where

v

and

v

We

show

that

there

exists

a

yellow

second

connection

of

T

that

encloses

one

of

and

This

contradicts

the

fact

that

and

are

both

unmarked

Since

T

is

not

a

path

by

assumption

we

know

-LRB-

T

-RRB-

is

not

empty

Let

be

a

node

in

-LRB-

T

-RRB-

Let

P

pathT

-LRB-

v

-RRB-

By

definition

of

green

nodes

and

are

both

leaves

of

T

Since

v

w

w

u

v

v

u

w

w

u

v

v

u

w

w

u

v

v

u

-LRB-

a

-RRB-

-LRB-

b

-RRB-

-LRB-

c

-RRB-

Figure

and

are

all

distinct

it

follows

from

Lemma

that

neither

nor

is

in

P

Let

w

be

the

node

closest

to

v

in

P

such

that

w

is

not

in

P

Let

u

be

the

node

such

that

uw

is

an

edge

in

pathT

-LRB-

v

w

-RRB-

Clearly

u

is

in

P

and

uw

is

an

edge

in

T

T

Note

that

v

and

are

distinct

Since

and

are

leaves

of

T

it

follows

from

Lemma

that

P

does

not

contain

or

By

we

know

that

u

must

be

in

pathT

-LRB-

-RRB-

Therefore

uw

encloses

either

or

It

follows

from

that

uw

is

a

second

connection

of

T

We

show

that

uw

is

yellow

Since

uw

is

a

second

connection

of

T

and

u

P

w

is

a

leaf

of

T

by

the

property

of

second

connections

Since

w

-LRB-

T

-RRB-

we

know

w

Since

v

w

and

are

distinct

and

w

is

in

P

it

follows

from

Lemma

that

w

is

not

a

leaf

of

T

Therefore

w

is

yellow

and

so

is

uw

Step

Lemma

Let

v

and

be

two

distinct

marked

green

nodes

Then

y-node

-LRB-

v

-RRB-

y-node

-LRB-

-RRB-

Proof

Let

w

y-node

-LRB-

v

-RRB-

and

y-node

-LRB-

-RRB-

Let

uw

y-edge

-LRB-

v

-RRB-

and

y-edge

-LRB-

-RRB-

We

prove

the

lemma

for

the

following

four

cases

Note

that

KN

-LRB-

T

-RRB-

and

KS

-LRB-

T

-RRB-

are

disjoint

Therefore

uw

KN

-LRB-

T

-RRB-

if

and

only

if

v

-LRB-

T

-RRB-

and

uw

KS

-LRB-

T

-RRB-

if

and

only

if

v

-LRB-

T

-RRB-

Similar

relations

between

and

also

holds

uw

KN

-LRB-

T

-RRB-

We

know

v

and

are

both

in

-LRB-

T

-RRB-

If

uw

then

u

and

w

by

the

way

y-node

-LRB-

v

-RRB-

and

y-node

-LRB-

-RRB-

are

chosen

in

Step

Therefore

w

If

uw

assume

for

a

contradiction

that

w

One

can

easily

verify

that

-LRB-

fuw

fuv

-RRB-

is

a

for

T

as

shown

in

Figure

-LRB-

a

-RRB-

This

contradicts

the

fact

that

T

is

a

uw

KS

-LRB-

T

-RRB-

By

definition

every

second

connection

of

T

encloses

exactly

one

node

in

-LRB-

T

-RRB-

Since

v

we

know

uw

Assume

for

a

contradiction

that

w

It

follows

that

-LRB-

fuw

fuv

-RRB-

is

a

for

T

as

shown

in

Figure

-LRB-

b

-RRB-

This

contradicts

the

fact

that

T

is

a

uw

KN

-LRB-

T

-RRB-

and

KS

-LRB-

T

-RRB-

Since

KN

-LRB-

T

-RRB-

and

KS

-LRB-

T

-RRB-

are

disjoint

we

know

uw

Assume

for

a

contradiction

that

w

One

can

easily

verify

that

-LRB-

fuw

fuv

-RRB-

is

a

for

T

as

shown

in

Figure

-LRB-

c

-RRB-

This

contradicts

the

fact

that

T

is

a

uw

KS

-LRB-

T

-RRB-

and

KN

-LRB-

T

-RRB-

The

lemma

can

be

proved

in

a

way

similar

to

case

Step

In

this

subsection

we

prove

the

following

lemma

Lemma

The

number

of

unmarked

green

nodes

of

T

is

at

most

the

number

of

unmarked

yellow

nodes

of

T

Clearly

Lemma

holds

trivially

if

T

does

not

have

any

green

long

Suppose

Pm

are

the

green

long

of

T

for

some

m

We

prove

Lemma

by

identifying

a

yellow

cross

connection

of

T

for

every

i

m

such

that

If

Pi

is

darkgreen

then

both

endpoints

of

are

yellow

and

unmarked

Otherwise

at

least

one

endpoint

of

is

yellow

and

unmarked

If

i

j

m

then

and

do

not

share

any

common

endpoint

In

order

to

identify

those

yellow

cross

connections

of

T

we

are

going

to

find

m

spanning

trees

of

G

denoted

Tm

that

satisfy

the

following

property

T

Ti

Ti

-LRB-

feig

fuiwig

-RRB-

for

every

i

m

where

-LRB-

i

-RRB-

ei

is

an

edge

of

T

such

that

Pi

contains

at

least

one

endpoint

of

ei

-LRB-

ii

-RRB-

uiwi

is

a

yellows

cross

connection

of

T

that

encloses

all

nodes

in

Pi

and

-LRB-

iii

-RRB-

ei

is

in

pathT

-LRB-

uiwi

-RRB-

We

need

the

following

lemmas

to

describe

the

procedure

for

identifying

those

m

spanning

trees

and

m

yellow

cross

connections

of

T

Lemma

Suppose

Pi

is

darkgreen

for

some

i

m.

Suppose

Ti

are

spanning

trees

of

G

that

satisfy

and

Let

v

and

be

the

unmarked

green

nodes

in

Pi

Then

pathTi

-LRB-

v

-RRB-

contains

a

yellow

cross

connection

of

T

that

encloses

v

and

Moreover

both

endpoints

of

are

yellow

and

unmarked

Lemma

Suppose

Pi

is

green

for

some

i

m.

Suppose

Ti

are

spanning

trees

of

G

that

satisfy

and

Let

v

be

an

unmarked

green

node

in

Pi

Suppose

and

are

the

nodes

adjacent

to

v

in

T

Then

either

pathTi

-LRB-

v

-RRB-

or

pathTi

-LRB-

v

-RRB-

contains

a

cross

connection

of

T

that

encloses

v.

Moreover

at

least

one

endpoint

of

is

yellow

and

unmarked

Here

is

the

procedure

for

identifying

those

m

spanning

trees

and

m

yellow

cross

connections

of

T

Let

T

For

i

to

m

do

If

Pi

is

darkgreen

then

Let

v

and

be

the

unmarked

green

nodes

in

Pi

Let

be

a

cross

connection

of

T

in

pathTi

-LRB-

v

-RRB-

that

encloses

v

and

v

v

v

v

v

v

v

v

v

-LRB-

a

-RRB-

T

-LRB-

b

-RRB-

-LRB-

c

-RRB-

Figure

Let

ei

else

-LRB-

i.e.

Pi

is

green

but

not

darkgreen

-RRB-

Let

v

be

the

unmarked

green

node

in

Pi

Let

and

be

the

nodes

adjacent

to

v

in

T

Let

be

a

cross

connection

of

T

in

pathTi

-LRB-

v

-RRB-

-LSB-

pathTi

-LRB-

v

-RRB-

that

encloses

v

If

is

in

pathTi

-LRB-

v

-RRB-

then

Let

ei

else

-LRB-

i.e.

is

in

pathTi

-LRB-

v

-RRB-

-RRB-

Let

ei

If

i

m

then

Let

Ti

Ti

-LRB-

feig

-RRB-

An

example

is

shown

in

Figure

The

dark

lines

in

-LRB-

a

-RRB-

forms

a

T

where

the

dashed

lines

are

the

connections

of

T

Let

the

picture

in

-LRB-

b

-RRB-

be

T

which

is

also

known

as

in

the

the

procedure

Note

that

is

leaf

of

T

but

it

is

not

green

since

it

is

not

in

a

long

of

T

Also

note

that

is

a

marked

green

node

of

T

where

y-node

-LRB-

-RRB-

and

y-edge

-LRB-

-RRB-

Therefore

and

are

the

unmarked

green

nodes

of

T

By

definition

of

green

long

is

darkgreen

and

is

green

Since

is

the

only

cross

connection

of

T

in

pathT

-LRB-

-RRB-

that

encloses

and

we

have

that

and

It

follows

that

is

the

tree

in

-LRB-

c

-RRB-

Since

is

the

only

cross

connection

of

T

in

-LRB-

-RRB-

that

encloses

v

we

have

that

Since

is

in

-LRB-

-RRB-

but

not

in

-LRB-

v

-RRB-

it

follows

that

Clearly

both

endpoints

of

are

yellow

and

unmarked

and

is

darkgreen

one

endpoint

of

is

yellow

and

unmarked

and

is

green

but

not

darkgreen

Proof

of

Lemma

u

v

w

P

i

e

i

Figure

A

We

first

prove

that

the

described

in

the

procedure

do

exist

and

satisfy

This

can

be

shown

by

proving

the

following

claim

for

every

i

n.

ffl

The

described

in

the

ith

iteration

exists

and

satisfies

ffl

The

Ti

described

in

the

first

i

iterations

are

spanning

trees

of

G

that

satisfy

and

We

prove

the

claim

by

induction

on

i

When

i

Ti

T

Clearly

satisfies

-LRB-

and

-RRB-

It

follows

from

Lemma

and

Lemma

that

the

described

in

the

first

iteration

of

the

procedure

does

exist

Clearly

satisfies

-LRB-

i

-RRB-

of

since

v

which

is

in

in

the

first

iteration

is

an

endpoint

of

By

Lemma

and

Lemma

we

know

that

satisfies

-LRB-

ii

-RRB-

of

One

can

verify

that

and

satisfy

-LRB-

iii

-RRB-

of

since

is

a

cross

connection

of

T

that

encloses

all

nodes

in

It

follows

from

-LRB-

iii

-RRB-

of

that

is

a

spanning

tree

of

G.

Therefore

and

are

spanning

trees

of

G

that

satisfy

and

Assume

the

claim

holds

for

every

i

k

for

some

k

We

show

that

the

claim

holds

for

i

k.

By

the

inductive

hypothesis

we

know

that

Tk

are

spanning

trees

of

G

that

satisfy

and

By

Lemma

and

Lemma

we

know

that

the

described

in

the

kth

iteration

of

the

procedure

does

exist

Clearly

ek

satisfies

-LRB-

i

-RRB-

of

since

v

which

is

in

Pk

in

the

kth

iteration

is

an

endpoint

of

ek

By

Lemma

and

Lemma

we

know

that

satisfies

-LRB-

ii

-RRB-

of

One

can

verify

that

and

ek

satisfies

-LRB-

iii

-RRB-

of

since

is

a

cross

connection

of

T

that

encloses

all

nodes

in

Pk

It

follows

from

-LRB-

iii

-RRB-

of

that

Tk

which

is

Tk

-LRB-

ek

-RRB-

is

a

spanning

tree

of

G.

Therefore

Tk

are

spanning

trees

of

G

that

satisfy

and

B

We

show

satisfy

Let

i

and

j

be

two

distinct

indices

where

i

j

m

We

first

show

that

By

definition

of

the

procedure

we

know

that

is

not

an

edge

of

Ti

Since

each

of

ei

ej

is

an

edge

of

T

and

is

not

an

edge

of

T

it

follows

that

is

still

not

an

edge

of

Tj

Note

that

is

an

edge

of

Tj

Therefore

Assume

for

a

contradiction

that

and

share

a

common

endpoint

Let

be

an

edge

in

Pi

and

let

be

an

edge

in

Pj

One

can

easily

verify

that

-LRB-

g

-RRB-

is

a

for

T

since

Pi

Pj

The

of

T

is

contradicted

Therefore

is

satisfied

by

The

lemma

is

proved

The

following

lemma

which

is

illustrated

in

Figure

is

useful

to

proving

Lemma

and

Lemma

Lemma

Let

be

a

cross

connection

of

T

that

encloses

the

nodes

of

a

green

long

Pi

of

T

Suppose

w

is

a

marked

yellow

endpoint

of

Let

u

and

v

be

the

nodes

such

that

w

y-node

-LRB-

v

-RRB-

and

uw

y-edge

-LRB-

v

-RRB-

Then

uw

is

a

second

connection

of

T

that

encloses

v

and

both

u

and

v

are

in

Pi

Moreover

u

and

v

are

the

only

two

nodes

in

Pi

Proof

Assume

for

a

contradiction

that

u

is

not

in

Pi

Then

v

is

not

in

Pi

by

the

way

that

y-node

-LRB-

v

-RRB-

and

y-edge

-LRB-

v

-RRB-

are

chosen

in

Step

and

Step

It

follows

that

-LRB-

uwg

fe

uvg

-RRB-

is

a

for

T

where

e

could

be

any

edge

in

Pi

The

of

T

is

thus

contradicted

Therefore

we

know

that

u

is

in

Pi

which

implies

that

v

is

also

in

Pi

and

uw

is

a

second

connection

of

T

that

encloses

v.

By

definition

of

second

connections

we

know

that

uv

is

an

edge

of

Pi

Let

be

the

node

other

than

v

that

is

adjacent

to

u

in

T

Let

be

the

node

other

than

u

that

is

adjacent

to

v

In

other

words

is

a

path

of

T

We

know

that

is

not

in

Pi

since

otherwise

-LRB-

uwg

fuv

-RRB-

would

be

a

for

T

and

the

of

T

would

be

contradicted

By

definition

of

of

second

connections

we

know

that

is

not

in

Pi

since

otherwise

uw

would

enclose

two

nodes

i.e.

v

and

of

Pi

Since

Pi

does

not

contain

and

it

follows

that

u

and

v

are

the

only

two

nodes

in

Pi

It

remains

to

prove

Lemma

and

Lemma

Proof

of

Lemma

By

definition

of

green

nodes

v

and

are

both

leaves

of

T

One

can

easily

verify

that

Pi

does

not

contain

the

endpoints

of

ej

and

for

every

j

i

Therefore

v

and

are

both

leaves

of

Ti

Let

P

pathT

-LRB-

v

-RRB-

Clearly

v

and

are

not

adjacent

in

Ti

since

otherwise

they

have

to

be

the

only

two

nodes

of

G

which

contradicts

the

assumption

that

T

is

not

a

path

By

Lemma

we

know

that

v

and

are

adjacent

in

T

In

order

to

connect

v

and

in

Ti

P

must

contain

either

a

second

connection

of

T

that

encloses

one

of

v

and

or

a

cross

connection

of

T

that

encloses

both

v

and

Assume

for

a

contradiction

that

P

contains

a

second

connection

uw

of

T

that

encloses

one

of

v

and

say

v.

By

definition

of

second

connections

one

endpoint

of

uw

say

u

is

a

special

leaf

of

T

Since

u

is

in

P

u

is

not

a

leaf

of

Ti

One

can

easily

verify

that

the

degree

of

u

in

Ti

is

no

more

than

that

in

T

since

u

is

not

an

endpoint

of

ei

It

follows

that

u

is

not

a

leaf

of

T

Therefore

uw

is

a

yellow

second

connection

of

T

that

encloses

v

This

contradicts

the

fact

that

v

is

an

unmarked

green

nodes

Therefore

P

has

to

contain

a

cross

connection

of

T

that

encloses

v

and

We

show

that

each

endpoint

of

is

not

a

leaf

of

T

Assume

for

a

contradiction

that

an

endpoint

u

of

is

a

leaf

of

T

If

u

is

not

an

endpoint

of

then

u

is

a

leaf

of

If

u

is

an

endpoint

of

we

know

u

must

be

a

normal

leaf

of

T

since

is

an

edge

of

T

and

the

other

endpoint

of

is

in

a

long

of

T

Hence

u

must

be

an

endpoint

of

and

thus

u

is

still

a

leaf

of

For

exactly

the

same

reason

we

know

that

u

is

a

leaf

of

Ti

This

contradicts

the

fact

that

u

is

in

P

by

Lemma

since

u

v

u

and

P

pathT

-LRB-

v

-RRB-

Therefore

both

endpoints

of

are

yellow

We

show

that

both

endpoints

of

are

unmarked

Assume

for

a

contradiction

that

an

endpoint

w

of

is

marked

It

follows

from

Lemma

that

Pi

contains

only

two

nodes

and

one

of

them

-LRB-

i.e.

the

v

in

Lemma

-RRB-

is

green

and

marked

This

contradicts

the

fact

that

Pi

is

darkgreen

Proof

of

Lemma

Let

P

pathTi

-LRB-

-RRB-

Clearly

E

-LRB-

P

-RRB-

E

-LRB-

pathTi

-LRB-

v

-RRB-

-LSB-

pathTi

-LRB-

v

-RRB-

-RRB-

We

show

that

P

contains

a

cross

connection

of

T

such

that

encloses

v

and

at

least

one

endpoint

of

is

yellow

and

unmarked

By

definition

of

green

nodes

v

is

a

leaf

of

T

One

can

easily

verify

that

Pi

does

not

contain

any

endpoints

of

ej

and

for

every

j

i

Therefore

v

is

also

a

leaf

of

Ti

It

follows

that

-LRB-

a

-RRB-

-LRB-

b

-RRB-

Figure

v

is

not

in

P

by

Lemma

since

v

and

v

In

order

to

connect

and

in

Ti

P

must

contain

either

a

second

connection

of

T

that

encloses

v

or

a

cross

connection

of

T

that

encloses

v.

Assume

for

a

contradiction

that

P

contains

a

second

connection

uw

of

T

that

encloses

v.

By

the

property

of

second

connections

one

endpoint

of

uw

say

u

is

a

special

leaf

of

T

Therefore

u

and

are

distinct

and

thus

u

is

not

a

leaf

of

Ti

by

Lemma

Since

u

is

a

special

leaf

of

T

we

know

that

u

is

not

an

endpoint

of

ei

Therefore

the

degree

of

u

in

Ti

is

no

more

than

that

in

T

It

follows

that

u

is

not

a

leaf

of

T

Therefore

uw

is

a

yellow

second

connection

of

T

that

encloses

v

This

contradicts

the

fact

that

v

is

an

unmarked

green

node

Therefore

P

must

contain

a

cross

connection

of

T

that

encloses

v

We

show

that

an

endpoint

of

is

not

a

leaf

of

T

Since

Pi

is

a

long

of

T

and

v

is

in

Pi

at

least

one

of

and

say

is

in

Pi

It

follows

that

is

not

an

endpoint

of

Since

is

in

pathT

-LRB-

-RRB-

it

follows

from

Lemma

that

at

least

one

endpoint

w

of

is

not

a

leaf

of

Ti

We

show

that

w

is

not

a

leaf

of

T

as

follows

Let

us

consider

Ti

first

If

w

is

not

an

endpoint

of

ei

then

w

is

not

a

leaf

of

Ti

If

w

is

an

endpoint

of

ei

we

know

that

w

must

be

a

normal

leaf

of

T

since

ei

is

an

edge

of

T

and

the

other

endpoint

of

ei

is

in

Pi

a

long

of

T

It

follows

that

w

is

also

an

endpoint

of

Therefore

w

is

not

a

leaf

of

Ti

For

exactly

the

same

reason

we

know

that

w

is

not

a

leaf

of

Ti

Ti

Therefore

w

is

yellow

If

w

is

unmarked

then

the

lemma

is

proved

Suppose

w

is

marked

We

show

that

u

is

yellow

and

unmarked

where

u

is

the

endpoint

of

other

than

w

Since

w

is

yellow

and

unmarked

it

follows

from

Lemma

that

Pi

contains

only

two

nodes

which

must

be

v

and

where

vw

is

a

second

connection

of

T

that

encloses

Therefore

v

is

the

only

node

in

Pi

that

is

green

and

unmarked

Assume

for

a

contradiction

that

u

is

not

yellow

Since

u

is

a

leaf

of

T

it

follows

from

Lemma

that

u

Therefore

which

is

is

a

normal

connection

of

T

since

u

has

to

be

a

normal

leaf

of

T

One

can

easily

verify

that

-LRB-

vwg

-RRB-

is

a

for

T

which

contradicts

the

of

T

Therefore

u

is

yellow

Assume

for

a

contradiction

that

u

is

marked

It

follows

from

Lemma

that

y-node

-LRB-

v

-RRB-

u

This

contradicts

the

fact

that

v

is

green

and

unmarked

Tightness

of

the

Performance

Ratio

Figure

shows

the

that

we

believe

to

have

the

worst

performance

ratio

which

is

only

Therefore

we

conjecture

the

following

Conjecture

The

performance

guarantee

of

is

-LRB-

a

-RRB-

-LRB-

b

-RRB-

Figure

The

tree

in

-LRB-

a

-RRB-

is

an

example

of

k-LOT

with

performance

ratio

k

k

This

tree

is

not

a

-LRB-

k

-RRB-

LOT

since

the

tree

in

-LRB-

b

-RRB-

can

be

obtained

by

applying

a

-LRB-

k

-RRB-

improvement

Concluding

Remarks

We

have

discovered

a

series

of

examples

of

graphs

in

which

the

performance

ratio

of

a

k-LOT

can

be

as

bad

as

k

k

for

each

k

We

present

a

typical

such

graph

in

Figure

However

we

have

no

proof

that

this

is

the

worst

possible

performance

of

k-LOTs

on

any

graph

Since

Maximum

Leaf

Spanning

Tree

is

MAX

SNP-complete

-LSB-

-RSB-

there

must

be

a

constant

such

that

there

is

no

-LRB-

-RRB-

approximation

for

Maximum

Leaf

Spanning

Tree

unless

P

NP

It

follows

that

the

graph

shown

in

Figure

is

certainly

not

the

worst

example

of

k-LOTs

for

some

k

It

remains

open

to

determine

what

the

is

and

construct

examples

of

k-LOTs

whose

performance

ratio

is

worse

than

k

k

for

some

k

We

mention

in

passing

that

the

directed

version

of

the

problem

we

considered

in

which

where

we

seek

an

arborescence

rooted

at

some

fixed

node

with

the

maximum

number

of

leaves

is

also

NP-complete

This

can

be

easily

deduced

using

a

reduction

from

the

undirected

problem

We

note

that

locally

optimal

algorithms

like

those

we

presented

for

the

undirected

case

perform

badly

in

the

directed

case

Consider

the

closely

related

Minimum

Leaf

Spanning

Tree

problem

where

we

seek

a

spanning

tree

with

the

minimum

number

of

leaves

It

is

easy

to

see

that

this

problem

is

NP-complete

by

a

simple

reduction

from

the

Hamiltonian

path

problem

It

is

natural

to

ask

whether

the

techniques

similar

to

ours

can

be

used

to

approximate

this

problem

as

well

We

can

use

recent

results

on

hardness

of

approximating

the

longest

path

problem

-LSB-

-RSB-

to

infer

that

the

Minimum

Leaf

Spanning

Tree

problem

is

hard

to

approximate

The

key

observation

is

that

a

spanning

tree

on

n

nodes

with

at

most

leaves

contains

a

path

with

at

least

edges

This

can

be

shown

by

considering

an

E?uler

tour

of

the

tree

and

estimating

the

longest

length

of

a

path

between

any

two

consecutive

leaves

in

the

tour

A

Hamiltonian

graph

has

a

spanning

tree

with

at

most

two

leaves

Any

polynomial-time

approximation

algorithm

with

a

multiplicative

performance

guarantee

of

f

can

be

used

to

obtain

a

spanning

tree

of

a

Hamiltonian

graph

with

at

most

leaves

By

our

observation

above

this

yields

a

path

in

the

graph

of

size

at

least

n

f

The

results

of

Karger

Motwani

and

Ramkumar

-LSB-

-RSB-

imply

that

f

is

at

least

-LRB-

n

-RRB-

for

any

ffl

unless

P

NP

where

P

stands

for

the

complexity

class

Deterministic

Quasi-polynomial

time

or

DTIME

-LSB-

npolylog

n

-RSB-

Acknowledgments

We

gratefully

acknowledge

the

support

guidance

and

encouragement

of

our

advisor

Prof.

Philip

Klein

References

-LSB-

-RSB-

S.

Arora

C.

Lund

R.

Motwani

M.

Sudan

and

M.

Szegedy

Proof

verification

and

the

hardness

of

approximation

problems

Proceedings

of

the

Thirty-third

Annual

IEEE

Symposium

on

Foundations

of

Computer

Science

-LRB-

-RRB-

pp.

-LCB-

-LSB-

-RSB-

H.

L.

Bodlaender

On

linear

time

minor

tests

and

depth

first

search

Proceedings

of

Workshop

on

Algorithms

and

Data

Structures

-LRB-

-RRB-

pp.

-LCB-

-LSB-

-RSB-

G.

A.

Croes

A

method

for

solving

traveling-salesman

problems

Operations

Research

-LRB-

Nov.-Dec

-RRB-

pp.

-LCB-

-LSB-

-RSB-

E.

W.

Dijkstra

Self-stabilizing

systems

in

spite

of

distributed

control

Communications

of

ACM

-LRB-

Nov.

-RRB-

pp.

-LCB-

-LSB-

-RSB-

M.

R.

Fellows

and

M.

A.

Langston

On

well-partial-order

theory

and

its

applications

to

combinatorial

problems

of

VLSI

design

Technical

Report

Dept.

of

Computer

Science

Washington

State

University

-LSB-

-RSB-

M.

F?urer

and

B.

Raghavachari

Approximating

the

minimum

degree

spanning

tree

to

within

one

from

the

optimal

degree

Proceedings

of

the

Third

Annual

ACM-SIAM

Symposium

on

Discrete

Algorithms

-LRB-

-RRB-

pp.

-LCB-

-LSB-

-RSB-

M.

F?urer

and

B.

Raghavachari

Approximating

the

minimum-degree

Steiner

tree

to

within

one

of

optimal

Journal

of

Algorithms

-LRB-

-RRB-

p.

-LCB-

-LSB-

-RSB-

G.

Galbiati

F.

Maffioli

and

A.

Morzenti

A

short

note

on

the

approximability

of

the

maximum

leaves

spanning

tree

problem

Information

Processing

Letters

-LRB-

-RRB-

pp.

-LCB-

-LSB-

-RSB-

M.

R.

Garey

and

D.

S.

Johnson

Computers

and

Intractability

A

guide

to

the

theory

of

NP

completeness

W.

H.

Freeman

San

Francisco

-LRB-

-RRB-

-LSB-

-RSB-

J.

R.

Griggs

D.

J.

Kleitman

and

A.

Shastri

Spanning

trees

with

many

leaves

in

cubic

graphs

Journal

of

Graph

Theory

-LRB-

-RRB-

pp.

-LCB-

-LSB-

-RSB-

M.

M.

Halld?orsson

Approximating

discrete

collections

via

local

improvements

Proceedings

of

the

Sixth

Annual

ACM-SIAM

Symposium

on

Discrete

Algorithms

-LRB-

-RRB-

pp.

-LCB-

-LSB-

-RSB-

D.

S.

Johnson

C.

H.

Papadimitriou

and

M.

Yannakakis

How

easy

is

local

search

Journal

of

Computer

and

System

Sciences

-LRB-

-RRB-

pp.

-LCB-

-LSB-

-RSB-

D.

Karger

R.

Motwani

and

G.

D.

S.

Ramkumar

On

approximating

the

longest

path

in

a

graph

Proceedings

of

Workshop

on

Algorithms

and

Data

Structures

-LRB-

-RRB-

pp.

-LCB-

-LSB-

-RSB-

R.

M.

Karp

Reducibility

among

combinatorial

problems

in

R.

E.

Miller

and

J.

W.

Thatcher

-LRB-

eds

-RRB-

Complexity

of

Computer

Computations

Plenum

Press

New

York

-LRB-

-RRB-

pp.

-LCB-

-LSB-

-RSB-

B.

W.

Kernighan

and

S.

Lin

An

efficient

heuristic

procedure

for

partitioning

graphs

BSTJ

-LRB-

-RRB-

pp.

-LCB-

-LSB-

-RSB-

D.

J.

Kleitman

and

D.

B.

West

Spanning

trees

with

many

leaves

SIAM

Journal

on

Discrete

Mathematics

-LRB-

Feb.

-RRB-

pp.

-LCB-

-LSB-

-RSB-

S.

Lin

and

B.

W.

Kernighan

An

effective

heuristic

algorithm

for

the

Traveling-Salesman

Problem

Operations

Research

-LRB-

-RRB-

pp

-LCB-

-LSB-

-RSB-

H.-I

Lu

and

R.

Ravi

The

power

of

local

optimization

approximation

algorithms

for

maximum-leaf

spanning

tree

Proceedings

of

the

Thirtieth

Annual

Allerton

Conference

on

Communication

Control

and

Computing

-LRB-

Oct.

-RRB-

pp.

-LCB-

-LSB-

-RSB-

C.

H.

Papadimitriou

A.

A.

Sch?affer

and

M.

Yannakakis

On

the

complexity

of

local

search

-LRB-

Extended

Abstract

-RRB-

Proceedings

of

the

Twenty-second

Annual

ACM

Symposium

on

the

Theory

of

Computing

-LRB-

-RRB-

pp.

-LCB-

-LSB-

-RSB-

C.

H.

Papadimitriou

and

K.

Steiglitz

Combinatorial

Optimization

Algorithms

and

Complexity

Prentice-Hall

Inc.

-LRB-

-RRB-

-LSB-

-RSB-

C.

H.

Papadimitriou

and

M.

Yannakakis

Optimization

Approximation

and

Complexity

classes

Proceedings

of

the

Twentieth

Annual

ACM

Symposium

on

the

Theory

of

Computing

-LRB-

-RRB-

pp.

-LCB-

-LSB-

-RSB-

C.

Payan

M.

Tchuente

and

N.

H.

Xuong

Arbres

avec

un

nombres

maximum

de

sommets

pendants

Discrete

Mathematics

-LRB-

-RRB-

pp

-LCB-

-LSB-

-RSB-

A.

A.

Sch?affer

and

M.

Yannakakis

Simple

local

search

problems

that

are

hard

to

solve

SIAM

Journal

on

Computing

-LRB-

-RRB-

-LRB-

-RRB-

pp

-LCB-

-LSB-

-RSB-

R.

Ravi

B.

Raghavachari

and

P.

N.

Klein

Approximation

through

local

optimality

Designing

networks

with

small

degree

Proceedings

Twelfth

Annual

Conference

on

Foundations

of

Software

Technology

and

Theoretical

Computer

Science

-LRB-

Dec.

-RRB-

LNCS

pp.

-LCB-

-LSB-

-RSB-

J.

A.

Storer

Constructing

full

spanning

trees

for

cubic

graphs

Information

Processing

Letters

-LRB-

-RRB-

pp.

-LCB-

-LSB-

-RSB-

M.

Tchuente

Sur

l'auto-stabilisation

dans

un

r?eseau

d'odinateurs

R.

A.

I.

R.

O.

Informatique

Th?eorique

-LRB-

-RRB-

pp

-LCB-

-LSB-

-RSB-

M.

Yannakakis

The

analysis

of

local

search

problems

and

their

heuristics

STACS

-LRB-

-RRB-

pp.

-LCB-

approximation algorithms
local optimization
performance ratio

combinatorial algorithms
communication network design
np-complete problems