Greed is go o d Approximating indep endent sets in sparse and

y

b oundeddegree graphs

z x

Magns M Halldrsson Jaikumar Radhakrishnan

Abstract

The minimumdegree greedy algorithm or Greedy for short is a simple and well studied

metho d for nding indep endent sets in graphs We show that it achieves a p erformance ratio

of (
+ 2)3 for approximating indep endent sets in graphs with degree b ounded by
The

analysis yields a precise characterization of the size of the indep endent sets found by the

algorithm as a function of the indep endence numb er as well as a generalization of Turns

b ound We also analyze the algorithm when run in combination with a known prepro cessing

d + 3)5 p erformance ratio on graphs with average technique and obtain an improved (2

degree d improving on the previous b est (d + 1)2 of Ho chbaum Finally we present an

ecient parallel and distributed algorithm attaining the p erformance guarantees of Greedy

Intro duction

An independent set in a graph is a collection of vertices that are mutually nonadjacent The

problem of nding an indep endent set of maximum cardinality is one of the fundamental com

binatorial problems It is known to b e NP complete even for b oundeddegree graphs and

therefore no ecient algorithms are in sight

Given the hardness of exact computation we are interested in approximation algorithms for

the indep endent set problem in b oundeddegree graphs In particular we seek an algorithm

with a go o d performance ratio which is a b ound on the maximum ratio b etween the optimal

solution size ie the indep endence numb er and the size of the solution found by the heuristic

One of the most ubiquitous heuristic metho ds for this problem is the greedy algorithm which

selects a vertex of minimum degree deletes that vertex and all of its neighb ors from the graph

and rep eats this pro cess until the graph b ecomes empty As a delightfully simple and ecient

algorithm the Greedy metho d deserves a particularly detailed analysis It is already known to

p ossess several imp ortant qualities attaining the Turn b ound and its generalization in terms

of degree sequences almost always obtaining a solution at least half the size of an optimal

solution in a random graph yielding a nontrivial graph coloring approximation and a

light coloring with a small chromatic sum when applied iteratively as a coloring metho d

and nding optimal indep endent sets in trees seriesparallel cographs and graphs of degree at

most

While the p erformance ratio of Greedy has b een analyzed b efore to some extent the true

extent of its p erformance has apparently not b een determined b efore The b est ratio previously

Gordon Gekko

y

A preliminary version of this pap er app eared in the th ACM Symp osium on Theory of Computing

z

Science Institute University of Iceland IS Reykjavik Iceland email mmhrhihiis Work done at

Japan Advanced Institute of Science and Technology Hokuriku

x

Theoretical Computer Science Group Tata Institute of Fundamental Research Bombay India email

jaikumartcstifrresin Work done while visiting JAIST

claimed for Greedy was on graphs with maximum degree and d on graphs of

average degree d

Our main result is that Greedy is much b etter than previously claimed We obtain a tight

p erformance ratio of in terms of maximum degree and an asymptotically optimal

d in terms of average degree In the pro cess we give a natural extension b ound of

of Turns b ound that incorp orates the actual indep endence numb er of the graph and give a

general tight expression of the size of the solution found as a function of the indep endenc e

numb er and the numb er of vertices

We further analyze Greedy extended with a prepro cessing metho d of Ho chbaum We use

d but it to improve the b est p erformance ratio known in terms of average degree to

show it to b e of limited use in terms of maximum degree

It follows from our analysis that globally minimum degree is not required for Greedy to achieve

the p erformance guarantees claimed ab ove in fact it holds for any vertex whose degree is at

most the average of the degrees of it and its neighb ors This is a lo cally evaluated prop erty that

naturally leads to a parallel and distributed algorithm inheriting the approximative prop erties

of Greedy for the rst nontrivial such approximations known to us

The remainder of the pap er is organized as follows In Section we present the Greedy

algorithm and some of its prop erties including the Turn b ound and review other results on

approximating this problem We analyze Greedy in detail in Section starting with a general

ization of the Turn b ound in Section followed by tight p erformance ratios in Section

and limitations on its p erformance in Section In Section we consider improvements ob

tained by additionally applying a prepro cessing technique and describ e in Section a parallel

algorithm attaining the b ounds proved for Greedy

Notation

We use fairly standard graph notation and terminology For the graph in question usually

denoted G V E n denotes the numb er of vertices the maximum degree d the average

degree the indep endence numb er the size of the largest indep endent set and the inde

p endence fraction that is n For a vertex v dv denotes the degree of v and N v the set

of neighb ors of v

For an indep endent set algorithm A AG is the size of the solution obtained by A on graph

G V E The p erformance ratio of A is dened by

A

G

n max

A A

GjGj=n AG

We fo cus on Greedy denoted by Gr

The Greedy Algorithm and Related Results

The Greedy Algorithm

The minimumdegree greedy algorithm or Greedy for short incrementally constructs an inde

p endent set by selecting some vertex of minimum degree removing it and its neighb ors from

the graph and iterating on the remaining graph until empty

GreedyG

I

while G do

Cho ose v such that dv min dw

w V (G)

I I fv g

G G fv g N v

o d

Output I

end

The algorithm can b e implemented in time linear in the numb er of edges and vertices

indep endent of the degree

We call a no de critical if its degree is at most the average of the degrees of it and its neighb ors

That is v is critical if it satises

X

2

d v dw

w N (v )

A vertex of minimum degree is critical hence such a no de always exists Although we state

Greedy with this minimumdegree pivoting rule the only prop erty that we shall use is that the

selected vertex is critical

Consider an execution of the algorithm to b e a sequence of reductions each corresp onding

to an iteration In a reduction a vertex is selected added to the solution and then removed

along with its neighb orho o d from the graph Let t denote the numb er of reductions and d the

i

degree in the remaining graph of the ith vertex selected i t The numb er of vertices

removed in the ith reduction is thus d

i

The main prop erty of the algorithm that we shall b e using in our analysis is that the sum

of the degrees of the d vertices removed in the ith reduction must b e at least d d

i i i

This allows us to b ound from b elow the numb er of edges removed in each step

Previous results on Greedy

A classical theorem in graph theory due to Turn states that for any graph G

G n d

and that the inequality is tight only for the graph consisting of G disjoint cliques of size as

equal as p ossible Wei see proved an extension

X

G dv

v V

which he showed was attained by Greedy This result actually follows from an earlier theorem

of Erds see which states that if a graph contains no indep endent set of size k

then there is graph consisting of k disjoint cliques whose degree sequence dominates that of the

original graph The pro of of his theorem can also b e seen to indirectly refer to Greedy

We include a pro of here of the fact that Greedy attains the Turn b ound since the pro ofs of

our central results build directly up on it

n

Theorem Gr

d

Proof The pro of is a variation of the one given by Ho chbaum The Greedy algorithm

p erforms a sequence of t reductions each time picking a vertex and deleting it and its neighb ors

from the graph We count the numb er of vertices and edges deleted in each reduction

The removal of vertices in each reduction partitions the vertex set thus

t

X

d n

i

i=1

Since Greedy always selects a critical vertex the sum of the degrees of the vertices deleted in

step i is at least d d and thus the numb er of edges deleted is at least half that amount

i i

Summing over all the reductions

t

X

d d

i

n jE j

i=1

We now add and twice and obtain

t

X

2

d n d

i

i=1

Using the CauchySchwarz inequality and we get

2

d n n t

Rearranging the inequality we obtain the desired b ound on t which is precisely the numb er of

vertices found by Greedy

Another simple algorithm op erates by a rule that is the inverse of Greedy it deletes a

vertex of maximum degree until no edge remains Surprisingly this algorithm also attains the

Turn b ound as proved indep endently by Griggs and Chvtal and McDiarmid Its

approximative prop erties are however weaker On the graph with s vertices that is complete

bipartite less a single p erfect matching the algorithm may nd only a two vertex indep endent

set for an approximation ratio of s As a result we do not consider this metho d

further

An upp er b ound of of the p erformance ratio of Greedy with the minimumdegree rule

can b e obtained by rudimentary arguments Observe that in a graph with minimum degree

the indep endence numb er is b ounded from ab ove by n This is b ecause at least

jI j edges must exit an indep endent set I while at most n jI j edges can b e incident on

the remaining vertices Thus in a regular connected comp onent the indep endence numb er is at

most n for a ratio of This is at most for and we also know that

Greedy is optimal when

Consider now the case of a nonregular comp onent In each step there is a vertex of degree

at most so Greedy nds at least n indep endent vertices As long as Greedy selects a

vertex of degree one it pro ceeds optimally since the optimal solution can contain at most one of

the two vertices deleted in each step Thus we may assume without loss of generality that the

minimum degree is at least two Hence by the previous argument the indep endenc e numb er is

2

at most n Combined this yields a ratio of which is also at most

The ab ove argument may b e what is alluded to in p

Related results

Any maximal indep endent set is of size at least n which results in a trivial p erformance

ratio of In fact a ratio of holds since an optimal solution can contain at most jI j

vertices not already in a maximal solution I

The theorem of Bro oks is an early result that states that any connected graph can b e

colored with colors unless is consists of clique or an o dd cycle Since we can disp ose of

the exceptions optimally this yields a stronger b ound of n on the size of the indep endent set

we nd Lovsz gave an elegant pro of that can b e turned into an ecient algorithm

Ho chbaum intro duced a prepro cessing technique whose eect was to obtain stronger up

p er b ounds on the size of the optimal solution The technique was based on results of Nemhauser

and Trotter on solutions of the linear programming relaxation of the indep endent set prob

lem We describ e this approach in more detail in Section She obtained approximations of

weighted indep endent sets by applying coloring heuristics and selecting the heaviest color as a

solution In particular she obtained ratios of and dG where dG is the largest

minimum degree of any induced subgraph of G as well as a ratio of d in the unweighted

case

A decade passed with little eect Indep endent of the current work Berman and Frer

followed by Berman and Fujito gave signicantly improved ratios of The draw

10

back of their metho ds is the exorbitant time complexity of more than exp log n Their

approach is a lo cal search metho d that seeks a larger solution primarily by deleting a mo derate

numb er of vertices while adding a greater numb er

Khanna et al also considered a simpler version of this lo cal search strategy that merely

deletes one while adding two or more They obtained ratios that improved on the b ound

of

Alongside with the current work we studied subgraph removal metho ds that are motivated

by results in graph theory that state that graphs without small cliques contain provably larger

indep endent sets This allows us to obtain p erformance ratios of o for graphs of

small to fairly large degree as well as an asymptotic ratio of O log log We have

also analyzed further the lo cal search metho d of the previous two pap ers decreasing the time

complexity needed for the result of and obtained a p erformance ratio of with

an ecient version illustrating further timeapproximation tradeo p ossibilities

Halldrsson and Yoshihara have used ideas from the current pap er to analyze mo died

versions of the greedy algorithm schema In particular for graphs of maximum degree three

they give a lineartime algorithm that attains a ratio of

The preceding has fo cused on p ossibility results but nonapproximability results have made

great strides in recent years The indep endent set problem in b oundeddegree graphs for each

xed b elongs to the class of MAX SNPcomplete problems intro duced by Papadimitriou and

Yannakakis The groundbreaking results on the theory of interactive pro ofs resulting in the

landmark pap er of Arora et al show that there is a xed such that approximating the

indep endent set problem in b ounded degree graphs within a factor of is NP hard Even

stronger results hold when degree restrictions are lifted the indep endent set problem cannot

b e approximated with n for some unless P NP Alon et al have recently b een

able to scale the latter results down to show that approximation on b oundeddegree graphs

is similarly NP hard

Analysis of Greedy

Relative size of Greedy solutions

We start by strengthening the constructive version of Turns theorem by expressing the size

of the obtainable indep endent set as a function of the indep endence numb er Observe that

our b ound dominates Turns yielding strict improvements whenever the indep endence numb er

exceeds the promise of Turns b ound Recall that is the indep endence fraction n

2

Theorem Gr n

d

Proof Our pro of follows that of Theorem while we now additionally keep count the numb er

of vertices deleted that b elong to some maximum cardinality indep endent set

Fix an indep endent set of maximum cardinality and let k b e the numb er of vertices among

i

the d vertices deleted in reduction i that are also contained in that maximum indep endent

i

set Then

t

X

k

i

i=1

Recall that the sum of the degrees of the vertices deleted in step i is at least d d Note

i i

that no edge can have b oth its endp oints in the maximum indep endent set Thus the numb er

d +1 k

i i

of edges deleted is at least Hence we obtain the following strengthening of ineq

2 2

t

X

d k d

i i

n jE j

i=1

We now add and twice and apply the CauchySchwarz inequality to obtain

t

X

2 2 2 2

d n d k n t

i

i

i=1

Rearranging the inequality we obtain the desired b ound on t

We now turn our attention to b oundeddegree graphs using techniques similar to the pre

ceding pro of to obtain b ounds parametrized by the maximum degree

Theorem Gr n

Proof We extend the pro of of Theorem In the ith step d vertices and all edges incident

i

on them are deleted Of these edges let x have only one end in these d vertices the

i i

remaining edges have b oth ends among the d vertices of these let y have one end in the

i i

indep endent set and one outside and z have b oth ends outside Then we have

i

x y z d d

i i i i i

y k d k

i i i i

Multiply by reversing the inequality and add it to to obtain

x y z d d k d k

i i i i i i i i

d d

i i

k d k

i i i

d k d k

i i i i

Since the numb er of edges deleted in the ith step is precisely x y z we have the following

i i i

extension of

t

X

d k d k

i i i i

jE j z

i

i=1

We also count the total degree of vertices outside the maximum indep endent set which

entails counting edges incident on the indep endent set vertices once but those fully outside the

indep endent set twice

t

X

z jE j n

i

i=1

Now add twice and twice to obtain

t

X

n d d k k d k d k

i i i i i i i i

i=1

P

t

d k d k and comp ensate for To simplify the right hand side we add

i i i i

i=1

this by adding n to the left hand side invoking Then

t

X

2 2 2

d k d k n n

i i i

i

i=1

Using and the CauchySchwarz inequality we obtain

2 2 2

n n t

The claim follows from this

Performance guarantees

The following b ound on the p erformance of Greedy on sparse graphs follows easily We use the

b ound of Theorem in the denominator of the p erformance ratio function use the identity

n in the numerator and observe that the ratio is maximized when

d

Corollary

Gr

For b oundeddegree graphs the general expression obtained in Theorem almost but not

quite yields our main claim ab out the p erformance ratio of Greedy We now pro ceed to analyze

the p erformance of Greedy using a ner scalp el

Let t b e the numb er of reductions p erformed by Greedy where a vertex of degree d was

dk

chosen and exactly k vertices of the indep endent set were removed More precisely for d

and k maxd we dene t jfi d d and k k gj With this

dk i i

notation we may rewrite the constraints and as

X

d t n

dk

dk

X

k t

dk

dk

X

2 2 2

d k d k t n

dk

dk

P

We wish to extract from these constraints the b est p ossible lower b ound for t t

dk

dk

For this we use the metho d of multipliers describ ed in Chvtal page

Theorem

Gr

Proof We need to consider two cases based on the value of

Case mo d ie mo d We construct the linear combination of

the constraints and with multipliers and resp ectively and obtain

X

2 2 2

d d k d k t

dk

dk

Let

2 2 2

d min d k d k

k

and

C d d d

P

Set t t and conclude from that

d dk

k

X

C dt

d

d

We show b elow that for d C d That with then gives

t

as required It remains only to establish the following claim

Claim For d C d

Proof of claim It can easily b e veried that

3

2

d if d is o dd

2

d

1 3

2

d if d is even

2 2

Let f and f b e dened by

0 1

2

f x x x f x f x

0 1 0

Note

f d if d is o dd

0

C d

f d if d is even

1

Now f x f x i x and f x f x Thus b oth f and f are

0 1

0 1 0 1

concave functions that achieve their unique maximum at x Since f and f are

0 1

p olynomials of degree in x f x f x and f x f x for all Thus to

0 0 1 1

establish the claim it is enough to verify that f at the nearest even integer to x and f at the

0 1

nearest o dd integer to x are at most

Let m r where r since mo d The nearest even integer

to x is m and the nearest o dd integer to x is m r Plugging in we nd that

2

f m f m r m mr

0 1

establishing the claim

Case mo d This time we construct a linear combination of the constraints

and with multipliers and resp ectively and obtain

X

C d k t

dk

dk

where

2 2 2

C d k d k d k d k

d d k d k

Let C d max C d k To obtain the an upp er b ound on C d in terms of we consider

k

the functions f and f dened as follows

0 1

2

f x x x x x x

0

2

x x x f x x x

1

It can b e veried that

f d if d is o dd

0

C d

f d if d is even

1

We wish to b ound f d for o dd values of d and f d for even values of d Now f x f x

0 1

0 1

i x and f x f x thus f and f attain their unique maximum at x

0 1

0 1

Since f and f are p olynomials of degree in x it is enough to b ound f at the nearest o dd

0 1 0

integer to x ie and f at the nearest even integer to x ie Plugging

1

in we nd that

f

0

f

1

and using we obtain t Thus C d

Limitations

The p erformance ratios proved ab ove cannot b e improved

2

Theorem O n for every

Gr

Proof We give a detailed construction for mo d Consider the following family of

graphs H We have a chain of rep etitions of a pair of subgraphs a clique on vertices

followed by an indep endent set on vertices The two subgraphs are completely connected

while the connections b etween the indep endent set and the clique of the following pair miss only

Figure Initial p ortion of a hard graph for Greedy

a single p erfect matching ie each vertex in the indep endent set is adjacent to vertices in

the following clique The chain ends with one additional clique

An instance of this graph with is shown in Figure with the vertices picked by Greedy

shown in black and the maximum indep endent set vertices in grey

The essential prop erty of the graph is that the degree of the indep endent set vertices equals

the degree of the vertices of the rst clique of the chain We can therefore assume that Greedy

will pick one of the vertices from the rst clique and remove the remaining vertices from the

pair reducing the graph to an identical chain with one fewer pairs

Thus Greedy selects one vertex from each pair plus one from the nal clique for a total

of n The optimal solution contains all the indep endent set vertices for a total of

n This yields a ratio of

2

H n

Gr

To relate that to the degree measures we have that and

2

d

Thus

2

O n

Gr

and

d

2

d n O

Gr

even when

For the case of mo d we need more complicated chains of groups of six subgraphs

For mo d the elements are of the form

K K K K K K

1 1 1

K denotes a clique indep endent set on s vertices resp ectively In all cases a where K

s s

clique is completely connected to the following indep endent set while connections from the

indep endent set to the next clique miss a single p erfect matching In addition edges from

the rst indep endent set the second subgraph go towards the rst clique in the next group

rather than to the one immediately following The chain is nished with an additional

clique

Figure Initial p ortion of a hard graph for Greedy

An example for is given in Figure

The graphs are designed so that minimum degree will stay as and equal the degree of

the leftmost remaining clique Hence we may assume that Greedy will pick exactly one vertex

p er clique or no des p er group while the numb er of vertices in the maximum indep endent

set is p er group Hence ignoring the end of the chain the approximation ratio is

since maximum degree is

The case of mo d is similar with each element of the form

K K K K K K

1 +1

and edges going from the second to the fth subgraph We leave the details to the curious

reader

We now show that the b ound on the p erformance ratio in terms of average degree Corol

lary is optimal in an asymptotic sense

d

Theorem O d

Gr

Proof For each value of we describ e an innite family of graphs for which the claimed

ratio holds The graphs consists of chains of pairs of subgraphs as in the previous example with

the cliques reduced to single vertices Each pair consists of a vertex adjacent to a element

indep endent set each of whose no des are adjacent to the following single vertex cliques

The chain then ends with a single clique that the vertices of the last indep endent sets

are adjacent to An example is given in Figure

Figure Hard graph for Greedy in terms of average degree

2

There are edges for each pair consisting of vertices Thus if we ignore the end of

2

the chain the average degree is and the ratio obtained is

d

d

2

The end of the chain increases the average degree by roughly O n which disapp ears into

the lower order term

Variations of the ab ove constructions show that the b ounds of Theorems and are tight

for a range of values of This involves varying the size of the cliques relative to the size of

the indep endent sets p ossibly by connecting each indep endent set to several subsequent cliques

We omit the details

Greedy with prepro cessing

Finding a maximum indep endent set of a graph G V E can b e formulated as an integer

P

programming problem which maximizes x over the solutions of the system of linear

i

inequalities

x for each v V

i i

and

x x for each v v E

i j i j

If the integrality condition on the solutions is dropp ed we obtain the fractional independent

set problem or the linear programming relaxation of the indep endent set problem As shown

by Edmonds and Pulleyblank see this can b e solved eciently via a bipartite graph

constructed as follows Form two copies of the vertex set V and let vertices in dierent

copies b e adjacent i they corresp ond to adjacent vertices in G Given a characteristic vector

y y y y y y of a maximum indep endent set of this bipartite graph an optimal

1 2 n

1 2 n

LP solution of G is given by

x y y

i i

i

By the KnigEgervary theorem p the indep endence numb er of a bipartite graph equals

the numb er of vertices less the numb er of edges of a maximum matching The bipartite matching

p

njE j by an algorithm of Hop croft and Karp and problem can b e computed in time O

hence the LP relaxation of the indep endent set problem

We can partition V into the sets One Half and Zero corresp onding to the values of the

vertices in a given arbitrary but xed LP solution Nemhauser and Trotter showed that

the set One is contained in some optimal indep endent set O of G Furthermore Zero cannot

b e contained in O as otherwise the value of those vertices could b e unilaterally increased

Based on these results Ho chbaum prop osed a prepro cessing metho d to improve the

p erformance of approximate indep endent set algorithms Compute an LP solution and apply

the approximation algorithm only on the graph H induced by Half To the solution found

by the algorithm we can now add the vertices of One p ossibly resulting in a considerable

improvement The indep endence numb er of H can b e at most the sum of the values of the LP

solution or n so this approach is particularly valuable when the indep endence numb er of G

is greater than that

Theorem Greedy with preprocessing attains a ratio of which is tight when

mo d

Proof The upp er b ound of already follows from Theorem but here it can also b e

obtained directly from Theorem by arguing that This is b ecause b oth the approximate

and the optimal solution consist of One along with the resp ective solutions on the graph H

and in H the indep endence fraction is at most

For mo d the tightness of this ratio is demonstrated by the graphs given in the

pro of of Theorem They have the prop erty that for any indep endent set I the numb er of

neighb ors of the vertices in I exceeds the size of I It then follows from Theorem of that

any LP solution has all values equal to Half ie prepro cessing is of no help so H G

We can improve on this b ound slightly when mo d to

For instance we have a tight ratio of when Also for we get a ratio of

down from the promised by Theorem while there is a graph that forces a

ratio of We omit the details

Average degree

A greater care is needed for a result in terms of average degree Once prepro cessing has b een

applied the average degree may have changed for the worse and we cannot immediately apply

the b ounds proved on Greedy Nevertheless a closer lo ok shows that the b ounds will comple

ment each other as hop ed for Ho chbaum showed that the Turn b ound on Greedy can b e

complemented with the promise to yield a p erformance ratio b ound of d We

follow her lead to obtain a similar result as exp ected from Theorem

d

Theorem

Gr +P r e

Proof Let d n denote the average degree indep endence fraction and numb er of vertices

of H resp ectively Recall that H is the subgraph induced by Half and note that

Following we use two prop erties The rst is that jOnej jZero j as otherwise the LP

solution value is smaller than if Half V The second prop erty is that the numb er of edges in

d

G is at least n jOnej jZeroj since we may assume that G is a connected graph and that

2

n is p ositive Thus the average degree of G is b ounded by

d n jOnej

d

n jOnej

2

1+

n jOnej invoking Theorem and The size of the greedy solution will b e at least

d +1+

the size of the optimal solution n jOnej Our goal is now to argue that

2 3

d n jOnej

n jOnej

5 5

2

1+

n jOnej

n jOnej

d +1+

from which the theorem follows Crossmultiplying we nd that this entails establishing

d d

2

and if jOnej

2

d

d

Ineq holds b ecause the left hand side is monotone increasing with for To establish

2

we may note that the function f x x x d x is monotone increasing

for x Thus it suces to replace with in to obtain

d

d

which is true since ab ba is always at least when a b are p ositive

This again is tight for mo d by Theorem Ineq

Parallel algorithm

The minimumdegree greedy algorithm stipulates that in each step a vertex of globally minimum

degree b e selected added to the solution and removed from the graph along with its neighb ors

As such it lo oks imp ossible to parallelize and oers little freedom for heuristic improvements

Fortunately this is a case where the analysis guides us towards the design of b etter andor more

general algorithms

As observed in Section it suces to select a critical vertex ie one satisfying This

has some interesting implications For one it op ens up the p ossibility of the design of heuristics

using secondary selection rules or ordering heuristics while retaining the p erformance ratios of

Greedy

Another application is a straightforward derivation of a parallel as well as a distributed

approximation algorithm attaining the same p erformance ratios Vertices can b e selected in

parallel as long as the selection of one do esnt aect the ab ove criteria for the other In particular

vertices with disjoint and nonadjacent neighb orho o ds ie of distance four or greater can b e

selected and pro cessed concurrently

3

This suggests a natural approach to a parallel algorithm Let G denote the graph obtained

3

by taking the adjacency matrix of G to the third p ower two vertices are adjacent in G if they

are within distance three in G The adjacency matrices used here are assumed to contain s on

the diagonal For a vertex subset S let N S denote the set of vertices adjacent to some no de

in S

ParallelGreedyG

I

while V G

W f v V G v is critical g

3

H subgraph of G induced by W

MIS maximal indep enden t set of H

I I MIS

G G MIS N MIS

o d

return I

end

We assume the PRAM mo del with a ner distinction dep ending on the MIS algorithm used

We remark that this algorithm can also b e implemented in the distributed mo del of computation

The following lemma due to Alon and Szegedy private communication shows that a signif

icant fraction of the vertices must simultaneously satisfy prop erty

4

Lemma At least n vertices are critical

2

+4

P

2

Proof Let D denote dw dv We shall show that

v

w N (v )

PrD

v

2

v

which implies the lemma As observed by Shearer E D The value of D is b ounded

v v v

2

ab ove by dv dv and since it is integral it must dier from zero by at least one

when negative Thus

2

PrD PrD

v v

The claim now follows

Theorem ParallelGreedy nds an independent set of size and performance satisfying Theo

rems and in time O log n min pol y log n log using n processors in the EREW

model

Proof Each vertex added to the solution will satisfy prop erty regardless of the order of

removal of the simultaneously chosen vertices Hence the results of the theorems apply to this

algorithm

Let us now estimate the time complexity starting with the numb er of iterations From

2

Lemma the size of W is at least n Every vertex in W is reachable in G from MIS

N MIS by a not necessarily simple path of length Since the numb er of vertices reachable

2

from a xed vertex in two steps is at most we have

2 2

jMIS N MIS j jW j n

4

Thus the numb er of deleted vertices that is jMIS N MIS j is at least nD el ta Hence

4

the numb er of iterations is at most log n

Also notice that for any vertex v in the graph some vertex u of distance at most is

critical Either u is selected or some vertex within distance of u and thus within distance at

+1

most of v There are at most such no des and hence numb er of rounds A more

careful counting shows that the numb er of rounds is at most

The only nontrivial step in each round is the computation of a maximal indep endent set

of the graph H An algorithm of Goldb erg et al nds a maximal indep endent set in time

2

O log H H log n using a linear numb er of pro cessors

6 4

The combined time complexity is therefore b ounded by O log nlog min log n

Note that this is O log n on constant degree graphs

While the ab ove algorithm yields a solution satisfying Theorem its time complexity is not

well b ounded in terms of the average degree d Fortunately this can b e attained by rst deleting

all vertices of high degree

Lemma Theorem and Corol lary hold even if we rst eliminate simultaneously al l

vertices of degree at least d

d Let n d Proof Consider the subgraph induced by vertices of degree less than

denote the numb er of vertices average degree and indep endence fraction resp ectively Thus

represents the fraction of the vertices that were of high degree The prop osition is that

2

2

LH S n n RH S

d

d

At least d n edges are deleted so

d jE j jE jn n

d

n n

The indep endence numb er of the remaining graph is at least the original value less the numb er

of deleted vertices and thus We shall consider two cases dep ending on the value of

Case d LH S is minimized when is at its minimum which by Turns

theorem is d Thus we plug in for for

2

n

d

2 2

The numerator is at least which is at least Also

is at most since Hence is at least RH S

Case d LH S is at least

2 2

n n n

d

d d

d its value is On the other hand RH S is monotone increasing with so when

at most

2 2

d

n

d d d

Since is at least holds

By deleting rst all highdegree vertices in parallel b efore applying ParallelGreedy we obtain

d an ecient approximation in terms of

Corollary There is a EREW paral lel algorithm that nds an independent set satisfying

Theorem and Corol lary in time O log n minpol y d log n d using n processors

The prepro cessing metho d of last section requires the solution of bipartite matching for

which no deterministic parallel algorithms are known However randomized parallel algorithms

are known and thus the same applies to Theorem An ecient deterministic approach

that nearly matches the b ound of Theorem is to use either the complement of a maximal

matching or the output of the ab ove algorithm whichever indep endent set is larger As we

argue in this approach yields a p erformance ratio of d

Acknowledgments We thank Noga Alon and Mario Szegedy for kindly proving Lemma

Vask Chvtal for comments and helpful advice Dorit Ho chbaum for comments and corrections

and the anonymous referees for thorough imp ortant corrections

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