Approximate Graph Coloring by Semidenite Programming

y z

David Karger

Abstract

We consider the problem of coloring k colorable graphs with the fewest p ossible colors We

present a randomized p olynomial time which colors a colorable graph on n vertices

12 12

13 14

with minfO log log n O n log ng colors where is the maximum degree of

any vertex Besides giving the b est known approximation ratio in terms of n this marks the

rst nontrivial approximation result as a function of the maximum degree This result can

12

12k

b e generalized to k colorable graphs to obtain a coloring using minfO log log n

12

13(k +1)

O n log ng colors Our results are inspired by the recent work of Go emans and

Williamson who used an algorithm for semidenite optimization problems which generalize lin

ear programs to obtain improved approximations for the MAX CUT and MAX SAT problems

An intriguing outcome of our work is a duality relationship established b etween the value of

the optimum solution to our semidenite program and the Lovasz function We show lower

b ounds on the gap b etween the optimum solution of our semidenite program and the actual

chromatic numb er by duality this also demonstrates interesting new facts ab out the function

MIT Lab oratory for Computer Science Cambridge MA Email kargermitedu URL

httptheorylcsmitedukarger Supp orted by a Hertz Foundation Graduate Fellowship by NSF Young In

vestigator Award CCR with matching funds from IBM Schlumb erger Foundation Shell Foundation and

Xerox Corp oration and by NSF Career Award CCR

y

Department of Computer Science Stanford CA rajeevcsstanfordedu Sup

p orted by an Alfred P Sloan Research Fellowship an IBM Faculty Development Award and NSF Young Investigator

Award CCR with matching funds from IBM Mitsubishi Schlumb erger Foundation Shell Foundation and

Xerox Corp oration

z

MIT Lab oratory for Computer Science Cambridge MA Email madhulcsmitedu Work done when

this author was at IBMs Thomas J Watson Research Center Yorktown Heights NY

Intro duction

A legal vertex coloring of a graph GV E is an assignment of colors to its vertices such that no

two adjacent vertices receive the same color Equivalently a legal coloring of G by k colors is a

partition of its vertices into k indep endent sets The minimum numb er of colors needed for such

a coloring is called the chromatic numb er of G and is usually denoted by G Determining the

chromatic numb er of a graph is known to b e NPhard cf

Besides its theoretical signicance as a canonical NPhard problem graph coloring arises natu

rally in a variety of applications such as register allo cation and timetableexamination

scheduling In many applications which can b e formulated as graph coloring problems it

suces to nd an approximately optimum graph coloringa coloring of the graph with a small

though nonoptimum numb er of colors This along with the apparent imp ossibili ty of an exact

solution has led to some interest in the problem of approximate graph coloring The analysis of

approximation for graph coloring started with the work of Johnson who shows

that a version of the greedy algorithm gives an O n log napproximation algorithm for k coloring

k

Wigderson improved this b ound by giving an elegant algorithm which uses O n colors

to legally color a k colorable graph Subsequently other p olynomial time algorithms were provided

by Blum which use O n log n colors to legally color an nvertex colorable graph This

k

result generalizes to coloring a k colorable graph with O n log n colors The b est

known p erformance guarantee for general graphs is due to Halldorsson who provided a p oly

nomial time algorithm using a numb er of colors which is within a factor of O nlog log n log n

of the optimum

Recent results in the hardness of approximations indicate that it may b e not p ossible to sub

stantially improve the results describ ed ab ove Lund and Yannakakis used the results of Arora

Lund Motwani Sudan and Szegedy and Feige Goldwasser Lovasz Safra and Szegedy

to show that there exists a small constant such that no p olynomial time algorithm can

approximate the chromatic numb er of a graph to within a ratio of n unless P NP The current

hardness result for the approximation of the chromatic numb er is due to Feige and Kilian and

Hastad who show that approximating it to within n for any would imply NPRP

RP is the class of probabilistic p olynomial time algorithms making onesided error However

none of these hardness results apply to the sp ecial case of the problem where the input graph is

guaranteed to b e k colorable for some small k The b est hardness result in this direction is due to

Khanna Linial and Safra who show that it is not p ossible to color a colorable graph with

colors in p olynomial time unless P NP

In this pap er we present improvements on the result of Blum In particular we provide a

randomized p olynomial time algorithm which colors a colorable graph of maximum degree

with min fO log log n O n log ng colors moreover this can b e generalized to k

k k

colorable graphs to obtain a coloring using O log log n or O n log n colors

Besides giving the b est known approximations in terms of n our results are the rst nontrivial

approximations given in terms of Our results are based on the recent work of Go emans and

Williamson who used an algorithm for semidenite optimization problems cf to obtain

improved approximations for the MAX CUT and MAX SAT problems We follow their basic

paradigm of using algorithms for semidenite programming to obtain an optimum solution to a

relaxed version of the problem and a randomized strategy for rounding this solution to a feasible

but approximate solution to the original problem Motwani and Naor have shown that the

approximate graph coloring problem is closely related to the problem of nding a CUT COVER

of the edges of a graph Our results can b e viewed as generalizing the MAX CUT approximation

algorithm of Go emans and Williamson to the problem of nding an approximate CUT COVER In

fact our techniques also lead to improved approximations for the MAX k CUT problem We

also establish a duality relationship b etween the value of the optimum solution to our semidenite

program and the Lovasz function We show lower b ounds on the gap b etween the

optimum solution of our semidenite program and the actual chromatic numb er by duality this

also demonstrates interesting new facts ab out the function

Alon and Kahale use related techniques to devise a p olynomial time algorithm for coloring

random graphs drawn from a hard distribution on the space of all colorable graphs Recently

Frieze and Jerrum have used a semidenite programming formulation and randomized rounding

strategy essentially the same as ours to obtain improved approximations for the MAX k CUT

problem with large values of k Their results required a more sophisticated version of our analysis

but for the coloring problem our results are tight up to p olylogarithmic factors and their analysis

do es not help to improve our b ounds

Semidenite programming relaxations are an extension of the linear programming relaxation

approach to approximately solving NPcomplete problems We thus present our work in the style

of the classical LPrelaxation approach We b egin in Section by dening a relaxed version of

the coloring problem Since we use a more complex relaxation than standard linear programming

we must show that the relaxed problem can b e solved this is done in Section We then show

relationships b etween the relaxation and the original problem In Section we show that in a

sense to b e dened later the value of the relaxation b ounds the value of the original problem

Then in Sections and we show how a solution to the relaxation can b e rounded to make

it a solution to the original problem Combining the last two arguments shows that we can nd

a go o d approximation Section Section and Sections are in fact indep endent and can b e

read in any order after the denitions in Section In Section we investigate the relationship

b etween our fractional relaxations and the Lovasz function showing that they are in fact dual to

one another We investigate the approximation error inherent in our formulation of the chromatic

numb er via semidenite programming in Section

A Vector Relaxation of Coloring

In this section we describ e the relaxed coloring problem whose solution is in turn used to approx

imate the solution to the coloring problem Instead of assigning colors to the vertices of a graph

we consider assigning ndimensional unit vectors to the vertices To capture the prop erty of a

coloring we aim for the vectors of adjacent vertices to b e dierent in a natural way The vector

k coloring that we dene plays the role that a hyp othetical fractional k coloring would play in a

classical linearprogramming relaxation approach to the problem Our relaxation is related to the

concept of an orthonormal representation of a graph

Denition Given a graph G V E on n vertices and a real number k a vector k

n

coloring of G is an assignment of unit vectors u from the space to each vertex i V such that

i

for any two adjacent vertices i and j the dot product of their vectors satises the inequality

hu u i

i j

k

The denition of an orthonormal representation requires that the given dot pro ducts

b e equal to zero a weaker requirement than the one ab ove

Solving the Vector Coloring Problem

In this section we show how the vector coloring relaxation can b e solved using semidenite pro

gramming The metho ds in this section closely mimic those of Go emans and Williamson

To solve the problem we need the following auxiliary denition

Denition Given a graph G V E on n vertices a matrix k coloring of the graph is an

n n symmetric positive semidenite matrix M with m and m k if fi j g E

ii ij

We now observe that matrix and vector k colorings are in fact equivalent cf Thus to

solve the vector coloring relaxation it will suce to nd a matrix k coloring

Fact A graph has a vector k coloring if and only if it has matrix k coloring Moreover a vector

k coloring can be constructed from a matrix k coloring in time polynomial in n and log

Note that an exact solution cannot b e found as some of the values in it may b e irrational

Pro of Given a vector k coloring fv g the matrix k coloring is dened by m hv v i For the

i ij i j

other direction it is well known that for every symmetric p ositive denite matrix M there exists a

T T

square matrix U such that UU M where U is the transp ose of U The rows of U are vectors

n

that form a vector k coloring of G fu g

i

i

An close approximation to the matrix U can b e found in time p olynomial in n and log

can b e found using the Incomplete Cholesky Decomposition Here by close we mean a

T

matrix U such that U U M has L norm less than This in turn gives a vector k

coloring of the graph provided is chosen appropriately

Lemma If a graph G has a vector k coloring then a vector k coloring of the graph can

be constructed in time polynomial in k n and log

Pro of Our pro of is similar to those of Lovasz and Go emansWilliamson We construct

a semidenite optimization problem SDP whose optimum is k when k is the smallest

real numb er such that a matrix k coloring of G exists The optimum solution also provides a matrix

k coloring of G

minimize

where fm g is p ositive semidenite

ij

sub ject to m if i j E

ij

m m

ij j i

m

ii

Consider a graph which has a vector and matrix k coloring This means there is a solution to the

ab ove semidenite program with k The ellipsoid metho d or other interior p oint based

metho ds can b e employed to nd a feasible solution where the value of the ob jective is at

most k in time p olynomial in n and log This implies that for all fi j g E m is

ij

at most k which is at most k for k provided k

Thus a matrix k coloring can b e found in time p olynomial in k n and log From the

matrix coloring the vector coloring can b e found in p olynomial time as was noted in the previous

lemma

Relating Original and Relaxed Solutions

In this section we show that our vector coloring problem is a useful relaxation b ecause the solution

to it is related to the solution of the original problem In order to understand the quality of the

relaxed solution we need the following geometric lemma

n

Lemma For al l positive integers k and n such that k n there exist k unit vectors in

such that the dot product of any distinct pair is k

Pro of Clearly it suces to prove the lemma for n k For other values of n we make the

co ordinates of the vectors in all but the rst k co ordinates We b egin by proving the claim for

k k k k

k

such that hv v i k v n k We explicitly provide unit vectors v

i j

k

q

k

for i j The vector v is in all co ordinates except the ith co ordinate In the ith

i

k k

q

k

k

It is easy to verify that the vectors are unit length and that their dot co ordinate v is

i

k

pro ducts are exactly

k

As given the vectors are in a k dimensional space Note however that the dot pro duct of

each vector with the alls vector is This shows that all k of the vectors are actually in a

kdimensional hyp erplane of the kdimensional space This proves the lemma

Corollary Every k colorable graph G has a vector k coloring

Pro of Bijectively map the k colors to the k vectors dened in the previous lemma

Note that a graph is vector colorable if and only if it is colorable Lemma is tight in

that it provides the b est p ossible value for minimizing the maximum dotpro duct among k unit

vectors This can b e seen from the following lemma

Lemma Let G be vector k colorable and let i be a vertex in G The induced subgraph on the

neighbors of i is vector k colorable

Pro of Let v v b e a vector k coloring of G and assume without loss of generality that

n

obtained by pro jecting v onto v Asso ciate with each neighb or j of i a vector v

j i

j

co ordinates through n and then scaling it up so that v has unit length It suces to show that

j

i k v for any two adjacent vertices j and j in the neighb orho o d of i hv

j

j

Observe rst that the pro jection of v onto the rst co ordinate is negative and has magnitude

j

k

p

is at least at least k This implies that the scaling factor for v Thus

j

k k

k

i hv v i hv v

j j

j j

k k k k

A simple induction using the ab ove lemma shows that any graph containing a k clique is

not k vector colorable Thus the vector chromatic numb er lies b etween b etween the clique and

chromatic numb er This also shows that the analysis of Lemma is tight in that k is

the minimum p ossible value of the maximum of the dotpro ducts of k vectors

In the next few sections we prove the harder part namely if a graph has a vector k coloring

k k

then it has an O and an O n coloring

Semicolorings

Given the solution to the relaxed problem our next step is to show how to round the solution

to the relaxed problem in order to get a solution to the original problem Both of the rounding

techniques we present in the following sections pro duce the coloring by working through an almost

legal semicoloring of the graph as dened b elow

Denition A k semicoloring of a graph G is an assignment of k colors to the at least half it

vertices such that no two adjacent vertices are assigned the same color

An algorithm for semicoloring leads naturally to a coloring algorithm as shown by the following

lemma The algorithm uses up at most a logarithmic factor more colors than the semicoloring

algorithm Furthermore we do not even lose this logarithmic factor if the semicoloring algorithm

uses a p olynomial numb er of colors which is what we will show we use

Lemma If an algorithm A can k semicolor any ivertex subgraph of graph G in randomized

i

polynomial time where k increases with i then A can be used to O k log ncolor G Furthermore

i n

if there exists such that for al l i k i then A can be used to color G with O k colors

i n

Pro of We show how to construct a coloring algorithm A to color any subgraph H of G A

starts by using A to semicolor H Let S b e the subset of vertices which have not b een assigned

a color by A Observe that jS j jV H j A xes the colors of vertices not in S and then

recursively colors the induced subgraph on S using a new set of colors

Let c b e the maximum numb er of colors used by A to color any ivertex subgraph Then c

i i

satises the recurrence

c c k

i i

i

It is easy to see that this any c satisfying this recurrence must satisfy c k log i In particular

i i i

this implies that c O k log n Furthermore for the case where k i the ab ove recurrence

n n i

is satised only when c k

i i

Using the ab ove lemma we devote the next two sections to algorithms for transforming vector

colorings into semicolorings

Rounding via Hyp erplane Partitions

We now fo cus our attention on vector colorable graphs leaving the extension to general k for later

Let b e the maximum degree in a graph G In this section we outline a randomized rounding

log

3

scheme for transforming a vector coloring of G into an O semicoloring and thus into an

log

3

O log ncoloring of G Combining this metho d with a technique of Wigderson yields an

O n coloring of G The metho d is based on and is weaker than the metho d we describ e

in the following section however it intro duces several of the ideas we will use in the more p owerful

algorithm

n

Assume we are given a vector coloring fv g Recall that the unit vectors v and v asso ciated

i i j

i

with an adjacent pair of vertices i and j have a dot pro duct of at most implying that the

angle b etween the two vectors is at least radians degrees

Denition Consider a hyperplane H We say that H separates two vectors if they do not lie

on the same side of the hyperplane For any edge fi j g E we say that the hyperplane H cuts

the edge if it separates the vectors v and v

i j

In the sequel we use the term random hyperplane to denote the unique hyp erplane containing

the origin and having as its normal a random unit vector v uniformly distributed on the unit sphere

S The following lemma is a restatement of Lemma of Go emansWilliamson

n

Lemma Go emansWilliamson Given two vectors at an angle of the probability

that they are separated by a random hyperplane is exactly

We conclude that give a vector coloring for any edge fi j g E the probability that a random

hyp erplane cuts the edge is exactly It follows that the exp ected fraction of the edges in G

which are cut by a random hyp erplane is exactly Supp ose that we pick r random hyp erplanes

r

indep endently Then the probability that an edge is not cut by one of these hyp erplanes is

r

and the exp ected fraction of the edges not cut is also

We claim that this gives us a go o d semicoloring algorithm for the graph G Notice that r

n r

hyp erplanes can partition into at most distinct regions For r n this is tight since r

r

hyp erplanes create exactly regions An edge is cut by one of these r hyp erplanes if and only if

the vectors asso ciated with its endp oints lie in distinct regions Thus we can asso ciate a distinct

r

color with each of the regions and give each vertex the color of the region containing its vector

r

The exp ected numb er of edges whose endp oints have the same color is m where m is the

numb er of edges in E

log

3

Theorem If a graph has a vector coloring then it has an O semicoloring which can

be constructed from the vector coloring in polynomial time with high probability

Pro of We use the random hyp erplane metho d just describ ed Fix r dlog e and note

r r log

3

that and that O As noted ab ove r hyp erplanes chosen indep endently

at random will cut an edge with probability Thus the exp ected numb er of edges which

are not cut is m n n since the numb er of edges is at most n By Markovs

inequality cf page the probability that the numb er of uncut edges is more than twice

the exp ected value is at most Thus with probability at least we get a coloring with at

most n uncut edges Delete one endp oint of each such edge leaves a set of n colored vertices

with no uncut edgesie a semicoloring

log

3

Rep eating the entire pro cess t times means that we will nd a O semicoloring with

t

probability at least

Noting that log and that n this theorem and Lemma implies a semicoloring

using O n colors

By varying the numb er of hyp erplanes we can arrange for a tradeo b etween the numb er of

colors used and the numb er of edges that violate the resulting coloring This may b e useful in some

applications where a nearly legal coloring is go o d enough

Wigdersons Algorithm

Our coloring can b e improved using the following idea due to Wigderson Fix a threshold

value If there exists a vertex of degree greater than pick any one such vertex and color its

neighb ors its neighb orho o d is vector colorable and hence colorable The colored vertices are

removed and their colors are not used again Rep eating this as often as p ossible or until half the

vertices are colored brings the maximum degree b elow at the cost of using at most n colors

Thus we can obtain a semicoloring using O n colors The optimum choice of is around

n which implies a semicoloring using O n colors This semicoloring can b e used to legally

color G using O n colors by applying Lemma

Corollary A colorable graph with n vertices can be colored using O n colors by a poly

nomial time randomized algorithm

The b ound just describ ed is marginally weaker than the guarantee of a O n coloring due

to Blum We now improve this result by constructing a semicoloring with fewer colors

Rounding via Vector Pro jections

In this section we start by proving the following more p owerful version of Theorem A simple

application of Wigdersons technique to this algorithm yields our nal coloring algorithm

Lemma For every integer function k k n a vector k colorable graph with maximum degree

p

k

ln colors in probabilistic polynomial time can be semicolored with at most O

As in the previous section this has immediate consequences for approximate coloring

Given a vector k coloring we show that it is p ossible to extract an indep endent set of size

p

k

n ln If we assign one color to this set and recurse on the rest we will end up

p

k

using O ln colors in all to assign colors to half the vertices and the result follows To

nd such a large indep endent set we give a randomized pro cedure for selecting an induced subgraph

p

k

ln It follows that with with n vertices and m edges such that E n m n

a p olynomial numb er of rep eated trials we have a high probability of cho osing a subgraph with

p

k

n m n ln Given such a graph we can delete one endp oint of each edge

p

k

ln as desired leaving an indep endent set of size n m n

We now give the details of the construction Supp ose we have a vector k coloring assigning unit

vectors v to the vertices We x a parameter c c to b e sp ecied later We cho ose a random

i k

ndimensional vector r according to a distribution to b e sp ecied so on The subgraph consists of

all vertices i with v r c Intuitively since endp oints of an edge have vectors p ointing away from

i

each other if the vector asso ciated with a vertex has a large dot pro duct with r then the vector

corresp onding to an adjacent vertex will not have such a large dot pro duct with r and hence will

not b e selected Thus only a few edges are likely to b e in the induced subgraph on the selected set

of vertices

To complete the sp ecication of this algorithm and to analyze it we need some basic facts ab out

n

some probability distributions in

n

Probability Distributions in <

2

x

p

Recall that the standard normal distribution has the density function x e with

distribution function x mean and variance A random vector r r r is said to

n

have the ndimensional standard normal distribution if the comp onents r are indep endent random

i

variables each comp onent having the standard normal distribution It is easy to verify that this

distribution is spherically symmetric in that the direction sp ecied by the vector r is uniformly

distributed Refer to Feller v I I Knuth v and Renyi for further details ab out

the higher dimensional normal distribution

Subsequently the phrase random ddimensional vector will always denote a vector chosen

from the ddimensional standard normal distribution A crucial prop erty of the normal distribution

which motivates its use in our algorithm is the following theorem paraphrased from Renyi see

also Section I I I of Feller v I I

Theorem Theorem IV Let r r r be a random ndimensional vector

n

The projections of r onto two lines and are independent and normal ly distributed if and

only if and are orthogonal

Alternatively we can say that under any rotation of the co ordinate axes the pro jections of r

along these axes are indep endent standard normal variables In fact it is known that the only

distribution with this strong spherical symmetry prop erty is the ndimensional standard normal

distribution The latter fact is precisely the reason b ehind this choice of distribution in our

algorithm In particular we will make use of the following corollary to the preceding theorem

n

Corollary Let u be any unit vector in Let r r r be a random vector of

n

iid standard normal variables The projection of r along u given by dot product hu ri is dis

tributed according to the standard dimensional normal distribution

It turns out that even if r is a random ndimensional unit vector the ab ove corollary still holds

in the limit as n grows the pro jections of r on orthogonal lines approach scaled indep endent

normal distributions Thus using a random unit vectors for our pro jection turns out to b e equivalent

to using random normal vectors in the limit but is messier to analyze

Let N x denote the tail of the standard normal distribution Ie

Z

N x y dy

x

We will need the following wellknown b ounds on the tail of the standard normal distribution See

for instance Lemma VI I of Feller v I

Lemma For every x

x N x x

x x x

Pro of The pro of is immediate from insp ection of the following equations relating the three

quantities in the desired inequality to integrals involving x and the fact xx is nite for

every x

Z

dy x y

x x y

x

Z

y dy N x

x

Z

x y dy

x y

x

1

Readers familiar with physics will see the connection to Maxwells law on the distribution of velo cities of molecules

3 3

in < Maxwell started with the assumption that in every Cartesian co ordinate system in < the three comp onents

of the velo city vector are mutually indep endent and had exp ectation zero Applying this assumption to rotations of

the axes we conclude that the velo city comp onents must b e indep endent normal variables with identical variance

This immediately implies Maxwells distribution on the velo cities

The Analysis

We are now ready to complete the sp ecication of the coloring algorithm Recall that our goal is

to rep eatedly identify color and delete large indep endent sets from the graph We actually set an

easier intermediate goal nd an induced subgraph with a large numb er n of edges and a numb er

m n of vertices Since each edge only covers vertices the induced subgraph has n m

vertices with no incident edges These vertices form an indep endent set that can b e colored and

removed

As discussed ab ove to nd this sparse graph we cho ose a random vector r and take all vertices

whose dot pro duct with r exceeds a certain value c Let the induced subgraph on these vertices

have n vertices and m edges We show that for suciently larger c n m and we get an

indep endent set of size roughly n Intuitively this is true for the following reason Any particular

vertex has some particular probability p pc of landing near r and thus b eing captured into

our set However if two vertices are adjacent the probability that they both land near r is quite

small b ecause the vector coloring has placed them far apart

For example in the case of coloring when the probability that a vertex is chosen is p the

probability that b oth endp oints of an edge are chosen is roughly p It follows that we end up

capturing in exp ectation a set of pn vertices that contains in exp ectation only p m p n

edges in a degree graph In such a set at least pn p n of the vertices have no incident edges

and thus form an indep endent set We would like this indep endent set to b e large Clearly we need

to make p small enough to ensure p n pn meaning p Taking p much smaller only

decreases the size of the indep endent set so it turns out that our b est choice is to take p

yielding an indp endent set of size n Rep eating this capture pro cess many times therefore

achieves an O coloring

We now formalize this intuitive argument The vector r will b e a random ndimensional vector

We precisely compute the exp ectation of n the numb er of vertices captured and the exp ectation

of m the numb er of edges in the induced graph of the captured vertices We rst show that when

r is a random normal vector and our pro jection threshold is c the exp ectation of n m exceeds

nN c N ac for a certain constant a dep ending on the vector chromatic numb er We also

2

a

show that N ac grows roughly as N c For the case of coloring we have a and thus if

N c p then N ac p By picking a suciently large c we can nd an indep endent set of size

N c In the following lemma n and m are functions of c we do not make this dep endence

explicit

q

k

Lemma Let a Then for any c

k

E n m nN c N ac

Pro of We rst b ound E n from b elow Consider a particular vertex i with assigned vector

v The probability that it is in the selected set is just P v r c By Corollary v r is

i i i

normally distributed and thus this probability is N c By linearity of exp ectations the exp ected

numb er of selected vertices E n nN c

Now we b ound E m from ab ove Consider an edge with endp oint vectors v and v The

probability that this edge is in the induced subgraph is the probability that b oth endp oints are

selected which is

P v r c and v r c P v v r c

c v v

r P

kv v k kv v k

c

N

kv v k

where the expression follows from Corollary applied to the preceding probability expression

We now observe that

q

v v v v kv v k

q

k

q

k k

a

It follows that the probability that b oth endp oints of an edge are selected is at most N ac If the

graph has maximum degree then the total numb er of edges is at most n Thus the exp ected

numb er of selected edges E m is at most nN ac

Combining the previous arguments we deduce that

nN c nN ac E n m

We now determine the a c such that N ac N c This will give us an exp ectation of at

least N c in the ab ove lemma Using the b ounds on N x in Lemma we nd that

2

c

e

N c

3

c

c

2 2

a c

N ac

e ac

2 2

a c

e a

c

p

2 2

a c

e

c

p

p

The last equation holds since a k k Thus if we cho ose c so that c

2 2

a c

p

and e then we get N ac N c Both conditions are satised for suciently

large if we set

s

k

ln c

k

For smaller values of we can use the greedy coloring algorithm to get a color the graph

with a b ounded numb er of colors where the b ound is indep endent of n

For this choice of c we nd that the indep endent set that is found has size at least

E n m nN c

2

c

ne

c c

n

p

2

k

ln

as desired This concludes the pro of of Lemma

Adding Wigdersons Technique

To conclude we now determine absolute approximation ratios indep endent of This involves

another application of Wigdersons technique If the graph has any vertex of large degree then

we use the fact that its neighb orho o d is large and is vector k chromatic to nd a large

indep endent set in its neighb orho o d If no such vertex exists then the graph has small maximum

degree so we can use Lemma to nd a large indep endent set in the graph After extracting

such an indep endent set we recurse on the rest of the graph The following lemma describ es the

details and the correct choice of the threshold degree

Lemma For every integer function k k n any vector k colorable graph on n vertices can

k

be semicolored with O n log n colors by a probabilistic polynomial time algorithm

Pro of Given a vector k colorable graph G we show how to nd an indep endent set of size

k

n log n in the graph Assume by induction on k that there exists a constant c

k

st we can nd an indep endent set of size ci log i in any k vector chromatic graph on

i no des for k k We now prove the inductive assertion for k

k k

Let n n If G has a vertex of degree greater than n then we nd

k k k

a large indep endent set in the neighb orho o d of G By Lemma the neighb orho o d is vector

k colorable Hence we can nd in this neighb orho o d an indep endent set of size at least

k k

c log cn log n If G do es not have a vertex of degree greater than

k k

k

n then by Lemma we can nd an indep endent set of size at least cn log

k k k

k

cn log n in G This completes the induction

By now assigning a new color to each such indep endent set we nd that we can color at least

k

n vertices using up at most O n log n colors

The semicolorings guaranteed by Lemmas and can b e converted into colorings using

Lemma yielding the following theorem

Theorem Any vector k colorable graph on n nodes with maximum degree can be colored in

p p

k k

probabilistic polynomial time using min fO ln log n O n ln ng colors

Duality Theory

The most intensively studied relaxation of a semidenite programming formulation to date is the

Lovasz function This relaxation of the clique numb er of a graph led to the rst

p olynomialtime algorithm for nding the clique and chromatic numb ers of p erfect graphs We

now investigate a connection b etween and a close variant of the vector chromatic numb er

Intuitively the clique and coloring problems have a certain duality since large cliques prevent

a graph from b eing colored with few colors Indeed it is the equality of the clique and chromatic

numb ers in p erfect graphs which lets us compute b oth in p olynomial time We pro ceed to for

malize this intuition The duality theory of linear programming has an extension to semidenite

programming With the help of Eva Tardos and David Williamson we have shown that in fact the

function and a close variant of the vector chromatic numb er are semidenite programming duals

to one another and are therefore equal

We rst dene the variant

Denition Given a graph G V E on n vertices a strict vector k coloring of G is an

n

assignment of unit vectors u from the space to each vertex i V such that for any two

i

adjacent vertices i and j the dot product of their vectors satises the equality

hu u i

i j

k

As usual we say that a graph is strictly vector k colorable if it has a strict vector k coloring

The strict vector chromatic numb er of a graph is the smallest real numb er k for which it has a

strict vector k coloring It follows from the denition that the strict vector chromatic numb er of

any graph is lower b ounded by the vector chromatic numb er

G Theorem The strict vector chromatic number of G is equal to

Pro of The dual of our strict vector coloring semidenite program is as follows cf

X

maximize p

ii

where fp g is p ositive semidenite

ij

X

sub ject to p

ij

i j

p p

ij j i

p for i j E and i j

ij

By duality the value of this SDP is k where k is the strict vector chromatic numb er Our

goal is to prove k As b efore the fact that fp g is p ositive semidenite means we can nd

ij

vectors v such that p hv v i The last constraint says that the vectors v form an orthogonal

i ij i j

labeling ie that hv v i for i j E We now claim that the ab ove optimization problem

i j

can b e reformulated as follows

P

hv v i

i i

P

maximize

hv v i

i j

i j

over all orthogonal lab elings fv g To see this consider an orthogonal lab eling and dene

i

P

hv v i Note this is the value of the rst constraint in the rst formulation of the dual that

i j

i j

is the constraint is and of the denominator in the second formulation Then in an optimum

solution to the rst formulation we must have since otherwise we can divide each v by

i

p

and get a feasible solution with a larger ob jective value Thus the optimum of the second

formulation is at least as large as that of the rst Similarly given any optimum fv g for the second

i

p

formulation v forms a feasible solution to the rst formulation with the same value Thus the

i

optima are equal We now manipulate the second formulation

P P

hv v i hv v i

i i i i

P P P

max max

hv v i hv v i hv v i

i j i j i i

i j ij

P P

hv v i hv v i

i j i i

ij

P

min

hv v i

i i

P

hv v i

i j

ij

P

min

hv v i

i i

P

hv v i

i j

ij

P

max

hv v i

i i

It follows from the last equation that the vector chromatic numb er is

P

hv v i

i j

ij

P

max

hv v i

i i

However by the same argument as used to reformulate the dual this is equal to problem of

P P

hv v i over all orthogonal lab elings such that hv v i This is simply Lovaszs maximizing

i j i i

ij

formulation of the function page

The Gap b etween Vector Colorings and Chromatic Numb ers

The p erformance of our randomized rounding approach seems far from optimum In this section

we ask why and show that the problem is not in the randomized rounding but in the gap b etween

the original problem and its relaxation We investigate the following question given a vector k

colorable graph G how large can its chromatic numb er b e in terms of k and n We will show that

a graph with chromatic numb er n can have b ounded vector chromatic numb er This implies

o

that our technique is tight in that it is not p ossible to guarantee a coloring with n colors on all

vector colorable graphs

Denition The Kneser graph K m r t is dened as fol lows the vertices are al l possible r sets

from a universe of size m and the vertices v and v are adjacent if and only if the corresponding

i j

r sets satisfy jS S j t

i j

We will need following theorem of Milner regarding intersecting hyp ergraphs Recall that

a collection of sets is called an antichain if no set in the collection contains another

Theorem Milner Let S S be an antichain of sets from a universe of size m such

that for al l i and j

jS S j t

i j

Then it must be the case that

m

mt

Notice that using all q sets for q m t gives a tight example for this theorem

The following theorem establishes that the Kneser graphs have a large gap b etween their vector

chromatic numb er and chromatic numb ers

m

Theorem Let n denote the number of vertices of the graph K m r t For r m

r

and t m the graph K m r t is vector colorable but has chromatic number at least n

Pro of We prove a lower b ound on the Kneser graphs chromatic numb er by establishing an

upp er b ound on its indep endence numb er It is easy to verify that the in Milners theorem is

exactly the indep endence numb er of the Kneser graph To b ound observe that

n

m

r

m

mt

m

m

m

m

m

o

om lg lg

m

for large enough m

In the ab ove sequence the fourth line uses the approximation

m

p

m lg lg

c m

m

for every where c is a constant dep ending only on Using the inequality

m

m

n

r

we obtain m lg n and thus

lg n lg

n n

Finally it remains to show that the vector chromatic numb er of this graph is This follows by

asso ciating with each vertex v an mdimensional vector obtained from the characteristic vector of

i

the set S In the characteristic vector represents an element present in S and represents

i i

elements absent from S The vector asso ciated with a vertex is the characteristic vector of S

i i

p

m to obtain a unit vector Given vectors corresp onding to sets S scaled down by a factor of

i

and S the dot pro duct gets a contribution of m for co ordinates in S S and m for the

j i j

others Here AB represents the symmetric dierence of the two sets ie the set of elements

which o ccur in exactly one of A or B Thus the dot pro duct of two adjacent vertices or sets with

intersection at most t is given by

jS S j jS j jS j jS S j r t

i j i j i j

m m m

This implies that the vector chromatic numb er is

More rened calculations can b e used to improve this b ound somewhat

Theorem There exists a Kneser graph K m r t which is vector colorable but has chromatic

m

number exceeding n where n denotes the number of vertices in the graph Further

r

for large k there exists a Kneser graph K m r t which is k vector colorable but has chromatic

number exceeding n

Pro of The basic idea is to improve the b ound on the vector chromatic numb er of the Kneser

graph using an appropriately weighted version of the characteristic vectors We use weights a and

to represent presence and absence resp ectively of an element in the set corresp onding to a

vertex in the Kneser graph with appropriate scaling to obtain a unit vector The value of a which

minimizes the vector chromatic numb er can b e found by dierentiation and is

mr mt

A

r r t r r t

Setting a A proves that the vector chromatic numb er is at most

mr t

r mt

At the same time using Milners Theorem proves that the exp onent of the chromatic numb er is at

least

m m

m t log m t log

mt mt

m m

m r log r log

mr r

By plotting these functions we have shown that there is a set of values with vector chromatic

numb er and chromatic numb er at least n For large constant vector chromatic numb ers

the limiting value of the exp onent of the chromatic numb er is roughly

Conclusions

The Lovasz numb er of a graph has b een a sub ject of active study due to the close connections b e

tween this parameter and the clique and chromatic numb ers In particular the following sandwich

theorem was proved by Lovasz see Knuth for a survey

G G G

This led to the hop e that the following question may have an armative answer Do there exist

such that for any graph G on n vertices

G

G G G G n

n

Our work in this pap er proves a weak but nontrivial upp er b ound on the the chromatic numb er of

G However this is far from achieving the b ound conjectured ab ove and subsequent G in terms of

to our work two results have ended up answering this question negatively Feige has shown that

for every there exist families of graphs for which G Gn Interestingly families of

graphs exhibited in Feiges work use the construction of Section as a starting p oint Even more

conclusively the results of Hastad and Feige and Kilian have shown that no polynomial

time computable function approximates the clique numb er or chromatic numb er to within factors

of n unless NPRP Thus no simple mo dication of the function is likely to provide a much

b etter approximation guarantee

In related results Alon and Kahale have also b een able to use the semidenite programming

technique in conjunction with our techniques to obtain algorithms for computing b ounds on the

clique numb er of a graph with linearsized cliques improving up on some results due to Boppana and

Halldorsson Indep endent of our results Szegedy has also shown that a similar construction

yields graphs with vector chromatic numb er at most but which are not colorable using n colors

Notice that the exp onent obtained from his result is b etter than the one in Section Alon has

obtained a slight improvement over Szegedys b ound by using an interesting variant of the Kneser

graph construction Finally the main algorithm presented here has b een derandomized in a recent

work of Maha jan and Ramesh

Acknowledgments

Thanks to David Williamson for giving us a preview of the MAXCUT result during a visit to

Stanford We are indebted to John Tukey and Jan Pedersen for their help in understanding multi

dimensional probability distributions Thanks to David Williamson and Eva Tardos for discussions

of the duality theory of SDP We thank Don Copp ersmith Jon Kleinb erg Laci Lovasz

and for useful discussions and the anonymous referees for the careful comments

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