Approximate Graph Coloring by Semidenite Programming y z David Karger Rajeev Motwani Madhu Sudan Abstract We consider the problem of coloring k colorable graphs with the fewest p ossible colors We present a randomized p olynomial time algorithm 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 University 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 algorithms 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