Approximation Algorithms and Hardness of Approximation April 19, Lecture 15

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1 Approximation Algorithms and Hardness of Approximation April 19, 2013 Lecture 15 Lecturer: Alantha Newman Scribes: Marwa El Halabi and Slobodan Mitrović 1 Coloring 3-Colorable Graphs In previous lectures, we illustrated how to formulate the max cut problem as a vector programs(or SDP), by relaxing the variables in the exact formulation from scalars to vectors. We showed that solving thesdpuptoanarbitrarilysmalladditiveerrorandthenroundingthevectorsolutiontoafeasible solution for the original problem provides a polynomial time approximation algorithm for this problem. In this lecture, we will continue exploring SDP by studying the problem of coloring a 3-colorable graph, which is a special instance of the general graph coloring problem. Inthegraphcoloringproblem,wearegivenanundirectedgraph G = (V,E).Ourgoalistofinda legal coloring(also called proper coloring, or simply coloring) of the graph with the minimum number ofcolors. Alegalcoloringofagraphisacoloringwhereeachvertexisassignedacolorsuchthatno twoadjacentverticeshavethesamecolor.thisproblemisknowntobe NP-hardandevenfindingan approximationwithafactorbetterthan n 1 ǫ,forany ǫ > 0is NP-hard.Instead,wefocusourattention onspecialinstancesofthegraphcoloringprobleminwhichwearegivenak-colorablegraph,andour goalistofindalegalcoloringofthisgraphwiththefewestnumberofcolors.inwhatfollows,wewill assume Gisa3-colorablegraph.Notethatitisstill NP-hardtodecideifagraphcanbecoloredwith threecolors.itisalso NP-hardtofindacoloringofa3-colorablegraphwithatmostfivecolors.Inour discussion, we will focus on finding a coloring better than n. 1.1 Coloringwith O( n)colors Wedenotethemaximumdegreeofagraphby. Lemma1Wecanefficientlycoloranygraphwith +1colors,bygreedilycoloringthegraph. Givena3-colorablegraph G,wecanfindalegalcoloringusingatmost O( n)colors. 1.If Ghasavertex vwithdegree d(v) n,weusethreenewcolorstocolorthisvertexandits neighbors δ(v). Since G is 3-colorable, the neighbors of any vertex v form a bipartite graph(since noneoftheseverticescanhavethesamecolorasvertex v).thus,wecancolortheset δ(v)using twocolors.weuseathirdcolortocolorvertex v.werepeatthisstepuntilnoverticeswithdegree higherthan nremain. 2.Once Ghas < n,wecanapplylemma??tocolortherestofthegraphusing ncolors. n NotethatStep1willbeexecutedatmost n -times,sinceateachiterationwearecoloringatleast nvertices.ateachoftheseiterationsweareusingthreenewcolors.soattheendofthisphase,we willhaveusedatmost 3 ncolors.inthesecondstep,weuseonly ncolors.sointotalwewillcolor thegraphwithatmost 4 ncolors. Inthefollowingsections,wenotethatwewilloftenuse Õ()notationtohide logfactors. 1.2 Õ(n )Colors Tofindacoloringofak-colorablegraph,weneedtopartitionthegraphintosetssuchthatalledgesare cut.wewanttoformulatethisproblemasansdp,sowerepresenteachvertexbyavector v i R n.as we have seen in the previous lecture, if we are partitioning the vertices randomly, two vertices are more 1

2 likely to be separated if their corresponding vectors are away from each other, which corresponds to a smallinnerproduct,soourgoalistominimizethequantity v i v j foranyedge (i,j) E.Weformulate the following vector program for the graph coloring problem for a k-colorable graph: min λ s.t v i v j λ (i,j) E v i v i = 1 i V v i R n When Gis3-colorable,thereexistsafeasiblesolutionoftheaboveSDPfor λ = 1 2.Toseethis,note that any legal 3-coloring of G will partition the vertex set into three independent sets(an independent setisasetwherenotwoverticesareconnectedbyanedge)whereeachsetcorrespondtoacolor.thus anoptimalexactsolutionfortheabovesdpwouldconsistofmappingeachindependentsettooneof the color vectors,whichareeachatanangleof 2π 3 fromeachother. Thus,anylegalcoloringofa 3-colorable graph is a feasible solution of the following SDP: v i v j 1 2 (i,j) E v i v i = 1 i V v i R n Togiveacoloringof G,wewillsolvethisSDPandthenapplyarandomizedroundingstrategyto colorthevertices.thegoalinroundingistoavoidusingtoomanycolorsandatthesametimeavoid introducing monochromatic edges. We will use the following notion of a semicoloring. Definition 2(k-Semicoloring) A k-semicoloring of G is a k-coloring of the vertices such that at most n 4 edgeshaveendpointsofthesamecolor. Note that any k-semicoloring will directly result in a k-coloring of the rest of the graph after removing allthemonochromaticedges,soweobtainak-coloringofatleast n 2 vertices. Lemma3Ifwecansemicoloragraph Gwith kcolors,thenwecancolor Gwith klogncolors. Proof Eachroundweuse kcolorsandremoveatleasthalfthevertices.thereare lognrounds,so weuseatmost klogncolors. Now we are ready to present our approximation algorithm: 1.SolvetheSDPon G. 2.Set t = 2 + log 3 andpick trandomvectors {r 1,r 2,,r t }whereeachentry r i isdrawnfrom the standard normal distribution N(0, 1). 3.These tvectorsdefine 2 t differentregionsbasedonthesignofthedotproductswitheachofthe t vectors. We color each of these regions with a different color. Lemma4Randomhyperplaneroundingwith thyperplanesproducesasemicoloringusing 4 log 3 2 colors with probability at least 1/2. Proof Thenumberofcolorsweareusingis 2 t = 4 2 log 3 = 4 log 3 2.Usingthetrivialupperbound of non,weseethatthisisatmost 4n colors. 2

3 Wewanttoshowthatourcoloringprocedureproducesasemicoloringon G,soweneedtoshowthat thenumberofmonochromaticedgesisatmost n 4withprobabilityatleast 1/2. Pr(edge (i,j)ismonochromatic) = (1 arccos(v i v j ) ) t π = (1 arccos( 0.5) ) t π = (1 1 2π π 3 )t = ( 1 3 )t = 1 9. Let m = E.Then m n 2.Let Xbearandomvariabledenotingthenumberofmonochromaticedges in our coloring. Then: E[X] = (i,j) E Pr(edge (i,j)ismonochromatic) n = n 18. By Markov s inequality, we have: Pr[X n 4 ] E[X] n/4 ThenLemma4holdswithprobabilityatleast ThereforethisalgorithmresultsinaO(n logn) = Õ(n0.631 )-approximationwhichisevenworst thanthefirstsimpleapproximationalgorithm,butitprovidesuswithawaytocombinetheabove algorithmwiththecombinatorialalgorithmtogetaõ(n0.387 )-approximationalgorithm. 1.3 Õ(n )Colors Let θ be some parameter that we define later. The following algorithm merges the two approximation algorithms we presented so far. 1.If Ghasavertex vwithdegree d(v) θ,weusethreenewcolorstocolorthevertex vandits neighbours δ(v). We repeat this step until no vertices with degree at least θ remain. 2.Once Ghas < θ,weusethesecondalgorithmtocolortherestofthegraphwith 4 θ log 3 2 logn colors. Step1willuseatmost 3n θ colors. Setting n θ = θlog 3 2,wehave n = θ log 3 6. Thenwedefine θas θ = n log 6 3 n Intotal,thisapproximationalgorithmuses Õ(n θ ) = Õ(n0.387 )colors. 1.4 Õ ( n 1/4) ColorsviaLargeIndependentSets Wenowshowhowtocolora3-colorablegraphwith Õ( n 1/4) colorsinpolynomialtime.todothis,we usethefactthatinalegalcoloring,asetofverticesofthesamecolorformanindependentset.weuse this fact to construct the following general algorithm: 1.Findanindependentset I. 2.Colortheverticesin Iwithanewcolor. 3

4 3.Remove I. 4.Repeatallthesteps. IfineachiterationofStep3oftheabovealgorithm,atleastaγ-fractionofverticesareremoved, thenafter kiterations,atmost (1 γ) k nverticesremain.ifweremoveverticesuntilthereisatleast onevertex,thenusing log(1 x) xfor x > 1,wederive k max,themaximumnumberofiterations, as follows: (1 γ) kmax n 1 k max 1 γ logn. Thus,ifwehaveanalgorithmthatcolorsaγ-fractionofthegraphateachiteration,thenthealgorithm willuse O( 1 γ lnn)colors. As mentioned previously, we would like to map all the vertices onto vectors in a 2-dimensional plane, sothatanytwovectorscorrespondingtothetwoendpointsofanedgeare120degreesapart.thereare three such vectors, each corresponding to a single color, which are uniquely determined up to a rotation. Since such a solution corresponds to an optimal integral solution and can therefore not solve such an SDP, we relaxed the dimension of vectors onto which the vertices are mapped. However, after the relaxation, there can be more than three vectors satisfying the constraints, and it is no longer straightforward to mapthesevectorstocolors.westillknowthattwovectors v i and v j representinganedgearefaraway fromeachother,i.e. v i v j = 1/2forevery (i,j) E. Tobenefitfromsuchastructuralguarantee, we randomly sample an n-dimensional vector r and consider vectors that are close to it. Intuitively, vectorsthatarecloseto rshouldalsobeclosetoeachotherandthereforeformanindependentset. More formally, let r = (r 1,...,r n )suchthat r i N(0,1), i {1,...,n}, and define S(ǫ) = {i V : v i r ǫ}, where ǫwillbedeterminedlater.notethat S(ǫ)mightnotbeanindependentset.Wealsodefine S (ǫ) = {i S(ǫ) : ihasnoneighborsin S(ǫ)}, whichisanindependentset.(observethata smarter waytodefine S (ǫ)wouldbetofindamaximal matc hing in S(ǫ) and delete the vertices defining the matching. However, it would make our analysis morecomplicated.)next,ourgoalistoestimatethesizeof S (ǫ). First we recall some basic properties about the normal distribution N(0, 1) such as its probability density and cumulative distribution functions: p(x) = 1 2π e x2 2, Φ(x) = x p(s)ds Φ(x) = 1 Φ(x) = p(s)ds. Observethat v i r,forsomevertex i V,doesnotdependonthevector v i sincethedistribution of rissphericallysymmetric.therefore,wecanrotate v i sothat v i isanyvectorandwithoutlossof generality,wecanassumethat v i = (1,0,...,0),andthus v i r = r 1. Then, v i risdistributedas N(0,1).Thisimpliesthat: Pr[i S(ǫ)] = Φ(ǫ) E[ S(ǫ) ] = n Φ(ǫ). (1) Now,weprovethefollowinglemma,whichgivesanupperboundontheprobabilitythatavertex i fails,i.e. isnotincludedin S (ǫ),theindependentsetpickedbyouralgorithm. Inotherwords,we wishtocomputetheprobabilitythatvertex idoesnotbelongto S (ǫ)giventhatitbelongsto S(ǫ). x 4

5 Lemma 5 Pr[i / S (ǫ) i S(ǫ)] Φ 3ǫ. (2) Proof We have Pr[i / S (ǫ) i S(ǫ)] = Pr[ (i,j) E : v j r ǫ v i r ǫ]. (3) From v i v j = 1/2and v i v i = v j v j = 1,weconcludethat v j canbewritteninthefollowingway v j = v i + 2 u,suchthat v i u = 0and u u = 1. The last equation implies u = 2 (v j + 1 ) 3 2 v i. If v i r ǫand v j r ǫ,then u r = 2 (v j r+ 1 ) 3 2 v i r 3 2 (ǫ+ 12 ) ǫ = 3ǫ. Thus,weseethatif i / S (ǫ)and i S(ǫ)(i.e.ifboth v i ṙand v j rareatleast ǫ),thenitmustbethe casethattheprojectionof ronto uisatleast 3ǫ.However,notethatif r u 3ǫ,thenthisdoes necessarilyimplythatboth r v i and r v j areatleast ǫ. Since u v i = 0, r uand r v i areindependentlydistributed.thus,wehave: Pr[v j r ǫ v i r ǫ] Pr[u r 3ǫ v i r ǫ] = Pr[u r 3ǫ] = Φ( 3ǫ). Toconcludetheproof,weuseaunionboundovertheneighborsof i.sincethereareatmost ofthem, we have: Pr[ (i,j) E : v j r ǫ v i r ǫ] Pr[v j r ǫ v i r ǫ] Φ 3ǫ, which implies the lemma. j:(i,j) E So,howshouldwechoose ǫ?intuitively,asmallervalueof ǫshouldresultinthesmallernumberof edgesin S(ǫ),sincetwoadjacentverticesshouldbefarfromeachother.Ontheotherhand,wewould like S(ǫ)tobelarge.Wewillset ǫsothat Φ 3ǫ 1/2. (4) Thiswillupperboundtheprobabilitythatavertexin S(ǫ)doesnotbelongto S (ǫ).notethat,along with(1)andlemma5,itwouldfurtherimply E[ S (ǫ) ] E[ S(ǫ) ] 2 = n Φ(ǫ) 2. (5) Beforewestatethemainresultofthissection,westatethefollowinglemmawithoutaproof. Lemma6For x > 0, x 1+x 2p(x) Φ(x) 1 x p(x). 5

6 Nowwecanstatethefollowingtheoremwhichlowerboundsthesizeof S (ǫ). Theorem 7 ( ) E[ S (ǫ) ] Ω n 1 3 (ln ) 1 2. Proof First,wederive ǫsothat(4)holds.bylemma6wehave Φ 3ǫ 1 1 e 3ǫ2 2. (6) 3ǫ 2π 2 From(4)and(6)weconcludeitissufficienttoset ǫ = 3 ln. Toestimatethesizeof S (ǫ),wemustlowerboundthevalueof Φ(ǫ).Weobtainthefollowingbound x by applying Lemma 6 and using inequality 1+x 1 2 2x,for x 1,asfollows ǫ Φ(ǫ) p(ǫ) 1+ǫ 2 1 ǫ = e ǫ2 2 2π 1+ǫ e ln 3 2π 2ǫ 1 3 (ln ) 1 2. Applying the inequality in Equation(5) concludes the proof. Theorem7saysthatineachiterationofthealgorithm,weareabletofindanindependentsetofsize atleast Õ( n 1/3). Thisimpliesthatwecancoloragiven3-colorablegraphwith O ( 1/3 logn ) = Õ ( 1/3) colors.wecansummarizethegivenapproach,whichwecallindependent-set-coloring, as follows: 1. Solve SDP. 2.Pickarandomvector r. 3.Pickaset S(ǫ)whichis ǫ-closeto r. 4.Let S (ǫ) S(ǫ)beverticeswithdegreezeroin S(ǫ). 5.Removetheverticesintheindependentset S (ǫ)fromthegraph. 6.RepeatfromStep2whilethegraphisnon-empty. NotethatthereisnoneedtoresolveSDPinStep1ineachiteration,sinceasolutionfortheinitial graph is valid for any subgraph. Finally, we combine the algorithm from Section 1.1 with the algorithm Independent-Set-Coloring. 1.If Ghasavertex vwithdegree d(v) n 3/4,weusethreenewcolorstocolorthevertex vandits neighbours δ(v).werepeatthisstepuntilnoverticeswithdegreeatleast n 3/4 remain. 2.Once Ghas < n 3/4,weusethealgorithmIndependent-Set-Coloringtocolortherestof thegraphwithatmost Õ((n3/4 ) 1/3 ) = Õ(n1/4 )colors. ObservethatStep1. willexecuteatmost n/n 3/4 = n 1/4 times,usingatmost n 1/4 colors.thus,the totalnumberofcolorsusedisatmost Õ(n1/4 ). 6

7 References [ACC06] Sanjeev Arora, Eden Chlamtac, and Moses Charikar. New approximation guarantee for chromatic number. In Jon M. Kleinberg, editor, STOC, pages ACM, [KMS98] David R. Karger, Rajeev Motwani, and Madhu Sudan. Approximate graph coloring by semidefinite programming. J. ACM, 45(2): , [WS11] David P. Williamson and David B. Shmoys. The Design of Approximation Algorithms. Cambridge University Press, This lecture was mainly based on the following sources:[kms98, ACC06] and Chapters 6 and 13 from [WS11]. 7