CONSTANT FACTOR LASSERRE INTEGRALITY GAPS FOR GRAPH PARTITIONING PROBLEMS

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1 CONSTANT FACTOR LASSERRE INTEGRALITY GAPS FOR GRAPH PARTITIONING PROBLEMS VENKATESAN GURUSWAMI, ALI KEMAL SINOP, AND YUAN ZHOU Abstract. Partitioig the vertices of a graph ito two roughly equal parts while miimizig the umber of edges crossig the cut is a fudametal problem called Balaced Separator that arises i may settigs. For this problem, ad variats such as the Uiform Sparsest Cut problem where the goal is to miimize the fractio of pairs o opposite sides of the cut that are coected by a edge, there are large gaps betwee the kow approximatio algorithms ad o-approximability results. While o costat factor approximatio algorithms are kow, eve APXhardess is ot kow either. I this work we prove that for balaced separator ad uiform sparsest cut, semidefiite programs from the Lasserre hierarchy which are the most powerful relaxatios studied i the literature have a itegrality gap bouded away from 1, eve for Ω levels of the hierarchy. This complemets recet algorithmic results i Guruswami ad Siop 011 which used the Lasserre hierarchy to give a approximatio scheme for these problems with rutime depedig o the spectrum of the graph. Alog the way, we make a observatio that simplifies the task of liftig polyomial costraits such as the global balace costrait i balaced separator to higher levels of the Lasserre hierarchy. Key words. balaced separator, uiform sparsest cut, Lasserre semidefiite programmig hierarchy, itegrality gaps AMS subject classificatios. 90C, 90C7 1. Itroductio. Partitioig a graph ito two balaced parts with few edges goig across them is a fudametal optimizatio problem. Graph partitios or separators are widely used i may applicatios such as clusterig, divide ad coquer algorithms, VLSI layout, etc. Two prototypical objectives of graph partitioig are BALANCED- SEPARATOR ad UNIFORMSPARSESTCUT, defied as follows. DEFINITION 1.1. Give a udirected graph G V, E ad 0 < τ < 0.5, the goal of the τ vs 1 τ BALANCEDSEPARATOR problem is to fid a set A V such that τ V A 1 τ V, while edgesa, V \ A is miimized. Here edgesa, B is the umber of edges i E that cross the cut A, B. The goal of the UNIFORMSPARSESTCUT problem is to fid a set A V such that the sparsity edgesa, V \ A A V \ A is miimized. Despite extesive research, there are still huge gaps betwee the kow approximatio algorithms ad iapproximability results for these problems. The best algorithms, based o semidefiite relaxatios SDPs with triagle iequalities, give a O log approximatio to both problems [ARV09]. O the iapproximability side, a Polyomial Time Approximatio Scheme PTAS is ruled out for both problems assumig 3-SAT does ot have radomized subexpoetial-time algorithms [AMS11]. I this paper, our focus is o the UNIFORMSPARSESTCUT problem; the geeral SPARSESTCUT problem has bee show to ot admit a costat-factor approximatio algorithm uder the Uique Games Cojecture [CKK + 06, KV05, Kho0]. It is kow that the SDP used i [ARV09] caot give a costat factor approximatio for UNIFORMSPARSES- TCUT [DKSV06]. Itegrality gaps are also kow for stroger SDPs: super-costat factor itegrality gaps for both BALANCEDSEPARATOR ad UNIFORMSPARSESTCUT are kow for the so-called Sherali-Adams + hierarchy for a super-costat umber of rouds [RS09]. There has bee much recet iterest i the power ad limitatios of various hierarchies of relaxatios i the quest for better approximatio algorithms for combiatorial optimizatio problems. These hierarchies are parameterized by a iteger r called rouds/levels which capture higher order correlatios betwee roughly r-tuples of variables the basic SDP captures oly pairwise correlatios, ad certai extesios like triage iqualities pose costraits o triples. Larger the r, tighter the relaxatio. There are several hierarchies of relaxatios that have bee studied i the literature, such as Sherali-Adams [SA90], Lovász-Schrijver hierarchy [LS91], mixed hierarchies combiig Sherali-Adams liear programs with SDPs, ad Computer Sciece Departmet, Caregie Mello Uiversity. Supported i part by a Packard Fellowship ad NSF CCF guruswami@cmu.edu Computer Sciece Departmet, Priceto Uiversity. Supported by NSF CCF ad MSR-CMU Ceter for Computatioal Thikig. asiop@cs.cmu.edu Computer Sciece Departmet, Caregie Mello Uiversity. Supported i part by NSF CCF ad US-Israel BSF grat yuazhou@cs.cmu.edu 1

2 GURUSWAMI ET AL. the Lasserre hierarchy [Las0]. Of these hierarchies, the most powerful oe is the Lasserre hierarchy see [Lau03] for a compariso. The potetial of SDPs from the Lasserre hierarchy i deliverig substatial improvemets to approximatio guaratees for several otorious optimizatio problems is ot well uderstood, ad is a importat ad active directio of curret research. Ideed, it is cosistet with curret kowledge that eve 4 rouds of the Lasserre hierarchy could improve the factor Goemas-Williamso algorithm for MAXCUT, ad therefore refute the Uique Games cojecture. Very recet work [BBH + 1, OZ13, DMN13, KOTZ14] has show that O1 levels of the Lasserre hierarchy ca succeed where ω1 levels of weaker SDP hierarchies fail; i particular, this holds for the hardest kow istaces of UNIQUEGAMES [BBH + 1]. For the graph partitioig problems of iterest i this paper BALANCEDSEPARATOR ad UNIFORMSPARSEST- CUT, itegrality gaps were ot kow eve for a small costat umber of rouds. It was ot ucoditioally ruled out, for example, that 1/ɛ O1 rouds of the hierarchy could give a 1 + ɛ-approximatio algorithm, thereby givig a PTAS. O the algorithmic side, [GS11] recetly showed that for these problems, SDPs usig Or/ε rouds of the Lasserre hierarchy have a itegrality gap at most 1 + ɛ/ mi{1, λ r }. Here λ r is the r-th smallest eigevalue of the ormalized Laplacia of the graph. This result implies a approximatio scheme for these problems with rutime parameterized by the graph spectrum. Give this situatio, it is atural to study the limitatios of the Lasserre hierarchy for these two fudametal graph partitioig problems. Several of the kow results o strog itegrality gap results for may rouds of the Lasserre hierarchy, startig with Schoeebeck s remarkable costructio [Sch08], apply i situatios where a correspodig NP-hardess result is already kow. Thus they are ot prescriptive of hardess. I fact, we are aware of oly the followig examples where a polyomial-roud Lasserre itegrality gap stroger tha the correspodig NP-hardess result is kow: Max k-csp, k-colorig [Tul09] ad Desest k-subgraph [BCG + 1]. The mai results of this paper, described ext, exted this body of results, by showig that Lasserre SDPs caot give a PTAS for BALANCEDSEPA- RATOR ad UNIFORMSPARSESTCUT Our results. I this paper, we study itegrality gaps for the Lasserre SDP relaxatios for BALANCEDSEPA- RATOR ad UNIFORMSPARSESTCUT. As metioed before, APX-hardess is ot kow for these two problems, eve assumig the Uique Games cojecture. Supercostat hardess results are kow based o a strog itractability assumptio cocerig the Small Set Expasio problem [RST1]. I cotrast, we show that liear-roud Lasserre SDP has a itegrality gap bouded away from 1, ad thus fails to give a factor α-approximatio for some absolute costat α > 1. Specifically, we prove the followig two theorems. THEOREM 1. iformal. For 0.45 < τ < 0.5, there are liear-roud Lasserre gap istaces for the τ vs 1 τ BALANCEDSEPARATOR problem, such that the itegral optimal solutio is at least 1 + ɛτ times the SDP solutio, where ɛτ > 0 is a costat depedet o τ. THEOREM 1.3 iformal. There are liear-roud Lasserre gap istaces for the UNIFORMSPARSESTCUT problem, such that the itegral optimal solutio is at least 1 + ɛ times the SDP solutio, for some costat ɛ > Our techiques. All of our gap results are based o Schoeebeck s igeious Lasserre itegrality gap for 3-XOR [Sch08]. For BALANCEDSEPARATOR ad UNIFORMSPARSESTCUT, we use the ideas i [AMS11] to build gadget reductios ad combie them with Schoeebeck s gap istace. [AMS11] desiged gadget reductios from Khot s quasi-radom PCP [Kho06] i order to show APX-Hardess of the two problems. If we view the Lasserre hierarchy as a computatioal model as suggested i [Tul09], we ca view Schoeebeck s costructio as playig the role of a quasi-radom PCP i the Lasserre model. Our gadget reductios, therefore, bear some resemblace to the oes i [AMS11], though the aalysis is differet due to differet radom structures of the PCPs. We feel our reductios are slightly simpler tha the oes i [AMS11], although we eed some additioal tricks to make the reductios have oly liear blowup. This latter feature is eeded i order to get Lasserre SDP gaps for a liear umber of rouds. We are also able to make the gap istace graphs have oly costat degree, while the reductios i [AMS11] give graphs with ubouded degree. Also, ulike 3-XOR, for balaced separator there is a global liear costrait stipulatig the balace of the cut, ad our Lasserre solutio must also satisfy a lifted form of this costrait [Las0]. We make a geeral observatio that such costraits ca be easily lifted to the Lasserre hierarchy whe the vectors i our costructio satisfy a related liear costrait. This observatio applies to costraits give by ay polyomials, ad to our kowledge, was ot made before. It simplifies the task of costructig legal Lasserre vectors i such cases.

3 LASSERRE GAPS FOR GRAPH PARTITIONING PROBLEMS 3. Lasserre SDPs. I this sectio, we begi with a geeral descriptio of semidefiite programmig relaxatios from the Lasserre hierarchy, followed by a useful observatio about costructig feasible solutios for such a SDP. We the discuss the specific SDP relaxatios for our problems of iterest. Fially, we recall Schoeebeck s Lasserre itegrality gaps [Sch08] i a form coveiet for our later use..1. Lasserre Hierarchy Relaxatio. Cosider a biary programmig problem with polyomial objective fuctio P ad a sigle costrait expressed as a polyomial Q:.1 Miimize/Maximize subject to T d P T [] j T x j T d QT [] j T x j 0, x i {0, 1} for all i []. It is easy to see that this captures all problems we cosider i this paper: BALANCEDSEPARATOR Sectio..1 ad UNIFORMSPARSESTCUT Sectio... We ow defie the Lasserre hierarchy semidefiite program relaxatio for the above iteger program. It is easily see that the below is a relaxatio by takig U A x A I ad Y A Qx U A where x {0, 1} is a feasible solutio to.1, x A i A x i, ad I is ay fixed uit vector. PROPOSITION.1. For ay positive iteger r d, r rouds of Lasserre Hierarchy relaxatio [Las0] of.1 is give by the followig semidefiite programmig formulatio:. Miimize/Maximize T P T U T subject to U 1, U A, U B U A B for all A, B with A B r, S d QS U S, U [] A B Y A, Y B, Y A, Y B Y A B for all A, B with A B r d. U A, Y B R Υ. Proof. Give y R r [], let My Sym [] r be the momet matrix whose rows ad colums correspod to subsets of size r. The etry at row S ad colum T of My is give by y S T. For ay multiliear polyomial P of degree-d, let P y R r d [] be the vector whose etry correspodig to subset S is give by T P T y S T. The Lasserre Hierarchy relaxatio [Las0] of.1 is give by:.3 Miimize/Maximize T P T y T subject to y 1, My 0, MQ y 0. Proof of..3. Give feasible solutio for., let y S U S ad z S Y S. We have y 1 ad T P T y T T P T U S. Observe that y S T U S T U S, U T therefore My 0. With a similar reasoig, we also have Mz 0. Fially, for ay S: Q y S T QT y S T T QT U T, U S Y S z S, which implies z Q y. Hece y is a feasible solutio for.3. Proof of.3.. Let y be a feasible solutio for.3. Defie z Q y. Sice My 0 resp. Mz 0, there exists a matrix U [U S ] S resp. Y [Y S ] S such that My U T U resp. Mz Y T Y. It is easy to see that U S, U T y S T ad Y S, Y T z S T. Therefore: T P T U T T P T y T. U y 1. U S, U T y S T U S T similar for Y. For ay S, T QT U T, U S T QT y S T Q y S z S Y S.

4 4 GURUSWAMI ET AL. Therefore U, Y is a feasible solutio for. with same objective value, completig our proof. Note that a straightforward verificatio of last two costraits requires the costructio of vectors Y A i additio to U A. Below we give a easier way to verify these last two costraits without havig to costruct Y A s. This greatly simplifies our task of costructig Lasserre vectors for the liftig of global balace costraits. THEOREM.. Give vectors U T for all T [] r satisfyig the first two costraits of., if there exists a o-egative real δ > 0 such that.4 QSU S δ U S [] d the these vectors form part of a feasible solutio to.. Proof. Cosider the followig vectors. For each A with A r, let Y A δ U A. By costructio, these vectors satisfy the Y A, Y B Y A B costraits sice U A, U B U A B. Now we verify the other costrait: QS U S, U A B QSU S, U A B δu, U A B δ U A, U B Y A, Y B. S [] d S [] d.. Lasserre SDP for graph partitioig problems. I light of Theorem., to show good solutios for the Lasserre SDP for our problems of iterest, we oly eed to show good solutios for the followig SDPs...1. BALANCEDSEPARATOR. The stadard iteger programmig formulatio of BALANCEDSEPARATOR is show i the left part of Figure.1. The r roud SDP relaxatio Ψ 1 show i the right part of Figure.1 has a vector U S for each subset S V with S r. I a itegral solutio, the iteded value of U {u} is x u U for some fixed uit vector U, ad that of U S is u S x u U. FIG..1. IP ad SDP relaxatios for BALANCEDSEPARATOR. We ca solve Ψ 1 by first eumeratig over all τ {1/, /,..., 1} [τ, 1 τ] ad the choosig τ which miimizes the objective fuctio. Note that the resultig relaxatio is stroger tha usual Lasserre Hierarchy relaxatio. IP SDP Relaxatio Ψ 1 miimize u,v E s.t. τ V u V x u {0, 1} x u x v x u 1 τ V u V miimize u,v E U {u} U {v} s.t. U S1, U S 0 for all S 1, S U S1, U S U S3, U S4 for all S 1 S S 3 S 4 U 1 U {v} τ V U for some τ τ 1 τ v... UNIFORMSPARSESTCUT. The UNIFORMSPARSESTCUT problem asks to miimize the value of the quadratic iteger program show i the left part of Figure. over all τ {1/, /,..., / }. The correspodig SDP relaxatio Ψ is to miimize the value of the SDP show i the right part of Figure. over all τ {1/, /,..., / }. Remark. Prior to our paper, kow lower bouds [DKSV06, KM13] o the itegrality gap of UNIFORMSPARS- ESTCUT problem used a weaker relaxatio, where the last two equality costraits i Ψ of Figure. are replaced by the followig istead: U {u} U {v} 1 u<v with the objective fuctio beig simply u,v E U {u} U {v}.

5 LASSERRE GAPS FOR GRAPH PARTITIONING PROBLEMS 5 FIG... IP ad SDP relaxatios for UNIFORMSPARSESTCUT. We ca solve Ψ by first eumeratig over all τ {1/, /,..., 1/} ad the choosig τ which miimizes the objective fuctio. Note that the resultig relaxatio is stroger tha usual Lasserre Hierarchy relaxatio. IP SDP Relaxatio Ψ miimize s.t. 1 V τ1 τ x u τ V u u,v E x u {0, 1} u V x u x v miimize u,v E 1 U V {u} U {v} τ1 τ s.t. U S1, U S 0 for all S 1, S U S1, U S U S3, U S4 for all S 1 S S 3 S 4 U 1 U {v} τ V U v.3. Lasserre Gaps for 3-XOR from [Sch08]. We start by defiig the 3-XOR problem. DEFINITION.3. A istace Φ of 3-XOR is a set of costraits C 1, C,, C m where each costrait C i is over 3 distict variables x i1, x i, ad x i3, ad is of the form x i1 x i x i3 b i for some b i {0, 1}. For each costrait C i ad each partial assigmet α that is valid o the variables x i1, x i, ad x i3, we use the otatio C i α 1 whe C i is satisfied by α ad C i α 0 otherwise. A radom istace of 3-XOR is sampled by choosig each costrait C i uiform idepedetly from the set of possible costraits. We will make use of the followig fudametal result of Schoeebeck. THEOREM.4 [Sch08]. For every large eough costat β > 1, there exists η > 0, such that with probability 1 o1, a radom 3-XOR istace Φ over m β costraits ad variables caot be refuted by the SDP relaxatio obtaied by η rouds of the Lasserre hierarchy, i.e. there are vectors W S,α for all S η ad all α : S {0, 1}, such that i the value of the solutio is perfect: m i1 α:{x i1,x i,x i3 } {0,1},C iα1 W {xi1,x i,x i3 },α m; ii W S1,α 1, W S,α 0 for all S 1, S, α 1, α ; iii W S1,α 1, W S,α 0 if α 1 S 1 S α S 1 S ; iv W S1,α 1, W S,α W S3,α 3, W S4,α 4 for all S 1 S S 3 S 4 ad α 1 α α 3 α 4. Here, whe α 1 S 1 S α S 1 S, α 1 α is aturally defied as the mappig from S 1 S to {0, 1} such that its restrictio to S 1 equals α 1 ad its restrictio to S equals α. We make similar defiitio for α 3 α 4. v α:s {0,1} W S,α 1 for all S. Note that ideed we have for every S, α:s {0,1} W S,α W,. This is because W, 1 ad W S,α, W, W S,α, W, W S,α 1. α:s {0,1} α:s {0,1} α:s {0,1} OBSERVATION.5. I the costructio of Theorem.4, the vectors W satisfy the followig property. For ay costrait C i over set of variables S i, the vectors correspodig to all satisfyig partial assigmets of S i sums up to W : W Si,α W. α:s i {0,1} C iα1 3. Gaps for BALANCEDSEPARATOR. I this sectio, we prove Theorem 1.. We state the theorem i detail as follows. THEOREM 3.1. Let M be a large eough itegral costat. For all 0.45 < τ < 0.5, ad for ifiitely may positive iteger N s, there is a N-vertex istace H Φ for the τ vs. 1 τ BALANCEDSEPARATOR problem, such

6 6 GURUSWAMI ET AL. that the optimal solutio is at least 43τ τ 3 /5 O1/M times the best solutio of the ΩN-roud Lasserre SDP relaxatio. Moreover, the solutio for Lasserre SDP relaxatio is a fractioal 0.5 O1/M vs O1/M balaced separator. The rest of this sectio is dedicated to the proof of Theorem 3.1. I Sectio 3.1, we will describe how to get a BALANCEDSEPARATOR istace from a 3-XOR istace. The, we will show that whe the 3-XOR istace is radom, the correspodig BALANCEDSEPARATOR istace is a desired gap istace. This is doe by showig there is a SDP solutio with good objective value completeess part, Lemma 3. i Sectio 3. while the istace i fact has ot great itegral solutio soudess part, Lemma 3.4 i Sectio 3.3. The completeess part relies Theorem.4 we use the 3-XOR vectors which exist for radom istaces by the theorem to costruct BALANCEDSEPARATOR vectors. I the soudess part, we first prove two pseudoradom structural properties exhibited i the radom 3-XOR istaces Lemma 3.3, ad the prove that ay 3-XOR with these two properties leads to a BALANCEDSEPARATOR istace with bad itegral optimum by our costructio. Fially, i Sectio 3.4, we slightly twist our gap istace i order to make its vertex degree bouded Reductio. Give a 3-XOR istace Φ with m β costraits ad variables, we build a graph H Φ V Φ, E Φ for BALANCEDSEPARATOR as follows. H Φ cosists of a almost bipartite graph H Φ L Φ, R Φ, E Φ obtaied by replacig each right vertex of a bipartite graph by a clique, a expader Z r, ad edges betwee L Φ ad Z r. The left side L Φ of H Φ cotais 4m 4β vertices, each correspods to a pair of a costrait ad a satisfyig partial assigmet for the costrait, i.e. L Φ {C i, α α : {x i1, x i, x i3 } {0, 1}, C i α 1}. The right side R Φ of H Φ cotais cliques, each is of size Mβ, ad correspods to oe of the literals, i.e. where R Φ j,α:{xj} {0,1}C xj,α, C xj,α {x j, α, t 1 t Mβ}. Call x j, α, 1 the represetative vertex of C xj,α. Besides the clique edges, we coect a left vertex C i, α ad a right represetative vertex x j, α, 1 if x j is accessed by C i ad α is cosistet with α, i.e. E Φ {clique edges} {{C i, α, x j, α, 1} x j {x i1, x i, x i3 }, αx j α x j }. Now we have fiished the defiitio of H Φ. To get H Φ, we add a OM-regular expader Z r of size m β ad edge expasio M. I.e. the degree of each vertex i Z r is OM, ad each subset T Z r T Z r / has at least T M edges coectig to Z r \ T. For more discuss o the defiitios ad applicatios of expader graphs, please refer to, e.g., [HLW06]. We coect each vertex i L Φ to two differet vertices i Z r, so that each vertex i Z r has the same umber of eighbors i L Φ this umber should be 4β /β 8. I other words, if we view each vertex i L Φ as a udirected edge betwee its two eighbors i Z r, the graph should be a regular graph. The whole costructio is show i Figure 3.1. Our costructio is very similar to the oe i [AMS11], but there are some techical differeces. Istead of havig cliques i R Φ, [AMS11] has clusters of vertices with o edges coectig them. Also, i our costructio, the vertices i L Φ are coected to the represetative vertices i R Φ oly, while i [AMS11], all the vertices i the right clusters could be coected to the left side. The most importat differece is that i our way, the cliques are of costat size, while the clusters i [AMS11] has supercostatly may vertices. This meas that our reductio blows up the istace size oly by a costat factor, therefore we are able to get liear roud Lasserre gap. Observe that there are L Φ + R Φ + Z r 4m + Mm + m M + 5m vertices i H Φ. I the followig two subsectios, we will prove the completeess lemma Lemma 3., which states that there is a SDP solutio with a good objective value ad the soudess lemma Lemma 3.4, which states that every itegral solutio has a bad objective value. Combiig the two lemmas, we prove our mai itegrality gap theorem for BALANCEDSEPARATOR as follows. Proof. [of Theorem 3.1 from Lemma 3. ad Lemma 3.4] Let β, M be large eough costats. Let Φ be a radom 3-XOR istace over m β costraits ad variables. For all 0.45 < τ < 0.5, we will show that the optimal

7 LASSERRE GAPS FOR GRAPH PARTITIONING PROBLEMS 7 L Φ R Φ represetative vertex the vertices correspodig to C i the clique C xj,α C i, α the expader Z r FIG The reductio for BALANCEDSEPARATOR. Note that the icidet edges are draw for oly oe of the vertices i L Φ, while others ca be draw similarly. solutio for H Φ is at least 43τ τ 3 /5 O1/ β + 1/M times the best solutio of the ΩN-roud Lasserre SDP relaxatio; ad the solutio for Lasserre SDP relaxatio is a fractioal 0.5 O1/M vs O1/M balaced separator. Oe may choose β M to get the statemet i the theorem. By Theorem.4 we kow that, with probability 1 o1, Φ admits a perfect solutio for Ω-roud Lasserre SDP relaxatio. Therefore, by Lemma 3., with probability 1 o1, Ω-roud SDP relaxatio Ψ 1 with parameter τ 0.5 O1/M for the BALANCEDSEPARATOR istace H Φ has a solutio of value 5m. O the other had, by Lemma 3.4, with probability 1 o1, for τ > 1/3, every τ vs. 1 τ balaced separator has at least 4m3τ τ 3 O1/ β O1/M edges i the cut. Therefore, with probability 1 o1, whe τ > 1/3, the ratio betwee the optimal itegral solutio to H Φ ad the optimal Ω-roud Ψ 1 solutio is at least 43τ τ 3 /5 O1/ β +1/M. This ratio is greater tha whe τ > 0.45 ad β ad M are large eough. By our observatio i Sectio..1, this gap also holds for the Lasserre SDP relaxatio. Let be the maximum umber of occurreces of ay variable i Φ. By our costructio, the graph has degree ΘM +. Whe β O1, we have Θlog / log log with probability 1 o1 see, e.g. [Go81]. This meas that our graph does ot have the desired costat-degree property. However, sice there are few edges icidet to vertices with supercostat degree, we ca simply remove all these edges to get a costat-degree graph, while the completeess ad soudess are still preserved. We will discuss this i more details i Sectio Completeess : good SDP solutio. LEMMA 3. Completeess. If the 3-XOR istace Φ admits perfect solutio for r-roud Lasserre SDP relaxatio, the the r/3-roud SDP relaxatio Ψ 1 i Figure.1 with parameter τ 0.5 O1/M for the BALANCED- SEPARATOR istace H Φ has a solutio of value 5m. Proof. We defie a set of vectors i.e. a solutio to Ψ 1 usig the vectors give i Theorem.4, as follows.

8 8 GURUSWAMI ET AL. For each set S L Φ R Φ Z r with S r/3, we defie the vector U S as follows. If S Z r, let U S 0. If S Z r, suppose that S L Φ cotais S R Φ cotais C i1, α 1, C i, α,, C ir1, α r1, x j1, α 1, t 1, x j, α, t,, x jr, α r, t r, we have r 1 + r S. Let S be the set of variables accessed by C i1, C i together with x j1,, x jr. Note that S 3r 1 + r 3 S r. If there is o cotradictio amog the partial assigmets α i s ad α i s i.e. there are ot two of them assigig the same variable to differet values, we ca defie α α 1 α α r1 α 1 α α r. ad let U S W S,α, otherwise we let U S 0. We first check that the first 3 costraits i relaxatio Ψ 1 are satisfied. For two sets S 1, S, either at least oe of the vectors U S1, U S is 0 therefore their ier-product is 0, or U S1 W S 1,α 1, U S W S,α for some S 1, S, α 1, α ad U S1, U S W S 1,α 1, W S,α 0. For ay S 1, S, S 3, S 4 such that S 1 S S 3 S 4, either the set of partial assigmets i S 1 S S 3 S 4 are cosistet with each other, i which case we have U S1 S U S3 S 4 W S,α where S is the uio of all the variables icluded i S 1 S ad α is the cocateatio of the partial assigmets i S 1 S ; or we have U S1 S U S3 S 4 0. U W, 1. Now we check that the balace coditio the last costrait i relaxatio Ψ 1 is satisfied. We will prove that U {v} M + 1mU. v Sice there are M + 5m vertices i H Φ, this shows that the solutio is feasible for Φ 1 with τ 0.5 O1/M. Usig Observatio.5, we see that C i,α L Φ U {Ci,α} C i U mu. Similarly x j,α,t R Φ U {xj,α,t} βm j1 α:{x j} {0,1} t1 βm U MmU. U {xj,α,t} βm j1 α:{x j} {0,1} U {xj,α,1} Thus U {v} v V U {v} U {Ci,α} + U {xj,α,t} M + 1mU. v L Φ R Φ Z r C i,α L Φ x j,α,t R Φ Now, we calculate the value of the solutio U {u} U {v} u,v E Φ m i1 α:{x i1,x i,x i3 } {0,1},C iα1 z1 + + m 3 U {Ci,α} U {xiz,α {xiz },1} i1 α:{x i1,x i,x i3 } {0,1},C iα1 v Z r:c i,α,v E Φ j1 α:{x j} {0,1} z 1,z [Mβ] U {xj,α,z 1} U {xj,α,z } + U {Ci,α} U {v} v 1,v Z r U {v1} U {v}

9 m i1 α:{x i1,x i,x i3 } {0,1},C iα1 m i1 α:{x i1,x i,x i3 } {0,1},C iα1 m i1 α:{x i1,x i,x i3 } {0,1},C iα1 m LASSERRE GAPS FOR GRAPH PARTITIONING PROBLEMS 9 3 U {Ci,α} U {xiz,α {xiz },1} + U {Ci,α} z1 3 W {xi1,x i,x i3 },α W {xiz },α {xiz } + z1 3 W{xiz },α {xiz z1 3 i1 α:{x i1,x i,x i3 } {0,1},C iα1 z1 m i1 z1 W {xiz },α {xiz } } m 3 W{xiz },{xiz 0} + W{xiz },{xiz 1} m 6m m 5m. W {xi1,x i,x i3 },α W {xi1,x i,x i3 },α + i1 α:{x i1,x i,x i3 } {0,1},C iα1 W {xi1,x i,x i3 },α W {xi1,x i,x i3 },α 3.3. Soudess : boud for itegral solutios. Let L {x j, α α : {x j } {0, 1}} be the set of literals. For each literal x j, α L, let degx j, α be the umber of left vertices that coect to the literal s represetative vertex x j, α, 1. For a set of literals L L, let degl x degx j,α L j, α. Also, give a subset L L, for left vertex C i, α, say C i, α is cotaied i L if all the three literals correspodig to the three eighbors of C i, α i H Φ are cotaied i L, i.e. {x i1, α xi1, x i, α xi, x i3, α xi3 } L. We first prove the followig lemma regardig the structure of H Φ, defied by a radom 3-XOR istace Φ. LEMMA 3.3. Over the choice of radom 3-XOR istace Φ, with probability 1 o1, the followig statemets hold. For each L L, L /3, we have degl 6m L /1 0/ β. For each L L, L /3, the umber of left vertices i L Φ cotaied i L is at most m L 3 / / β. Proof. Fix a literal x j, α, a radom costrait C i accesses x j with probability 3/. Oce C i accesses x j, there are vertices out of the 4 left vertices correspodig to C i adjacet to x j, α. Therefore, i expectatio, there are 6/ edges from the left vertices correspodig to C i to x j, α. By liearity of expectatio, for fixed L L, there are 6 L / edges from the left vertices correspodig to a radom costrait C i to L i expectatio. Now for each C i, let the radom variable X i be the umber of represetative vertices i L that is coected to left vertices correspodig to C i. By defiitio we have degl m i1 X i. Sice each left vertex correspodig to C i has 3 eighbors o the right side, ad there are 4 of such left vertices, we kow that X i [0, 1]. I the previous paragraph we have cocluded that E[X i ] 6 L / for all i 1,,..., m. It is also easy to see that X 1, X,..., X m are idepedet radom variables. Now assumig that L /3, we use Hoeffdig s iequality for the radom variables X 1, X,..., X m, ad get Pr[degL < 6m L /1 0/ [ β] Pr X i < 6m L /1 0/ ] β 0 β 6m L exp m 1 exp 00 i1 L exp 4. Sice there are at most such L s, by a uio boud, with probability at least 1, the first statemet holds. For the secod statemet, fix a L L, let a 0, a 1, a be the umber of variables that have 0, 1, correspodig literals i L, respectively. Note that a 0 + a 1 + a ad a 1 + a L Now, for a radom costrait C i, we are iterested i the expected umber of the four correspodig left vertices C i, α that are cotaied i L. Note that oce C i accesses a variable that correspods to a 0, oe of the four correspodig left vertices are cotaied i L.

10 10 GURUSWAMI ET AL. Now let us coditio o the case that, out of the 3 variables accessed by C i, t variables have two literals i L ad the other 3 t variables have oe literal i L. Observe that i expectatio which is over the radom choice of C i while coditioed o t, there are t 1 left vertices correspodig to C i cotaied i L. I all, the expected umber of the left vertices correspodig to C i that are cotaied i L is 3 t0 a1 a 3 t t 3 t 1 < t0 3 t a 1 / 3 t a / t t 1 for > 3 a 1 + a 3 / L 3 / 3. For each C i, let the radom variable X i be the umber of left vertices correspodig to C i that are cotaied i L. By the discuss above, we kow that E[X i ] < L 3 / 3. Now we are iterested i the probability that the total umber of left vertices cotaied i L i.e. m i1 X i is big. Sice X i s are always bouded by [0, 4], by stadard Cheroff boud, we have [ m Pr X i > m L 3 / / ] β Pr Pr exp i1 [ m i1 [ m i1 X i > m ] L 3 / / β / X i > m m L 3 / 3 exp 14 m L 3 / 3 80/ β 3 exp β L β exp / β 10/ / L 3 / 3 100/ β 10/ / ] for large eough β for β 1 sice L /3 Sice there are at most such L s, by a uio boud, with probability at least 1, the secod statemet holds. Now, we are ready to prove the soudess lemma. LEMMA 3.4 Soudess. For τ > 1/3, with probability 1 o1, the τ vs. 1 τ balaced separator has at least 4m3τ τ 3 O1/ β O1/M edges i the cut. Proof. We are goig to prove that, oce the two coditios i Lemma 3.3 hold, we have the desired upper boud for τ vs. 1 τ balaced separator. Let us assume that there is a balaced separator A, B such that edgesa, B 4m3τ τ 3 1m, we will show that edgesa, B 4m3τ τ 3 O1/ β O1/M. Based o A, B we build aother cut A, B such that A Z r A Z r ad A R Φ A R Φ. For each left vertex i L Φ, it has 5 edges goig to Z r ad R Φ. We assig the vertex to A if it has less tha 3 edges goig to B Z r R Φ, ad assig it to B otherwise. Note that edgesa, B edgesa, B, therefore we oly eed to show that edgesa, B m1τ τ 3 O1/ β O1/M. Sice L Φ cotais oly O1/M fractio of the total vertices, A, B is still τ O1/M vs. 1 τ + O1/M balaced. Sice edgesa, B 1m, for large eough costat M, we have the followig two statemets. 1 Oe of A Z r ad B Z r has at most 100/M Z r 100m/M vertices. Let C bad {x j, α : the clique C xj,α is broke by A, B}, the C bad 0/M. If 1 does ot hold, the we see there are at least 100/M Z r M 100m edges i Z r cut by A, B, by the expasio property. If does ot hold, for each clique C xj,α that is broke by A, B, at least βm 1 edges of the clique are i the cut. I all, there are at least βm 1 0/M > 1β 1m edges i the cut.

11 LASSERRE GAPS FOR GRAPH PARTITIONING PROBLEMS 11 Now, by 1, assume w.l.o.g. that A Z r is the smaller side havig at most 100/M Z r vertices, ad let L be the set of literals x j, α such that its represetative vertex x j, α, 1 is i A. To get a lower boud for L, ote that 3.1 A L + C bad Mβ + Z r + L Φ L Mβ + O1m. Also, sice A, B is a balaced separator, we have A τ O1/M Mm. Hece, by 3.1, we have L τ O1/M. Let L bad L Φ be the set of left vertices such that at least oe of the two eighbors i Z r falls ito A Z r. By the regularity of the graph where Z r is the set of vertices ad L Φ is the set of edges, we kow that L bad 8 100/M Z r Om/M. Now let us get a lower boud o edgesa, B. First, we have edgesa, B edgesa \ L bad, B \ L bad. Let L Φ L Φ \ L bad, we have edgesa \ L bad, B \ L bad edgesa L Φ R Φ Z r, B L Φ R Φ Z r edgesa L Φ, B Z r + edgesa R Φ, B L Φ edgesa L Φ, B Z r + edgesa R Φ, L Φ edgesa R Φ, A L Φ edgesa L Φ, B Z r + edgesa R Φ, L Φ L bad 3 edgesa R Φ, A L Φ. Cosider a left vertex C i, α L Φ. We claim that it is cotaied i L if ad oly if C i, α A. This is because if it is cotaied i L, the we have C i, α A because 3 out of 5 edges icidet to C i, α go to A side the three variable represetative vertices. If C i, α is ot cotaied i L, we have at least 3 out of the 5 edges goig to B side the two edges to B Z r ad at least oe of the variable represetative vertices, ad therefore we have C i, α B. By this claim, we kow the followig two facts. A L Φ is small. Sice τ > 1/3, we have L /3 O1/M > /3, ad by the secod property of Lemma 3.3, we have A L Φ m L 3 / / β. We have edgesa L Φ, B Z r A L Φ ad edgesa L Φ, A R Φ 3 A L Φ. For edgesa R Φ, L Φ, we kow that this is exactly degl. Agai, sice τ > 1/3, by the first property of Lemma 3.3, we kow this value is lower-bouded by 6m L /1 0/ β. I all, we have edgesa, B edgesa L Φ, B Z r + edgesa R Φ, L Φ L bad 3 edgesa R Φ, A L Φ A L Φ + degl L bad 3 3 A L Φ degl A L Φ Om/M 6m L /1 0/ β m L 3 / / β Om/M m 1γ 4γ 3 40γ + 400γ 3 / β O1/M let γ L / 4m 3τ τ 3 O1/ β O1/M. The last step follows because i 3γ γ 3 mootoically icreases whe γ [0, 1], ad ii γ τ O1/M Costat-degree itegrality gap istace. I this subsectio, we slightly modify the graph H Φ obtaied i the previous subsectios to get a itegrality gap istace with costat degree. Observe that i H Φ, whe M ad β are costats, the oly vertices whose degree might be supercostat are the represetative vertices i R Φ. Now cosider the edges coectig vertices i L Φ ad represetative vertices: there are 1m of them, each of them correspods to a combiatio of costrait C i, satisfyig assigmet α, ad oe of the variables i the costrait. Let E b be the set of these edges. For two edges e 1, e E b, let the radom variable Y {e1,e } 1 if they share the same represetative vertex, ad let Y {e1,e } 0 otherwise. Fially let Y e 1,e E b Y {e1,e }. By the simple secod momet method, we kow that with probability 1 o1, we have Y 1000m 1000β.

12 1 GURUSWAMI ET AL. For every edge e E b, if e E b \{e} Y {e,e } > βm, we remove e from the graph. I this way, we get a ew graph, amely H Φ. We claim the followig properties about H Φ. 1. The maximum degree of H Φ is OβM. This is because the maximum degree of vertices other tha represetative vertices i H Φ is OβM, ad after the edge removal process described above, the represetative vertices have degree OβM.. The umber of edges removed is at most Y/βM, ad therefore 000m/M with probability 1 o1. This is because wheever a edge is removed, we charge βm to Y. Sice each edge i Y ca be charge at most twice, there are at most Y/βM edges to be removed. 3. The SDP solutio i Lemma 3. is still feasible ad has objective value at most 5m sice we removed edges with probability 1 o1. 4. The soudess lemma Lemma 3.4 still holds sice we removed oly Om/M edges. Therefore, we claim that H Φ is a itegrality gap istace for Theorem 3.1 with costat degree. 4. Gaps for UNIFORMSPARSESTCUT. I this sectio, we provide the full aalysis of the gap istace for UNI- FORMSPARSESTCUT. We first describe our costructio of the gap istace for UNIFORMSPARSESTCUT as follows. We modify the gap istace we got for BALANCEDSEPARATOR to get a istace for the liear roud Lasserre relaxatio of UNIFORMSPARSESTCUT. The reductio coverts the gap istace for BALANCEDSEPARATOR to the gap istace for UNIFORMSPARSESTCUT i a almost black box style. I the BALANCEDSEPARATOR problem, we have the hard costrait that the cut is τ-balaced. I the reductio from BALANCEDSEPARATOR to UNIFORMSPARS- ESTCUT, we eed to use the sparsity objective to eforce this costrait. We do it as follows. Recall that give a 3-XOR istace Φ, the correspodig gap istace for BALANCEDSEPARATOR cosists of vertex set L Φ R Φ Z r ad edge set E Φ. To get a gap istace for UNIFORMSPARSESTCUT, we add two more OM-regular expaders with edge expasio 10 4 M D l ad D r of size 1000Mm where M is the same parameter defied i the previous sectios. Now, let the edge set E Φ cotai the edges i E Φ, i the expaders D l ad D r, ad the followig edges : for each vertex v L Φ R Φ Z r, itroduce ew edges icidet to it, oe to a vertex i D l say, v l ad the other oe to a vertex i D r say, v r. We arrage these edges betwee L Φ R Φ Z r ad D l, D r i a way so that each vertex i D l or D r has at most oe eighbor i L Φ R Φ Z r this ca be doe because L Φ + R Φ + Z r M + 5m < 1000Mm D l D r. Usig the istace described above, we will prove our mai itegrality gap theorem Theorem 1.3 for UNI- FORMSPARSESTCUT. We state the full theorem as follows. THEOREM 4.1. For large eough costats β, M where β is the same parameter as i previous sectios, ad ifiitely may positive iteger N s, there is a N-vertex istace for UNIFORMSPARSESTCUT problem, such that the optimal solutio is at least 1 + 1/100M times worse tha the optimal solutio of the ΩN-roud Lasserre SDP. Theorem 4.1 is directly implied by the followig completeess lemma Lemma 4. ad soudess lemma Lemma 4.3. LEMMA 4. Completeess. The value of relaxatio Ψ i Figure. is at most M + 10m/1001M + 1m for τ 1001M + 1/00M + 5. Proof. Give the SDP solutio {U S } S L Φ R Φ Z r, S r/3 i the completeess case of BALANCEDSEPARA- TOR, we exted it to the SDP solutio {U S } S LΦ R Φ Z r D l D r, S r/3 for UNIFORMSPARSESTCUT by puttig D l ad D r oe per side. That is, for each S L Φ R Φ Z r D l D r with S r/3, let S S L Φ R Φ Z r. Now we let U S 0 if S D r, ad let U S U S otherwise. We first check that {U S } S LΦ R Φ Z r D l D r, S r/3 is a feasible SDP solutio. We oly check that the balace costrait the last costrait i relaxatio Φ is met. We are goig to prove prove that u L Φ R Φ Z r D l D r U {u} 1001M + 1mU.

13 From the proof of Lemma 3., we kow that together with the fact that LASSERRE GAPS FOR GRAPH PARTITIONING PROBLEMS 13 u L Φ R Φ Z r U {u} M + 1mU, u D l, U {u} U, u D r, U {u} 0, we get the desired equality. Now we calculate the value of the solutio. First, we calculate the followig value. U {u} U {v} U {u} U {v} + U {u} U {v} u,v E Φ 5m + u,v E Φ U {u} U {v} + U {u} U {v} u,v E Φ \EΦ u,v D l + u L Φ R Φ Z r u,v D r U {u} U {vl } + U {u} U {vr}, Note that U u,v D l {u} U {v} + U u,v D r {u} U {v} 0, ad U {u} U {vl } + U {u} U {vr} u L Φ R Φ Z r u L Φ R Φ Z r u L Φ R Φ Z r U {u} + U {vl } + U {vr} U {u}, U {vl } U {u}, U {vr} U {u} U {u,vl } U {u,vr} u L Φ R Φ Z r 1 L Φ + R Φ + Z r M + 5m. Thus, we have u,v E Φ Sice τ < 1/, the value of the solutio is at most 1 L Φ R Φ Z r D l D r τ U {u} U {v} M + 10m. u,v E Φ by property of Lasserre vectors U {u} U {v} M + 10m/1001M + 1m. LEMMA 4.3 Soudess. For large eough M, the sparsity of the sparsest cut is at least γ 1 + 1/100M M + 10m/1001Mm. Proof. Let D l be the smaller part amog D l S ad D l S, ad D l be the larger part. Also, let D r be the smaller part amog D r S ad D r S ad D r be the larger part. Let T, T be the cut restricted to L Φ R Φ Z r the BALANCEDSEPARATOR istace, i.e. let T S L Φ R Φ Z r ad T S L Φ R Φ Z r. First, we show that to get a cut of sparsity better tha γ, D l M D l, ad the same is true for D r by the same argumet. This is because if D l > M D l, by the expasio property, there are at least 10 4 M D l > 1000Mm edges i the cut. Sice the graph has L Φ + R Φ + Z r + D l + D r 00M + 5m vertices, therefore the sparsity of the cut is at least 1000M m 500Mm 1 > 4 00M + 5m 1001Mm > γ,

14 14 GURUSWAMI ET AL. for M > 1/5. Secod, we show that D l ad D r should be o opposite sides of ay cut of sparsity better tha γ. Suppose ot, let S be the side of the cut which D l ad D r are o. Recall that T S L Φ R Φ Z r. We have edgess, S edgest, D l D r + edgesd l, D l + edgesd r, D r. Note that edgest, D l D r T D l D r as each vertex i D l, D r is coected to at most oe vertex i T. Also, by the expasio property, edgesd l, D l + edgesd r, D r 1000M D l + D r. Now, we have edgess, S T D l + D r M D l + D r Therefore, the sparsity of the cut T + D l + D r M 3 D l + D r T + D l + D r S. edgess, S S S S S S S 00M + 5m > γ. Third, we show that if the cut S, S has sparsity better tha γ, the the cut T, T defied above is a 0.49 vs 0.51 balaced cut, i.e. T / L Φ + R Φ + Z r [0.49, 0.51]. Supposig T, T is ot 0.49 vs 0.51 balaced, i.e. T T > 0.0 M + 5m, we have S S T T D l D r 0.0 M + 5m 1000M 1000Mm 0.04M m 0.01Mm, for large eough M. Therefore, S, S is ot vs balaced. Thus, S S < 00M + 5m < 1001Mm Sice D l ad D r are o opposite sides of S, S, we kow that edgess, S M + 5m D l D r M + 5m 1 1/M, ad therefore the sparsity of the cut edgess, S S S > M + 5m 1001Mm 1 1/M This value is greater tha γ whe M > Fially, sice T, T is a 0.49 vs 0.51 balaced cut, by Lemma 3.4, we kow that with probability 1 o1, edgest, T > 5.4 O1/ β O1/Mm. Therefore edgess, S S S edgest, T + M + 5m D l D r 00M + 5m O1/ β O1/Mm + M + 5m 1000Mm 00M + 5m M O1/ β O1/Mm 00M + 5m M 1000Mm 10 4 M M O1/ β O1/Mm 1001Mm 1 1/00M M m 1001Mm 1 1/00M for large eough β ad M M + 10m 1 + 1/30M1 1/00M for large eough M 1001Mm M + 10m 1 + 1/100M γ. 1001Mm

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