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16
Linear Level Lasserre Lower Bounds for Certain kCSPs
"... We show that for k ≥ 3 even the Ω(n) level of the Lasserre hierarchy cannot disprove a random kCSP instance over any predicate type implied by kXOR constraints, for example kSAT or kXOR. (One constant is said to imply another if the latter is true whenever the former is. For example kXOR constr ..."
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Cited by 33 (1 self)
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We show that for k ≥ 3 even the Ω(n) level of the Lasserre hierarchy cannot disprove a random kCSP instance over any predicate type implied by kXOR constraints, for example kSAT or kXOR. (One constant is said to imply another if the latter is true whenever the former is. For example kXOR constraints imply kCNF constraints.) As a result the Ω(n) level Lasserre relaxation fails to approximate such CSPs better than the trivial, random algorithm. As corollaries, we obtain Ω(n) level integrality gaps for the Lasserre hierarchy of 76 − ε for VertexCover, 2 − ε for kUniformHypergraphVertexCover, and any constant for kUniformHypergraphIndependentSet. This is the first construction of a Lasserre integrality gap. Our construction is notable for its simplicity. It simplifies, strengthens, and helps to explain several previous results.
A linear round lower bound for lovaszschrijver sdp relaxations of vertex cover
 In IEEE Conference on Computational Complexity. IEEE Computer Society
, 2006
"... We study semidefinite programming relaxations of Vertex Cover arising from repeated applications of the LS+ “liftandproject ” method of Lovasz and Schrijver starting from the standard linear programming relaxation. Goemans and Kleinberg prove that after one round of LS+ the integrality gap remains ..."
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Cited by 30 (9 self)
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We study semidefinite programming relaxations of Vertex Cover arising from repeated applications of the LS+ “liftandproject ” method of Lovasz and Schrijver starting from the standard linear programming relaxation. Goemans and Kleinberg prove that after one round of LS+ the integrality gap remains arbitrarily close to 2. Charikar proves an integrality gap of 2 for a stronger relaxation that is, however, incomparable with two rounds of LS+ and is strictly weaker than the relaxation resulting from a constant number of rounds. We prove that the integrality gap remains at least 7/6 − ε after cεn rounds, where n is the number of vertices and cε> 0 is a constant that depends only on ε.
Tight Integrality Gaps for LovászSchrijver LP Relaxations of Vertex Cover and Max Cut
 PROCEEDINGS OF THE 39TH SYMPOSIUM ON ACM SYMPOSIUM ON THEORY OF COMPUTING
, 2007
"... We study linear programming relaxations of Vertex Cover and Max Cut arising from repeated applications of the “liftandproject ” method of Lovasz and Schrijver starting from the standard linear programming relaxation. For Vertex Cover, Arora, Bollobas, Lovasz and Tourlakis prove that the integralit ..."
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Cited by 29 (9 self)
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We study linear programming relaxations of Vertex Cover and Max Cut arising from repeated applications of the “liftandproject ” method of Lovasz and Schrijver starting from the standard linear programming relaxation. For Vertex Cover, Arora, Bollobas, Lovasz and Tourlakis prove that the integrality gap remains at least 2 − ε after Ωε(log n) rounds, where n is the number of vertices, and Tourlakis proves that integrality gap remains at least 1.5 − ε after Ω((log n) 2) rounds. Fernandez de la Vega and Kenyon prove that the integrality gap of Max Cut is at most 1 2 + ε after any constant number of rounds. (Their result also applies to the more powerful SheraliAdams method.) We prove that the integrality gap of Vertex Cover remains at least 2 − ε after Ωε(n) rounds, and that the integrality gap of Max Cut remains at most 1/2 + ε after Ωε(n) rounds.
Convex Relaxations and Integrality Gaps
"... We discuss the effectiveness of linear and semidefinite relaxations in approximating the optimum for combinatorial optimization problems. Various hierarchies of these relaxations, such as the ones defined by Lovász and Schrijver [47], Sherali and Adams [55] and Lasserre [42] generate increasingly st ..."
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Cited by 21 (0 self)
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We discuss the effectiveness of linear and semidefinite relaxations in approximating the optimum for combinatorial optimization problems. Various hierarchies of these relaxations, such as the ones defined by Lovász and Schrijver [47], Sherali and Adams [55] and Lasserre [42] generate increasingly strong linear and semidefinite programming relaxations starting from a basic one. We survey some positive applications of these hierarchies, where their use yields improved approximation algorithms. We also discuss known lower bounds on the integrality gaps of relaxations arising from these hierarchies, demonstrating limits on the applicability of such hierarchies for certain optimization problems.
Integrality Gaps of Linear and Semidefinite Programming Relaxations for Knapsack
"... Recent years have seen an explosion of interest in lift and project methods, such as those proposed by Lovász and Schrijver [40], Sherali and Adams [49], Balas, Ceria and Cornuejols [6], Lasserre [36, 37] and others. These methods are systematic procedures for constructing a sequence of increasingly ..."
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Cited by 20 (0 self)
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Recent years have seen an explosion of interest in lift and project methods, such as those proposed by Lovász and Schrijver [40], Sherali and Adams [49], Balas, Ceria and Cornuejols [6], Lasserre [36, 37] and others. These methods are systematic procedures for constructing a sequence of increasingly tight mathematical programming relaxations for 01 optimization problems. One major line of research in this area has focused on understanding the strengths and limitations of these procedures. Of particular interest to our community is the question of how the integrality gaps for interesting combinatorial optimization problems evolve through a series of rounds of one of these procedures. On the one hand, if the integrality gap of successive relaxations drops sufficiently fast, there is the potential for an improved approximation algorithm. On the other hand, if the integrality gap for a problem persists, this can be viewed as a lower bound in a certain restricted model of computation. In this paper, we study the integrality gap in these hierarchies for the knapsack problem. We have two main results. First, we show that an integrality gap of 2 − ɛ persists up to a linear number of rounds of SheraliAdams. This is interesting, since it is well known that knapsack has a fully polynomial time approximation scheme [30, 39]. Second, we show that Lasserre’s hierarchy closes the gap quickly. Specifically, after t 2 rounds of Lasserre, the integrality gap decreases to t/(t − 1). Thus, we provide a second example of an integrality gap separation between Lasserre and Sherali Adams. The only other such gap we are aware of is in the recent work of Fernandez de la Vega and Mathieu [19] (respectively of Charikar, Makarychev and Makarychev [12]) showing that the integrality gap for MAXCUT remains 2 − ɛ even after ω(1) (respectively n γ) rounds of SheraliAdams. On the other hand, it is known that 2 rounds of Lasserre yields a relaxation as least as strong as the GoemansWilliamson SDP, which has an integrality gap of 0.878.
SheraliAdams relaxations of the matching polytope
 In STOC’2009
, 2009
"... We study the SheraliAdams liftandproject hierarchy of linear programming relaxations of the matching polytope. Our main result is an asymptotically tight expression 1 + 1/k for the integrality gap after k rounds of this hierarchy. The result is derived by a detailed analysis of the LP after k rou ..."
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We study the SheraliAdams liftandproject hierarchy of linear programming relaxations of the matching polytope. Our main result is an asymptotically tight expression 1 + 1/k for the integrality gap after k rounds of this hierarchy. The result is derived by a detailed analysis of the LP after k rounds applied to the complete graph K2d+1. We give an explicit recurrence for the value of this LP, and hence show that its gap exhibits a “phase transition, ” dropping from close to its maximum value 1 + 1 2d to close to 1 around the threshold k = 2d − √ d. We also show that the rank of the matching polytope (i.e., the number of SheraliAdams rounds until the integer polytope is reached) is exactly 2d − 1.
Exponential lower bounds and Integrality Gaps for Treelike LovászSchrijver Procedures
, 2007
"... The matrix cuts of Lovász and Schrijver are methods for tightening linear relaxations of zeroone programs by the addition of new linear inequalities. We address the question of how many new inequalities are necessary to approximate certain combinatorial problems with strong guarantees, and to solve ..."
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Cited by 10 (1 self)
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The matrix cuts of Lovász and Schrijver are methods for tightening linear relaxations of zeroone programs by the addition of new linear inequalities. We address the question of how many new inequalities are necessary to approximate certain combinatorial problems with strong guarantees, and to solve certain instances of Boolean satisfiability. We show that relaxations of linear programs, obtained by tightening via any subexponentialsize semidefinite LovászSchrijver derivation tree, cannot approximate maxkSAT to a factor better than 1+ 1 2k−1, maxkXOR to a factor better than 2 − ε, nor vertex cover to a factor better than 7/6. We prove exponential size lower bounds for treelike LovászSchrijver proofs of unsatisfiability for several prominent unsatisfiable CNFs, including random 3CNF formulas, random systems of linear equations, and the Tseitin graph formulas. Furthermore, we prove that treelike LS+ cannot polynomially simulate treelike cutting planes, and that treelike LS+ cannot polynomially simulate unrestricted resolution. All of our size lower bounds for derivation trees are based upon connections between the size and height of the derivation tree (its rank). The primary method is a treesize/rank tradeoff for LovászSchrijver refutations: Small tree size implies small rank. Surprisingly, this does not hold for derivations of arbitrary linear inequalities. We show that for LS0 and LS, there are examples with polynomialsize treelike derivations, but requiring linear rank.
SheraliAdams Relaxations and Indistinguishability in Counting Logics
, 2012
"... Two graphs with adjacency matrices A and B are isomorphic if there exists a permutation matrix P for which the identity P T AP = B holds. Multiplying through by P and relaxing the permutation matrix to a doubly stochastic matrix leads to the linear programming relaxation known as fractional isomorph ..."
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Two graphs with adjacency matrices A and B are isomorphic if there exists a permutation matrix P for which the identity P T AP = B holds. Multiplying through by P and relaxing the permutation matrix to a doubly stochastic matrix leads to the linear programming relaxation known as fractional isomorphism. We show that the levels of the SheraliAdams (SA) hierarchy of linear programming relaxations applied to fractional isomorphism interleave in power with the levels of a wellknown colorrefinementheuristic for graph isomorphism called the WeisfeilerLehman algorithm, orequivalently, with the levelsofindistinguishability in a logicwith counting quantifiers and a bounded number of variables. This tight connection has quite striking consequences. For example, it follows immediately from a deep result of Grohe in the context of logics with counting quantifiers, that a fixed number of levels of SA suffice to determine isomorphism of planar and minorfree graphs. We also offer applications both in finite model theory and polyhedral combinatorics. First, we show that certain properties of graphs, such as that of having a flowcirculation of a prescribed value, are definable in the infinitary logic with counting with a bounded number of variables. Second, we exploit a lower bound construction due to Cai, Fürer and Immerman in the context of counting logics to give simple explicit instances that show that the SA relaxations of the vertexcover and cut polytopes do not reach their integer hulls for up to Ω(n) levels, where n is the number of vertices in the graph.
Optimal SheraliAdams Gaps from Pairwise Independence
"... This work considers the problem of approximating fixed predicate constraint satisfaction problems (MAX kCSP(P)). We show that if the set of assignments accepted by P contains the support of a balanced pairwise independent distribution over the domain of the inputs, then such a problem on n variable ..."
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This work considers the problem of approximating fixed predicate constraint satisfaction problems (MAX kCSP(P)). We show that if the set of assignments accepted by P contains the support of a balanced pairwise independent distribution over the domain of the inputs, then such a problem on n variables cannot be approximated better than the trivial (random) approximation, even using Ω(n) levels of the SheraliAdams LP hierarchy. It was recently shown [3] that under the Unique Game Conjecture, CSPs for predicates satisfying this condition cannot be approximated better than the trivial approximation. Our results can be viewed as an unconditional analogue of this result in the restricted computational model defined by the SheraliAdams hierarchy. We also introduce a new generalization of techniques to define consistent “local distributions ” over partial assignments to variables in the problem, which is often the crux of proving lower bounds for such hierarchies.