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36
CSP Gaps and Reductions in the Lasserre Hierarchy
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 104 (2008)
, 2008
"... We study integrality gaps for SDP relaxations of constraint satisfaction problems, in the hierarchy of SDPs defined by Lasserre. Schoenebeck [25] recently showed the first integrality gaps for these problems, showing that for MAX kXOR, the ratio of the SDP optimum to the integer optimum may be as l ..."
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Cited by 42 (5 self)
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We study integrality gaps for SDP relaxations of constraint satisfaction problems, in the hierarchy of SDPs defined by Lasserre. Schoenebeck [25] recently showed the first integrality gaps for these problems, showing that for MAX kXOR, the ratio of the SDP optimum to the integer optimum may be as large as 2 even after Ω(n) rounds of the Lasserre hierarchy. We show that for the general MAX kCSP problem over binary domain, the ratio of SDP optimum to the value achieved by the optimal assignment, can be as large as 2 k /2k − ɛ even after Ω(n) rounds of the Lasserre hierarchy. For alphabet size q which is a prime, we give a lower bound of q k /q(q − 1)k − ɛ for Ω(n) rounds. The method of proof also gives optimal integrality gaps for a predicate chosen at random. We also explore how to translate gaps for CSP into integrality gaps for other problems using reductions, and establish SDP gaps for Maximum Independent Set, Approximate Graph Coloring, Chromatic Number and Minimum Vertex Cover. For Independent Set and Chromatic Number, we show integrality gaps of n/2 O( √ log nlog log n) even after 2
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 31 (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.
How to Round Any CSP
"... A large number of interesting combinatorial optimization ..."
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Cited by 26 (3 self)
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A large number of interesting combinatorial optimization
Lasserre hierarchy, higher eigenvalues, and approximation schemes for graph partitioning and quadratic integer programming with PSD objectives
 In Proceedings of 52nd Annual Symposium on Foundations of Computer Science (FOCS
, 2011
"... We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and ..."
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Cited by 26 (4 self)
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We present an approximation scheme for optimizing certain Quadratic Integer Programming problems with positive semidefinite objective functions and global linear constraints. This framework includes well known graph problems such as Minimum graph bisection, Edge expansion, Uniform sparsest cut, and Small Set expansion, as well as the Unique Games problem. These problems are notorious for the existence of huge gaps between the known algorithmic results and NPhardness results. Our algorithm is based on rounding semidefinite programs from the Lasserre hierarchy, and the analysis uses bounds for lowrank approximations of a matrix in Frobenius norm using columns of the matrix. For all the above graph problems, we give an algorithm running in time nO(r/ε 2) with approximation ratio 1+εmin{1,λr} , where λr is the r’th smallest eigenvalue of the normalized graph Laplacian L. In the case of graph bisection and small set expansion, the number of vertices in the cut is within lowerorder terms of the stipulated bound. Our results imply (1 + O(ε)) factor approximation in time nO(r
Improved approximation guarantees through higher levels of SDP hierarchies
 In Proceedings of the 11th International Workshop, APPROX
, 2008
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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.
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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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.
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.
Polynomial integrality gaps for strong SDP relaxations of Densest ksubgraph
"... The Densest ksubgraph problem (i.e. find a size k subgraph with maximum number of edges), is one of the notorious problems in approximation algorithms. There is a significant gap between known upper and lower bounds for Densest ksubgraph: the current best algorithm gives an ≈ O(n 1/4) approximatio ..."
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Cited by 15 (4 self)
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The Densest ksubgraph problem (i.e. find a size k subgraph with maximum number of edges), is one of the notorious problems in approximation algorithms. There is a significant gap between known upper and lower bounds for Densest ksubgraph: the current best algorithm gives an ≈ O(n 1/4) approximation, while even showing a small constant factor hardness requires significantly stronger assumptions than P ̸ = NP. In addition to interest in designing better algorithms, a number of recent results have exploited the conjectured hardness of Densest ksubgraph and its variants. Thus, understanding the approximability of Densest ksubgraph is an important challenge. In this work, we give evidence for the hardness of approximating Densest ksubgraph within polynomial factors. Specifically, we expose the limitations of strong semidefinite programs from SDP hierarchies in solving Densest ksubgraph. Our results include: • A lower bound of Ω ( n 1/4 / log 3 n) on the integrality gap for Ω(log n / log log n) rounds of the SheraliAdams relaxation for Densest ksubgraph. This also holds for the relaxation obtained from SheraliAdams with an added SDP constraint. Our gap instances are in
Minimum congestion mapping in a cloud
 In Proc. 30th PODC
, 2011
"... We study a basic resource allocation problem that arises in cloud computing environments. The physical network of the cloud is represented as a graph with vertices denoting servers and edges corresponding to communication links. A workload is a set of processes with processing requirements and mutua ..."
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Cited by 13 (1 self)
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We study a basic resource allocation problem that arises in cloud computing environments. The physical network of the cloud is represented as a graph with vertices denoting servers and edges corresponding to communication links. A workload is a set of processes with processing requirements and mutual communication requirements. The workloads arrive and depart over time, and the resource allocator must map each workload upon arrival to the physical network. We consider the objective of minimizing the congestion. We show that solving a subproblem (SingleMap) about mapping a single workload to the physical graph essentially suffices to solve the general problem. In particular, an αapproximation for SingleMap gives an O(α log nD) competitive algorithm for the general problem, where n is the number of nodes in the physical network and D is the maximum to minimum workload duration ratio. We also show how to solve SingleMap for two natural class of workloads, namely depthd trees and completegraph workloads. For depthd tree, we give an n O(d) time O(d 2 log(nd))approximation based on a strong LP relaxation inspired by the SheraliAdams hierarchy.