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17
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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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.
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.
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
On Linear and Semidefinite Programming Relaxations for Hypergraph Matching
"... The hypergraph matching problem is to find a largest collection of disjoint hyperedges in a hypergraph. This is a wellstudied problem in combinatorial optimization and graph theory with various applications. The best known approximation algorithms for this problem are all local search algorithms. I ..."
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Cited by 15 (0 self)
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The hypergraph matching problem is to find a largest collection of disjoint hyperedges in a hypergraph. This is a wellstudied problem in combinatorial optimization and graph theory with various applications. The best known approximation algorithms for this problem are all local search algorithms. In this paper we analyze different linear and semidefinite programming relaxations for the hypergraph matching problem, and study their connections to the local search method. Our main results are the following: • We consider the standard linear programming relaxation of the problem. We provide an algorithmic proof of a result of Füredi, Kahn and Seymour, showing that the integrality gap is exactly k − 1 + 1/k for kuniform hypergraphs, and is exactly k − 1 for kpartite hypergraphs. This yields an improved approximation algorithm for the weighted 3dimensional matching problem. Our algorithm combines the use of the iterative rounding method and the fractional local ratio method, showing a new way to round linear programming solutions for packing problems. • We study the strengthening of the standard LP relaxation by local constraints. We show that, even after linear number of rounds of the SheraliAdams liftandproject procedure on the standard LP relaxation, there are kuniform hypergraphs with integrality gap at least k − 2. On the other hand, we prove that for every constant k, there is a strengthening of the standard LP relaxation by only a polynomial number of constraints, with integrality gap at most (k+1)/2 for kuniform hypergraphs. The construction uses a result in extremal combinatorics. • We consider the standard semidefinite programming relaxation of the problem. We prove that the Lovász ϑfunction provides an SDP relaxation with integrality gap at most (k + 1)/2. The proof gives an indirect way (not by a rounding algorithm) to bound the ratio between any local optimal solution and any optimal SDP solution. This shows a new connection between local search and linear and semidefinite programming relaxations. 1
How well can PrimalDual and LocalRatio algorithms perform?
, 2007
"... We define an algorithmic paradigm, the stack model, that captures many primaldual and localratio algorithms for approximating covering and packing problems. The stack model is defined syntactically and without any complexity limitations and hence our approximation bounds are independent of the P v ..."
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We define an algorithmic paradigm, the stack model, that captures many primaldual and localratio algorithms for approximating covering and packing problems. The stack model is defined syntactically and without any complexity limitations and hence our approximation bounds are independent of the P vs NP question. We provide tools to bound the performance of primal dual and local ratio algorithms and supply a (log n + 1)/2 inapproximability result for set cover, a 4/3 inapproximability for min steiner tree, and a 0.913 inapproximability for interval scheduling on two machines.
SDP 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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Cited by 7 (0 self)
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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 after augmenting the natural semidefinite relaxation with Ω(n) levels of the SheraliAdams 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 a restricted computational model. 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.
Communication lower bounds via critical block sensitivity
 In Proc. 46th Annual ACM Symposium on Theory of Computing (STOC ’14
, 2014
"... Abstract. We use critical block sensitivity, a new complexity measure introduced by Huynh and Nordström (STOC 2012), to study the communication complexity of search problems. To begin, we give a simple new proof of the following central result of Huynh and Nordström: if S is a search problem with ..."
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Cited by 6 (0 self)
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Abstract. We use critical block sensitivity, a new complexity measure introduced by Huynh and Nordström (STOC 2012), to study the communication complexity of search problems. To begin, we give a simple new proof of the following central result of Huynh and Nordström: if S is a search problem with critical block sensitivity b, then every randomised twoparty protocol solving a certain twoparty lift of S requires Ω(b) bits of communication. Besides simplicity, our proof has the advantage of generalising to the multiparty setting. We combine these results with new critical block sensitivity lower bounds for Tseitin and Pebbling search problems to obtain the following applications. • Monotone circuit depth: We exhibit a monotone function on n variables whose monotone circuits require depth Ω(n / log n); previously, a bound of Ω( n) was known (Raz and Wigderson, JACM 1992). Moreover, we prove a tight Θ( n) monotone depth bound for a function in monotone P. This implies an averagecase hierarchy theorem within monotone P similar to a result of Filmus et al. (FOCS 2013). • Proof complexity: We prove new rank lower bounds as well as obtain the first length–space lower bounds for semialgebraic proof systems, including Lovász– Schrijver and Lasserre (SOS) systems. In particular, these results extend and simplify the works of Beame et al. (SICOMP 2007) and Huynh and Nordström.ar
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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Cited by 6 (1 self)
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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.
Tight Gaps for Vertex Cover in the SheraliAdams SDP Hierarchy
, 2011
"... We give the first tight integrality gap for Vertex Cover in the SheraliAdams SDP system. More precisely, we show that for every ɛ> 0, the standard SDP for Vertex Cover that is strengthened with the level6 SheraliAdams system has integrality gap 2 − ɛ. To the best of our knowledge this is the f ..."
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Cited by 6 (2 self)
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We give the first tight integrality gap for Vertex Cover in the SheraliAdams SDP system. More precisely, we show that for every ɛ> 0, the standard SDP for Vertex Cover that is strengthened with the level6 SheraliAdams system has integrality gap 2 − ɛ. To the best of our knowledge this is the first nontrivial tight integrality gap for the SheraliAdams SDP hierarchy for a combinatorial problem with hard constraints. For our proof we introduce a new tool to establish LocalGlobal Discrepancy which uses simple facts from highdimensional geometry. This allows us to give SheraliAdams solutions with objective value n(1/2 + o(1)) for graphs with small (2 + o(1)) vector chromatic number. Since such graphs with no linear size independent sets exist, this immediately gives a tight integrality gap for the SheraliAdams system for superconstant number of tightenings. In order to obtain a SheraliAdams solution that also satisfies semidefinite conditions, we reduce semidefiniteness to a condition on the Taylor expansion of a reasonably simple function that we are able to establish up to constantlevel SDP tightenings. We conjecture that this condition holds even for superconstant levels which would imply that in fact our solution is valid for superconstant level SheraliAdams SDPs.