Results 1  10
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15
Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds
, 2012
"... We solve a 20year old problem posed by Yannakakis and prove that there exists no polynomialsize linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope an ..."
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Cited by 54 (13 self)
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We solve a 20year old problem posed by Yannakakis and prove that there exists no polynomialsize linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the cut polytope and the stable set polytope. These results were discovered through a new connection that we make between oneway quantum communication protocols and semidefinite programming reformulations of LPs.
Integrality gaps for sheraliadams relaxations
 In Proceedings of the FortyFirst Annual ACM Symposium on Theory of Computing
, 2009
"... We prove strong lower bounds on integrality gaps of Sherali–Adams relaxations for MAX CUT, Vertex Cover, Sparsest Cut and other problems. Our constructions show gaps for Sherali–Adams relaxations that survive nδ rounds of lift and project. For MAX CUT and Vertex Cover, these show that even nδ rounds ..."
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Cited by 39 (2 self)
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We prove strong lower bounds on integrality gaps of Sherali–Adams relaxations for MAX CUT, Vertex Cover, Sparsest Cut and other problems. Our constructions show gaps for Sherali–Adams relaxations that survive nδ rounds of lift and project. For MAX CUT and Vertex Cover, these show that even nδ rounds of Sherali–Adams do not yield a better than 2 − ε approximation. The main combinatorial challenge in constructing these gap examples is the construction of a fractional solution that is far from an integer solution, but yet admits consistent distributions of local solutions for all small subsets of variables. Satisfying this consistency requirement is one of the major hurdles to constructing Sherali–Adams gap examples. We present a modular recipe for achieving this, building on previous work on metrics with a local–global structure. We develop a conceptually simple geometric approach to constructing Sherali–Adams gap examples via constructions of consistent local SDP solutions. This geometric approach is surprisingly versatile. We construct Sherali–Adams gap examples for Unique Games based on our construction for MAX CUT together with a parallel repetition like procedure. This in turn allows us to obtain Sherali–Adams gap examples for any problem that has a Unique Games based hardness result (with some additional conditions on the reduction from Unique Games). Using this, we construct 2 − ε gap examples for Maximum Acyclic Subgraph that rules out any family of linear constraints with support at most nδ. 1
Towards Sharp Inapproximability For Any 2CSP
"... We continue the recent line of work on the connection between semidefinite programmingbased approximation algorithms and the Unique Games Conjecture. Given any boolean 2CSP (or more generally, any nonnegative objective function on two boolean variables), we show how to reduce the search for a good ..."
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Cited by 32 (1 self)
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We continue the recent line of work on the connection between semidefinite programmingbased approximation algorithms and the Unique Games Conjecture. Given any boolean 2CSP (or more generally, any nonnegative objective function on two boolean variables), we show how to reduce the search for a good inapproximability result to a certain numeric minimization problem. The key objects in our analysis are the vector triples arising when doing clausebyclause analysis of algorithms based on semidefinite programming. Given a weighted set of such triples of a certain restricted type, which are “hard ” to round in a certain sense, we obtain a Unique Gamesbased inapproximability matching this “hardness ” of rounding the set of vector triples. Conversely, any instance together with an SDP solution can be viewed as a set of vector triples, and we show that we can always find an assignment to the instance which is at least as good as the “hardness ” of rounding the corresponding set of vector triples. We conjecture that the restricted type required for the hardness result is in fact no restriction, which would imply that these upper and lower bounds match exactly. This conjecture is supported by all existing results for specific 2CSPs. As an application, we show that MAX 2AND is hard to approximate within 0.87435. This improves upon the best previous hardness of αGW + ɛ ≈ 0.87856, and comes very close to matching the approximation ratio of the best algorithm known, 0.87401. It also establishes that balanced instances of MAX 2AND, i.e., instances in which each variable occurs positively and negatively equally often, are not the hardest to approximate, as these can be approximated within a factor αGW.
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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Cited by 16 (0 self)
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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.
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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Cited by 12 (4 self)
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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.
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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Cited by 10 (0 self)
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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.
Hardness Amplification in Proof Complexity
, 2009
"... We present a generic method for converting any family of unsatisfiable CNF formulas that require large resolution rank into CNF formulas whose refutation requires large rank for proof systems that manipulate polynomials or polynomial threshold functions of degree at most k. (The latter are known as ..."
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Cited by 8 (1 self)
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We present a generic method for converting any family of unsatisfiable CNF formulas that require large resolution rank into CNF formulas whose refutation requires large rank for proof systems that manipulate polynomials or polynomial threshold functions of degree at most k. (The latter are known as Th(k) proofs.) As special cases, such systems include: LovászSchrijver systems (LS, LS+), high degree analogues of LovászSchrijver (LS(k), LS+(k)), Cutting Planes and high degree versions of Cutting Planes (CP(k)), as well as SheraliAdams and Lasserre proofs. We introduce two very general families of proof systems, denoted by T cc (k) and R cc (k). The proof lines of T cc (k) are arbitrary Boolean functions, each of which can be evaluated by an efficient kparty randomized communication protocol. T cc (k) proofs are very powerful and include Th(k − 1) proofs as a special case. R cc (k) proofs generalize T cc (k) proofs and require only that each inference be checkable (in a certain weak sense) by an efficient kparty randomized communication protocol. Our main results are the following:
Integrality gaps of semidefinite programs for Vertex Cover and relations to ℓ1 embeddability of negative type metrics
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Approximating Sparsest Cut in Graphs of Bounded Treewidth
"... We give the first constantfactor approximation algorithm for SparsestCut with general demands in bounded treewidth graphs. In contrast to previous algorithms, which rely on the flowcut gap and/or metric embeddings, our approach exploits the SheraliAdams hierarchy of linear programming relaxation ..."
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Cited by 5 (2 self)
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We give the first constantfactor approximation algorithm for SparsestCut with general demands in bounded treewidth graphs. In contrast to previous algorithms, which rely on the flowcut gap and/or metric embeddings, our approach exploits the SheraliAdams hierarchy of linear programming relaxations.