Rank bounds and integrality gaps for cutting planes procedures

Buresh-Oppenheim, Joshua; Galesi, Nicola; Hoory, Shlomo; Magen, Avner; Pitassi, Toniann

doi:10.1109/sfcs.2003.1238206

Cited by 43 publications

(64 citation statements)

References 34 publications

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“…This differs from the aforementioned work in that the lower bounds are for the size of the proofs rather than for the rank (which corresponds to depth in the tree-like scenario). It also extends earlier results by Buresh-Oppenheim et al [8] on rank lower bounds, and builds on work of Grigoriev et al [16] and Kojevnikov and Itsykson [18] that proves lower bounds for LS + indirectly via the more powerful but complex proof system known as static positivstellensatz refutations.…”

Section: Introductionsupporting

confidence: 73%

“…They showed in particular that the integrality gap for Vertex Cover remains at least 2 − ε after Ω ε (log n) rounds of LS. Since then there has been a flurry of activity, proving larger gaps after fewer rounds for Vertex Cover and several other classical NP-hard optimization problems; see, e.g., [2,8,9,10,13,14,24,25,28]. Most of this work has focused on the LS and LS + hierarchies; exceptions are [10,13], which consider Sherali-Adams, and [24] which considers Lasserre.…”

Section: Introductionmentioning

confidence: 99%

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Sherali-adams relaxations of the matching polytope

Mathieu

Sinclair

2009

Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing

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We study the Sherali-Adams lift-and-project 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 K 2d+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 Sherali-Adams rounds until the integer polytope is reached) is exactly 2d − 1.

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Section: Introductionsupporting

confidence: 73%

Section: Introductionmentioning

confidence: 99%

Sherali-adams relaxations of the matching polytope

Mathieu

Sinclair

2009

Proceedings of the Forty-First Annual ACM Symposium on Theory of Computing

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“…Our tree-size tradeoff together with the rank lower bounds from [8] immediately give the following theorem.…”

Section: Tree-size Lower Bounds and Integrality Gapsmentioning

confidence: 90%

“…At present there are rank-based integrality gaps for LS and LS + for many important problems, including max-k-SAT, max-k-LIN, and vertex cover. (For example, see [5,13,25,26,27,8,20,2,10].) While these algorithms rule out a large collection of SDP algorithms, they do not rule out all polynomialtime SDP algorithms.…”

Section: Introductionmentioning

confidence: 99%

Exponential lower bounds and Integrality Gaps for Tree-like Lovász-Schrijver Procedures

Segerlind¹,

Pitassi²

2009

Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms

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The matrix cuts of Lovász and Schrijver are methods for tightening linear relaxations of zero-one programs by the addition of new linear inequalities. We address the question of how many new inequalities are necessary to approximate certain combinatorial problems, and to solve certain instances of Boolean satisfiability.Our first result is a size/rank tradeoff for tree-like Lovász-Schrijver refutations, showing that any refutation that has small size also has small rank. This allows us to immediately derive exponential size lower bounds for tree-like refutations of many unsatisfiable systems of inequalities where prior to our work, only strong rank bounds were known.Unfortunately, we show that this tradeoff does not hold more generally for derivations of arbitrary inequalities. We give a very simple example showing that derivations can be very small but nonetheless require maximal rank. This rules out a generic argument for obtaining a size-based integrality gap from the corresponding rankbased integrality gap. Our second contribution is to show that a modified argument can often be used to prove size-based integrality gaps from rank-based integrality gaps. We apply this method to prove size-based integrality gaps for several prominant examples where prior to our work, only rank-based integrality gaps were known.Our third contribution is to prove new separation results. Using our machinery for converting rank-based lower bounds and integrality gaps into size-based lower bounds, we show that tree-like LS + cannot polynomially simulate tree-like Cutting Planes, and that tree-like LS + cannot polynomially simulate resolution.We conclude by examining size/rank tradeoffs be- * Research supported by NSERC.yond the LS systems. We show that for Shirali-Adams and Lasserre systems, size/rank tradeoffs continue to hold, even in the general (non-tree) case. A full version of this paper is available at the Electronic Colloquium on Computational Complexity [23].

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“…(Note that "correction" phases of some sort or another can be found in many previous works [3,1,5,25,22,23].) We will construct the tensored vectors so that the vectors Y e i , Y (e 0 − e i ) have high saturation.…”

Section: Overview Of the Proofmentioning

confidence: 99%