Approximation Limits of Linear Programs (Beyond Hierarchies)

Braun, Gábor; Fiorini, Samuel; Pokutta, Sebastian; Steurer, David

doi:10.1109/focs.2012.10

Cited by 59 publications

(126 citation statements)

References 30 publications

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“…For instance, we prove that every polynomial-sized LP for Max Cut has an integrality gap of 1 2 , answering a question from [BFPS12]. As another example, every such LP for Max 3-Sat has an integrality gap of 7 8 , and every such LP for Max 2-Sat has an integrality gap of 3 4 .…”

Section: Introductionmentioning

confidence: 99%

Approximate Constraint Satisfaction Requires Large LP Relaxations

Chan

Lee

Raghavendra

et al. 2016

J. ACM

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We prove super-polynomial lower bounds on the size of linear programming relaxations for approximation versions of constraint satisfaction problems. We show that for these problems, polynomial-sized linear programs are no more powerful than programs arising from a constant number of rounds of the Sherali-Adams hierarchy.In particular, any polynomial-sized linear program for Max Cut has an integrality gap of

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Section: Introductionmentioning

confidence: 99%

Approximate Constraint Satisfaction Requires Large LP Relaxations

Chan

Lee

Raghavendra

et al. 2016

J. ACM

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“…Braun, Fiorini, Pokutta and Steurer [4] showed that to lower bound the size of extended formulations for any polytope sandwiched between inner polytope P and outer polytope Q, it suffices to lower bound the nonnegative rank of the corresponding slack matrix S P,Q .…”

Section: Slack Matrices and Non-negative Rankmentioning

confidence: 99%

“…Braun, Fiorini, Pokutta and Steurer [4] defined a notion of approximation for combinatorial optimization problems in terms of extended formulations. They proved that even approximating the value of any linear objective function over the Correlation polytope requires extended formulations of exponential size by proving non-negative rank lower bounds for unique disjointness when the intersecting entries are 1−ρ.…”

Section: Other Related Workmentioning

confidence: 99%

“…The above theorem is proven using the connection between extension complexity and non-negative rank of slack matrices that was established in the work of Yannakakis [24] and subsequently extended by Braun, Fiorini, Pokutta and Steurer [4] to handle approximations 1 See Appendix A for a proof.…”

Section: Introductionmentioning

confidence: 99%

“…Starting with the breakthrough work of Fiorini, Massar, Pokutta, Tiwary and de Wolf [13], a sequence of works [4,7,6] proved that the non-negative rank of any unique disjointness matrix is 2 Ω(ρn) .…”

Section: Introductionmentioning

confidence: 99%

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Lower Bounds for Approximating the Matching Polytope

Sinha

2018

Proceedings of the Twenty-Ninth Annual ACM-SIAM Symposium on Discrete Algorithms

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We prove that any linear program that approximates the matching polytope on n-vertex graphs up to a factor of (1 + ε) for any 2 n ≤ ε ≤ 1 must have at least n α/ε inequalities where 0 < α < 1 is an absolute constant. This is tight as exhibited by the (1 + ε) approximating linear program obtained by dropping the odd set constraints of size larger than (1 + ε)/ε from the description of the matching polytope. Previously, a tight lower bound of 2 Ω(n) was only known for ε = O 1 n [22, 5] whereas for 2 n ≤ ε ≤ 1, the best lower bound was 2 Ω(1/ε) [22]. The key new ingredient in our proof is a close connection to the non-negative rank of a lopsided version of the unique disjointness matrix.

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Strong Reductions for Extended Formulations

Braun

Pokutta

Roy

2016

Integer Programming and Combinatorial Optimization

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We generalize the reduction mechanism for linear programming problems and semidefinite programming problems from [BPZ15] in two ways (1) relaxing the requirement of affineness, and (2) extending to fractional optimization problems.As applications we provide several new LP-hardness and SDP-hardness results, e.g., for the SparsestCut problem, the BalancedSeparator problem, the MaxCut problem and the Matching problem on 3-regular graphs. We also provide a new, very strong Lasserre integrality gap for the IndependentSet problem, which is strictly greater than the best known LP approximation, showing that the Lasserre hierarchy does not always provide the tightest SDP relaxation.

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Approximation Limits of Linear Programs (Beyond Hierarchies)

Cited by 59 publications

References 30 publications

Approximate Constraint Satisfaction Requires Large LP Relaxations

Approximate Constraint Satisfaction Requires Large LP Relaxations

Lower Bounds for Approximating the Matching Polytope

Strong Reductions for Extended Formulations

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