Extension Complexity of Independent Set Polytopes

Göös, Mika; Jain, Rahul; Watson, Thomas

doi:10.1137/16m109884x

Cited by 30 publications

(26 citation statements)

References 19 publications

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“…A similar situation arises for the Max-Knapsack problem for which Bienstock [3] gave an LP of size ≈ n . An exponential lower bound for exact extended formulations for Max-Knapsack follows by a reduction from the unique disjointness matrix [2,20] (also see [14] which proves a better lower bound by using a different slack matrix). It is unclear how to extend the aforementioned reduction from unique disjointness to prove a strong lower bound for any (1−ε) approximation for Max-Knapsack.…”

Section: Other Related Workmentioning

confidence: 90%

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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Section: Other Related Workmentioning

confidence: 90%

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 proved in Section 5.1 that constraints (15)- (20) define an extension R G of D G (see Theorem 13). Claim 7 implies that Y G is an extension of D G .…”

Section: The Dominant Matching Polytope In R Mmentioning

confidence: 99%

“…Thus constraints (23)-(26) hold. Also, constraints (27) and (28) are implied by constraints (19) and (20). Thus if R G is the projection of R G obtained by projecting out coordinates corresponding to self-loops, then R G ⊆ Y G .…”

Section: The Dominant Matching Polytope In R Mmentioning

confidence: 99%

“…We also have α u ≥ c x (u, u) = 0 = − e∈δ G (u) x e for u ∈ U (the second equality is because every edge e incident to u ∈ U is in E \ E D and constraint (27) tells us that x e = 0). Thus constraints (15) (20). Thus the augmented (x, α) ∈ R G , in other words, the original (x, α) ∈ R G (the projection of R G defined above).…”

Section: The Dominant Matching Polytope In R Mmentioning

confidence: 99%

“…[20]). There exists an (explicitly constructed) family of graphs H = (V, E) with bounded degree such that xc(I H ) = 2 Ω(|V |/ log |V |) .…”

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confidence: 99%

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Quasi-popular Matchings, Optimality, and Extended Formulations

Faenza¹,

Kavitha²

2020

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

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Let G = (A ∪ B, E) be an instance of the stable marriage problem where every vertex ranks its neighbors in a strict order of preference. A matching M in G is popular if M does not lose a headto-head election against any matching. Popular matchings are a well-studied generalization of stable matchings, introduced with the goal of enlarging the set of admissible solutions, while maintaining a certain level of fairness. Every stable matching is a min-size popular matching. Unfortunately, when there are edge costs, it is NP-hard to find a popular matching of minimum cost -even worse, the min-cost popular matching problem is hard to approximate up to any factor. Let opt be the cost of a min-cost popular matching. Our goal is to efficiently compute a matching of cost at most opt by paying the price of mildly relaxing popularity. Our main positive results are two bi-criteria algorithms that find in polynomial time a near-popular or "quasi-popular" matching of cost at most opt. Moreover, one of the algorithms finds a quasi-popular matching of cost at most that of a min-cost popular fractional matching -this cost could be much smaller than opt. Key to the other algorithm are a number of results for certain polytopes related to matchings. In particular, we give a polynomial-size extended formulation for an integral polytope sandwiched between the popular and quasi-popular matching polytopes. We complement these results by showing that it is NP-hard to find a quasi-popular matching of minimum cost, and that both the popular and quasi-popular matching polytopes have near-exponential extension complexity.

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Extended Formulations from Communication Protocols in Output-Efficient Time

Aprile

Faenza

2019

Integer Programming and Combinatorial Optimization

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Deterministic protocols are well-known tools to obtain extended formulations, with many applications to polytopes arising in combinatorial optimization. Although constructive, those tools are not output-efficient, since the time needed to produce the extended formulation also depends on the number of rows of the slack matrix (hence, on the exact description in the original space). We give general sufficient conditions under which those tools can be implemented as to be output-efficient, showing applications to e.g. Yannakakis' extended formulation for the stable set polytope of perfect graphs, for which, to the best of our knowledge, an efficient construction was previously not known. For specific classes of polytopes, we give also a direct, efficient construction of extended formulations arising from protocols. Finally, we deal with extended formulations coming from unambiguous non-deterministic protocols.

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Extension Complexity of Independent Set Polytopes

Cited by 30 publications

References 19 publications

Lower Bounds for Approximating the Matching Polytope

Lower Bounds for Approximating the Matching Polytope

Quasi-popular Matchings, Optimality, and Extended Formulations

Extended Formulations from Communication Protocols in Output-Efficient Time

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