2014
DOI: 10.1007/s10107-014-0764-2
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On the extension complexity of combinatorial polytopes

Abstract: In this paper we extend recent results of Fiorini et al. on the extension complexity of the cut polytope and related polyhedra. We first describe a lifting argument to show exponential extension complexity for a number of NP-complete problems including subset-sum and three dimensional matching. We then obtain a relationship between the extension complexity of the cut polytope of a graph and that of its graph minors. Using this we are able to show exponential extension complexity for the cut polytope of a large… Show more

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Cited by 41 publications
(88 citation statements)
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“…Assuming the Sparse Graph Conjecture we would obtain that the extension complexity of polytopes (see, e.g., [12,13] for definitions) for important combinatorial problems considered in [1,12,13,19] including (among others) the stable set polytope, knapsack polytope, and the 3SAT polytope would have truly exponential extension complexity, that is 2 (n) extension complexity, where n is the dimension of the polytope.…”
Section: Discussionmentioning
confidence: 99%
“…Assuming the Sparse Graph Conjecture we would obtain that the extension complexity of polytopes (see, e.g., [12,13] for definitions) for important combinatorial problems considered in [1,12,13,19] including (among others) the stable set polytope, knapsack polytope, and the 3SAT polytope would have truly exponential extension complexity, that is 2 (n) extension complexity, where n is the dimension of the polytope.…”
Section: Discussionmentioning
confidence: 99%
“…A crucial part of the proof is a strong lower bound on the nondeterministic communication complexity of the unique disjointness matrix (UDISJ), which was initially obtained by [11] using [12]. An existence proof of a polytope with high extension complexity, or equivalently of a slack matrix with high nonnegative rank, was given in [13] via a beautiful counting argument and, by reductions, lower bounds have been also obtained for various other polytopes (see [14], [15]). …”
Section: Introductionmentioning
confidence: 98%
“…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: 95%
“…For more details on this interpretation, see [6]. 2 We adopt the convention that x log x = 0 at x = 0. The above implies that if A and B are independent then I (A : B) = 0.…”
Section: Proposition 42 ([6])mentioning
confidence: 99%