A note on the extension complexity of the knapsack polytope

Pokutta, Sebastian; Vyve, Mathieu Van

doi:10.1016/j.orl.2013.03.010

Cited by 39 publications

(37 citation statements)

References 5 publications

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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%

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Average case polyhedral complexity of the maximum stable set problem

2016

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We study the minimum number of constraints needed to formulate random instances of the maximum stable set problem via linear programs (LPs), in two distinct models. In the uniform model, the constraints of the LP are not allowed to depend on the input graph, which should be encoded solely in the objective function. There we prove a $2^{\Omega(n/ \log n)}$ lower bound with probability at least $1 - 2^{-2^n}$ for every LP that is exact for a randomly selected set of instances; each graph on at most n vertices being selected independently with probability $p \geq 2^{-\binom{n/4}{2}+n}$. In the non-uniform model, the constraints of the LP may depend on the input graph, but we allow weights on the vertices. The input graph is sampled according to the G(n, p) model. There we obtain upper and lower bounds holding with high probability for various ranges of p. We obtain a super-polynomial lower bound all the way from $p = \Omega(\log^{6+\varepsilon} / n)$ to $p = o (1 / \log n)$. Our upper bound is close to this as there is only an essentially quadratic gap in the exponent, which currently also exists in the worst-case model. Finally, we state a conjecture that would close this gap, both in the average-case and worst-case models

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Section: Discussionmentioning

confidence: 99%

“…For Eq. (13), i.e., in the case 1/n < p c/ 3 √ n, we neglect the exponential term in (19) for the choice of t:…”

Section: Corollary 45 For Every Fixedmentioning

confidence: 99%

Average case polyhedral complexity of the maximum stable set problem

2016

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“…A reduction mechanism for exact extended formulations has been considered in [3,36], where it already provided lower bounds on the exact extension complexity of various polytopes. Our generalization of (encoding dependent) extended formulations to encoding independent formulation complexity is closely related to [12] and [7].…”

Section: Related Workmentioning

confidence: 99%

Inapproximability of Combinatorial Problems via Small LPs and SDPs

Braun

Pokutta

Zink

2015

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

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Motivated by [12], we provide a framework for studying the size of linear programming formulations as well as semidefinite programming formulations of combinatorial optimization problems without encoding them first as linear programs. This is done via a factorization theorem for the optimization problem itself (and not a specific encoding of such). As a result we define a consistent reduction mechanism that degrades approximation factors in a controlled fashion and which, at the same time, is compatible with approximate linear and semidefinite programming formulations. Moreover, our reduction mechanism is a minor restriction of classical reductions establishing inapproximability in the context of PCP theorems. As a consequence we establish strong linear programming inapproximability (for LPs with a polynomial number of constraints) for several problems that are not 0/1-CSPs: we obtain a 3 2 − ε inapproximability for VertexCover (which is not of the CSP type) answering an open question in [12], we answer a weak version of our sparse graph conjecture posed in [6] showing an inapproximability factor of 1 2 +ε for bounded degree IndependentSet, and we establish inapproximability of Max-MULTI-k-CUT (a non-binary CSP). In the case of SDPs, we obtain relative inapproximability results for these problems.

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“…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: 99%

Common Information and Unique Disjointness

Braun,

Pokutta

2013

2013 IEEE 54th Annual Symposium on Foundations of Computer Science

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We provide a new framework for establishing strong lower bounds on the nonnegative rank of matrices by means of common information, a notion previously introduced in [1]. Common information is a natural lower bound for the nonnegative rank of a matrix and by combining it with Hellinger distance estimations we can compute the (almost) exact common information of UDISJ partial matrix. We also establish robustness of this estimation under various perturbations of the UDISJ partial matrix, where rows and columns are randomly or adversarially removed or where entries are randomly or adversarially altered. This robustness translates, via a variant of Yannakakis' Factorization Theorem, to lower bounds on the average case and adversarial approximate extension complexity. We present the first family of polytopes, the hard pair introduced in [2] related to the CLIQUE problem, with high average case and adversarial approximate extension complexity. We also provide an information theoretic variant of the fooling set method that allows us to extend fooling set lower bounds from extension complexity to approximate extension complexity.

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A note on the extension complexity of the knapsack polytope

Cited by 39 publications

References 5 publications

Average case polyhedral complexity of the maximum stable set problem

Average case polyhedral complexity of the maximum stable set problem

Inapproximability of Combinatorial Problems via Small LPs and SDPs

Common Information and Unique Disjointness

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