On the Existence of 0/1 Polytopes with High Semidefinite Extension Complexity

Briët, Jop; Dadush, Daniel; Pokutta, Sebastian

doi:10.1007/978-3-642-40450-4_19

Cited by 11 publications

(4 citation statements)

References 14 publications

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“…It is therefore reasonable to conjecture that the semidefinite extension complexity of polytopes such as the TSP polytope is exponential. This conjecture is supported by the counting argument of [19] (based on the earlier work of [56]) that shows that some 0 1 polytopes have large semidefinite extension complexity.…”

Section: Introductionmentioning

confidence: 86%

“…Hence this would imply that P = NP. Just the existence of a n poly( )-size semidefinite extension of n COR( ) would imply NP ⊆ P/poly, as follows from the results of Brïet, Dadush and Pokutta [19].…”

Section: N N Nmentioning

confidence: 93%

“…It is routine to obtain upper and lower bounds on δ i for each = … i n 1, , from the bounds on the other variables, using ( 13) and (19). We obtain…”

Section: Y Cmentioning

confidence: 99%

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Generalized probabilistic theories and conic extensions of polytopes

Fiorini

Massar

Patra

et al. 2014

J. Phys. A: Math. Theor.

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Generalized probabilistic theories (GPT) provide a general framework that includes classical and quantum theories. It is described by a cone C and its dual C * . We show that whether some one-way communication complexity problems can be solved within a GPT is equivalent to the recently introduced cone factorisation of the corresponding communication matrix M . We also prove an analogue of Holevo's theorem: when the cone C is contained in R n , the classical capacity of the channel realised by sending GPT states and measuring them is bounded by log n.Polytopes and optimising functions over polytopes arise in many areas of discrete mathematics. A conic extension of a polytope is the intersection of a cone C with an affine subspace whose projection onto the original space yields the desired polytope. Extensions of polytopes can sometimes be much simpler geometric objects than the polytope itself. The existence of a conic extension of a polytope is equivalent to that of a cone factorisation of the slack matrix of the polytope, on the same cone.We show that all 0/1 polytopes whose vertices can be recognized by a polynomial size circuit, which includes as a special case the travelling salesman polytope and many other polytopes from combinatorial optimisation, have small conic extension complexity when the cone is the completely positive cone.Using recent exponential lower bounds on the linear extension complexity of polytopes, this provides an exponential gap between the communication complexity of GPT based on the completely positive cone and classical communication complexity, and a conjectured exponential gap with quantum communication complexity.Our work thus relates the communication complexity of generalisations of quantum theory to questions of mainstream interest in the area of combinatorial optimisation.

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Section: Introductionmentioning

confidence: 86%

Section: N N Nmentioning

confidence: 93%

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Generalized probabilistic theories and conic extensions of polytopes

Fiorini

Massar

Patra

et al. 2014

J. Phys. A: Math. Theor.

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“…(A recent work shows that random 0/1 polytopes require exponential-size SDP relaxations [20], but these polytopes do not correspond to natural combinatorial optimization problems.) A concrete implication is that for every positive constant ε > 0, symmetric SDP relaxation require exponential size to achieve approximation ratio 7/8 + ε for Max 3-Sat.…”

Section: Introductionmentioning

confidence: 99%

On the Power of Symmetric LP and SDP Relaxations

Lee

Raghavendra

Steurer

et al. 2014

2014 IEEE 29th Conference on Computational Complexity (CCC)

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We study the computational power of general symmetric relaxations for combinatorial optimization problems, both in the linear programming (LP) and semidefinite programming (SDP) case. We show new connections to explicit LP and SDP relaxations, like those obtained from standard hierarchies.Concretely, for k < n/4, we show that k-rounds of sum-ofsquares / Lasserre relaxations of size k n k achieve best-possible approximation guarantees for Max CSPs among all symmetric SDP relaxations of size at most n k . This result gives the first lower bounds for symmetric SDP relaxations of Max CSPs, and indicates that the sum-of-squares method provides the "right" SDP relaxation for this class of problems.Moreover, for k < n/4, we show the existence of symmetric LP relaxations of size O(n 2k ) for the traveling salesman problem that achieve per instance best-possible approximation guarantees among all symmetric LP relaxations of size roughly n k .

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Worst-case results for positive semidefinite rank

2015

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We present various worst-case results on the positive semidefinite (psd) rank of a nonnegative matrix, primarily in the context of polytopes. We prove that the psd rank of a generic n-dimensional polytope with v vertices is at least (nv) 1 4 improving on previous lower bounds. For polygons with v vertices, we show that psd rank cannot exceed 4 v/6 which in turn shows that the psd rank of a p × q matrix of rank three is at most 4 min{p, q}/6 . In general, a nonnegative matrix of rank k+1 2 has psd rank at least k and we pose the problem of deciding whether the psd rank is exactly k. Using geometry and bounds on quantifier elimination, we show that this decision can be made in polynomial time when k is fixed.

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On the Existence of 0/1 Polytopes with High Semidefinite Extension Complexity

Cited by 11 publications

References 14 publications

Generalized probabilistic theories and conic extensions of polytopes

Generalized probabilistic theories and conic extensions of polytopes

On the Power of Symmetric LP and SDP Relaxations

Worst-case results for positive semidefinite rank

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