Approximate Constraint Satisfaction Requires Large LP Relaxations

Chan, Siu On; Lee, James R.; Raghavendra, Prasad; Steurer, David

doi:10.1145/2811255

Cited by 39 publications

(16 citation statements)

References 31 publications

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“…Combining the above with the connection between Sherali-Adams gaps and extended formulations by [7,21] also yields that the basic LP is at least as strong as any LP extended formulation of size n o(log n/ log log n) . Corollary 1.2.…”

Section: Our Resultsmentioning

confidence: 82%

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Ghosh¹,

Tulsiani²

2018

Theory of Comput.

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Section: Our Resultsmentioning

confidence: 82%

“…A connection between LP integrality gaps for the Sherali-Adams hierarchy, and lower bounds on the size of LP extended formulations, was first established by Chan et al [7] and later improved by Kothari et al [21]. In [21], the authors proved the following:…”

Section: Lower Bounds For Lp Extended Formulationsmentioning

confidence: 98%

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Ghosh¹,

Tulsiani²

2018

Theory of Comput.

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“…This parameter characterizes semidefinite representability in much the same way that nonnegative rank captures representation as a linear program. Recently, Lee, Raghavendra, and Steurer [25] building on techniques in [11] gave the first exponential lower bounds for general semidefinite programs, but there are still many open questions in this area about both proving quantitatively stronger lower bounds and proving lower bounds for approximation versions of these representability questions. In another direction, it is not known whether there is an algorithm to decide if the semidefinite rank of a nonnegative matrix M is at most r that runs in polynomial time for r = O(1) [13].…”

Section: Question 2 Can the Nonnegative Rank Of A Matrix M Be Certifmentioning

confidence: 99%

An Almost Optimal Algorithm for Computing Nonnegative Rank

Moitra¹

2016

SIAM J. Comput.

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Abstract. Here, we give an algorithm for deciding if the nonnegative rank of a matrix M of dimension m × n is at most r which runs in time (nm) O(r 2 ) . This is the first exact algorithm that runs in time singly exponential in r. This algorithm (and earlier algorithms) are built on methods for finding a solution to a system of polynomial inequalities (if one exists). Notably, the best algorithms for this task run in time exponential in the number of variables but polynomial in all of the other parameters (the number of inequalities and the maximum degree). Hence, these algorithms motivate natural algebraic questions whose solution have immediate algorithmic implications: How many variables do we need to represent the decision problem, and does M have nonnegative rank at most r? A naive formulation uses nr + mr variables and yields an algorithm that is exponential in n and m even for constant r. Arora et al. [Proceedings of STOC, 2012, pp. 145-162] recently reduced the number of variables to 2r 2 2 r , and here we exponentially reduce the number of variables to 2r 2 and this yields our main algorithm. In fact, the algorithm that we obtain is nearly optimal (under the exponential time hypothesis) since an algorithm that runs in time (nm) o(r) would yield a subexponential algorithm for 3-SAT [Proceedings of STOC, 2012, pp. 145-162]. Our main result is based on establishing a normal form for nonnegative matrix factorization-which in turn allows us to exploit algebraic dependence among a large collection of linear transformations with variable entries.Additionally, we also demonstrate that nonnegative rank cannot be certified by even a very large submatrix of M , and this property also follows from the intuition gained from viewing nonnegative rank through the lens of systems of polynomial inequalities.

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“…Subsequently Rothvoß [Rot14] proved a 2 Ω(n) lower bound for the perfect matching polytope, which via a reduction implies the same bound for TSP. Chan et al [CLRS13] obtained lower bounds on linear extension complexity for constraint satisfaction problems via a different route: roughly put, they showed that arbitrary linear extensions are not much more powerful than the specific linear extensions produced by the "Sherali-Adams Hierarchy"; hence they could obtain lower bounds on linear extension complexity from known bounds on the Sherali-Adams hierarchy.…”

Section: Motivation: Linear and Semidefinite Extension Complexitymentioning

confidence: 99%

“…Until recently, there were only a few lower bounds for "symmetric" psd extensions [LRST14,FSP13]. However, in a very recent breakthrough, Lee et al [LRS14] generalized the approach of [CLRS13] to show that arbitrary psd extensions are not much more powerful than the specific psd extensions produced by the "Lasserre Hierarchy". In particular they showed that the TSP polytope has psd extension complexity 2 Ω(n 1/13 ) .…”

Section: Motivation: Linear and Semidefinite Extension Complexitymentioning

confidence: 99%

Query Complexity in Expectation

Kaniewski

Lee

Wolf

2015

Automata, Languages, and Programming

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We study the query complexity of computing a function f : {0, 1}n → R + in expectation. This requires the algorithm on input x to output a nonnegative random variable whose expectation equals f (x), using as few queries to the input x as possible. We exactly characterize both the randomized and the quantum query complexity by two polynomial degrees, the nonnegative literal degree and the sum-ofsquares degree, respectively. We observe that the quantum complexity can be unboundedly smaller than the classical complexity for some functions, but can be at most polynomially smaller for functions with range {0, 1}.These query complexities relate to (and are motivated by) the extension complexity of polytopes. The linear extension complexity of a polytope is characterized by the randomized communication complexity of computing its slack matrix in expectation, and the semidefinite (psd) extension complexity is characterized by the analogous quantum model. Since query complexity can be used to upper bound communication complexity of related functions, we can derive some upper bounds on psd extension complexity by constructing efficient quantum query algorithms. As an example we give an exponentially-close entrywise approximation of the slack matrix of the perfect matching polytope with psd-rank only 2. Finally, we show there is a precise sense in which randomized/quantum query complexity in expectation corresponds to the Sherali-Adams and Lasserre hierarchies, respectively.

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Approximate Constraint Satisfaction Requires Large LP Relaxations

Cited by 39 publications

References 31 publications

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An Almost Optimal Algorithm for Computing Nonnegative Rank

Query Complexity in Expectation

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