Semidefinite bounds for nonbinary codes based on quadruples

Litjens, Bart; Polak, Sven; Schrijver, Alexander

doi:10.1007/s10623-016-0216-5

Cited by 28 publications

(56 citation statements)

References 8 publications

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“…Proposition 2 of [15] implies the following. (In reference [15], it is stated that G = S q and V = R q×q , but with a straightforward adaptation one obtains the following result. )…”

Section: A Representative Set For the Action Of G N S N On V ⊗Nmentioning

confidence: 89%

“…In this paper we consider the bound B L 3 (q, n, d). The method for obtaining a symmetry reduction, using representation theory of the dihedral and symmetric groups, is an adaptation of the method in [15].…”

Section: The Semidefinite Programming Boundmentioning

confidence: 99%

“…[12,15]). Then a straightforward calculation, highly similar to the one in Lemma 2 of [15], yields that…”

Section: Computations For |D| =mentioning

confidence: 99%

“…To add the empty code, one may add a new unit base vector ∅ to the matrix in the representative set (15)…”

Section: A Appendix: Formulas With Integersmentioning

confidence: 99%

“…For binary codes equipped with the Hamming distance, the Delsarte bound was generalized to a semidefinite programming bound based on triples of codewords by A. Schrijver [23], and later to a quadruple bound by Gijswijt, Mittelmann and Schrijver [10]. Also, nonbinary codes with the Hamming distance have been considered [11,15], codes with mixed alphabets [16] and constant weight (binary) codes [19,23]. In [5], the authors mention the possibility of applying semidefinite programming to Lee codes and they state that to their best knowledge, such bounds for Lee codes using triples have not yet been studied.…”

Section: Introductionmentioning

confidence: 99%

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Semidefinite programming bounds for Lee codes

Polak

2019

Discrete Mathematics

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For q, n, d ∈ N, let A L q (n, d) denote the maximum cardinality of a code C ⊆ Z n q with minimum Lee distance at least d, where Z q denotes the cyclic group of order q. We consider a semidefinite programming bound based on triples of codewords, which bound can be computed efficiently using symmetry reductions, resulting in several new upper bounds on A L q (n, d). The technique also yields an upper bound on the independent set number of the n-th strong product power of the circular graph C d,q , which number is related to the Shannon capacity of C d,q . Here C d,q is the graph with vertex set Z q , in which two vertices are adjacent if and only if their distance (mod q) is strictly less than d. The new bound does not seem to improve significantly over the bound obtained from Lovász theta-function, except for very small n.

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“…Proposition 2 of [15] implies the following. (In reference [15], it is stated that G = S q and V = R q×q , but with a straightforward adaptation one obtains the following result. )…”

Section: A Representative Set For the Action Of G N S N On V ⊗Nmentioning

confidence: 89%

Section: The Semidefinite Programming Boundmentioning

confidence: 99%

“…[12,15]). Then a straightforward calculation, highly similar to the one in Lemma 2 of [15], yields that…”

Section: Computations For |D| =mentioning

confidence: 99%

“…To add the empty code, one may add a new unit base vector ∅ to the matrix in the representative set (15)…”

Section: A Appendix: Formulas With Integersmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Semidefinite programming bounds for Lee codes

Polak

2019

Discrete Mathematics

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Breaking Symmetries to Rescue Sum of Squares: The Case of Makespan Scheduling

Verdugo

Verschae

2019

Integer Programming and Combinatorial Optimization

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The Sum of Squares (SoS) hierarchy gives an automatized technique to create a family of increasingly tight convex relaxations for binary programs. There are several problems for which a constant number of rounds of the hierarchy give integrality gaps matching the best known approximation algorithm. In many other, however, ad-hoc techniques give significantly better approximation ratios. Notably, the lower bounds instances, in many cases, are invariant under the action of a large permutation group. The main purpose of this paper is to study how the presence of symmetries on a formulation degrades the performance of the relaxation obtained by the SoS hierarchy. We do so for the special case of the minimum makespan problem on identical machines. Our first result is to show that a linear number of rounds of SoS applied over the configuration linear program yields an integrality gap of at least 1.0009. This improves on the recent work by Kurpisz et al. [40] that shows an analogous result for the weaker LS + and SA hierarchies. Then, we consider the weaker assignment linear program and add a well chosen set of symmetry breaking inequalities that removes a subset of the machine permutation symmetries. We show that applying the SoS hierarchy for O ε (1) rounds to this linear program reduces the integrality gap to (1 + ε). Our results suggest that for this classical problem the symmetries of the natural assignment linear program were the main barrier preventing the SoS hierarchy to give relaxations with integrality gap (1 + ε) after a constant number of rounds. We leave as an open question whether this phenomenon occurs for different problems where the SoS hierarchy yields weak relaxations.

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Symmetric sums of squares over k-subset hypercubes

et al. 2017

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Abstract. We consider the problem of finding sum of squares (sos) expressions to establish the non-negativity of a symmetric polynomial over a discrete hypercube whose coordinates are indexed by the k-element subsets of [n]. For simplicity, we focus on the case k = 2, but our results extend naturally to all values of k ≥ 2. We develop a variant of the Gatermann-Parrilo symmetryreduction method tailored to our setting that allows for several simplifications and a connection to flag algebras.We show that every symmetric polynomial that has a sos expression of a fixed degree also has a succinct sos expression whose size depends only on the degree and not on the number of variables. Our method bypasses much of the technical difficulties needed to apply the Gatermann-Parrilo method, and offers flexibility in obtaining succinct sos expressions that are combinatorially meaningful. As a byproduct of our results, we arrive at a natural representation-theoretic justification for the concept of flags as introduced by Razborov in his flag algebra calculus. Furthermore, this connection exposes a family of non-negative polynomials that cannot be certified with any fixed set of flags, answering a question of Razborov in the context of our finite setting.

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Semidefinite bounds for nonbinary codes based on quadruples

Cited by 28 publications

References 8 publications

Semidefinite programming bounds for Lee codes

Semidefinite programming bounds for Lee codes

Breaking Symmetries to Rescue Sum of Squares: The Case of Makespan Scheduling

Symmetric sums of squares over k-subset hypercubes

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