Semidefinite programming bounds for Lee codes

For $q,n,d \in \mathbb{N}$, let $A_q^L(n,d)$ denote the maximum cardinality of a code $C \subseteq \mathbb{Z}_q^n$ with minimum Lee distance at least $d$, where $\mathbb{Z}_q$ 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 $A_q^L(n,d)$. 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 $\mathbb{Z}_q$, 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 Lov\'asz theta-function, except for very small $n$.Comment: 14 pages. arXiv admin note: text overlap with arXiv:1703.0517

Semidefinite bounds for nonbinary codes based on quadruples

For nonnegative integers $q,n,d$, let $A_q(n,d)$ denote the maximum cardinality of a code of length $n$ over an alphabet $[q]$ with $q$ letters and with minimum distance at least $d$. We consider the following upper bound on $A_q(n,d)$. For any $k$, let \CC_k be the collection of codes of cardinality at most $k$. Then $A_q(n,d)$ is at most the maximum value of $\sum_{v\in[q]^n}x(\{v\})$, where $x$ is a function \CC_4\to R_+ such that $x(\emptyset)=1$ and $x(C)=0$ if $C$ has minimum distance less than $d$, and such that the \CC_2\times\CC_2 matrix (x(C\cup C'))_{C,C'\in\CC_2} is positive semidefinite. By the symmetry of the problem, we can apply representation theory to reduce the problem to a semidefinite programming problem with order bounded by a polynomial in $n$. It yields the new upper bounds $A_4(6,3)\leq 176$, $A_4(7,4)\leq 155$, $A_5(7,4)\leq 489$, and $A_5(7,5)\leq 87$

New lower bounds on crossing numbers of Km,nK_{m,n} from semidefinite programming

In this paper, we use semidefinite programming and representation theory to compute new lower bounds on the crossing number of the complete bipartite graph $K_{m,n}$, extending a method from de Klerk et al. [SIAM J. Discrete Math. 20 (2006), 189--202] and the subsequent reduction by De Klerk, Pasechnik and Schrijver [Math. Prog. Ser. A and B, 109 (2007) 613--624]. We exploit the full symmetry of the problem using a novel decomposition technique. This results in a full block-diagonalization of the underlying matrix algebra, which we use to improve bounds on several concrete instances. Our results imply that $\text{cr}(K_{10,n}) \geq 4.87057 n^2 - 10n$, $\text{cr}(K_{11,n}) \geq 5.99939 n^2-12.5n$, $\text{cr}(K_{12,n}) \geq 7.25579 n^2 - 15n$, $\text{cr}(K_{13,n}) \geq 8.65675 n^2-18n$ for all $n$. The latter three bounds are computed using a new and well-performing relaxation of the original semidefinite programming bound. This new relaxation is obtained by only requiring one small matrix block to be positive semidefinite.Comment: 17 pages, 3 figures, 3 tables. Revisions have been made based on comments of the referees. Accepted for publication in Mathematical Programmin

Approximate Pricing in Networks: How to Boost the Betweenness and Revenue of a Node

We introduce and study two new pricing problems in networks: Suppose we are given a directed graph G = (V, E) with non-negative edge costs (c_e)_{e in E}, k commodities (s_i, t_i, w_i)_{i in [k]} and a designated node u in V. Each commodity i in [k] is represented by a source-target pair (s_i, t_i) in V x V and a demand w_i>0, specifying that w_i units of flow are sent from s_i to t_i along shortest s_i, t_i-paths (with respect to (c_e)_{e in E}). The demand of each commodity is split evenly over all shortest paths. Assume we can change the edge costs of some of the outgoing edges of u, while the costs of all other edges remain fixed; we also say that we price (or tax) the edges of u. We study the problem of pricing the edges of u with respect to the following two natural objectives: (i) max-flow: maximize the total flow passing through u, and (ii) max-revenue: maximize the total revenue (flow times tax) through u. Both variants have various applications in practice. For example, the max flow objective is equivalent to maximizing the betweenness centrality of u, which is one of the most popular measures for the influence of a node in a (social) network. We prove that (except for some special cases) both problems are NP-hard and inapproximable in general and therefore resort to approximation algorithms. We derive approximation algorithms for both variants and show that the derived approximation guarantees are best possible

A note on the computational complexity of the moment-SOS hierarchy for polynomial optimization

The moment-sum-of-squares (moment-SOS) hierarchy is one of the most celebrated and widely applied methods for approximating the minimum of an n-variate polynomial over a feasible region defined by polynomial (in)equalities. A key feature of the hierarchy is that, at a fixed level, it can be formulated as a semidefinite program of size polynomial in the number of variables n. Although this suggests that it may therefore be computed in polynomial time, this is not necessarily the case. Indeed, as O'Donnell (2017) and later Raghavendra & Weitz (2017) show, there exist examples where the sos-representations used in the hierarchy have exponential bit-complexity. We study the computational complexity of the moment-SOS hierarchy, complementing and expanding upon earlier work of Raghavendra & Weitz (2017). In particular, we establish algebraic and geometric conditions under which polynomial-time computation is guaranteed to be possible.Comment: 10 page