Abstract Computing the edge expansion of a graph is a famously hard combinatorial problem for which there have been many approximation studies. We present two versions of an exact algorithm using semidefinite programming (SDP) to compute this constant for any graph. The SDP relaxation is used to first reduce the search space considerably. One version applies then an SDP-based branch-and-bound algorithm, along with heuristic search. The other version transforms the problem into an instance of a max-cut problem and solves this using a state-of-the-art solver. Numerical results demonstrate that we clearly outperform mixed-integer quadratic solvers as well as another SDP-based algorithm from the literature.

The famous open Vizing conjecture claims that the domination number of the Cartesian product graph of two graphs G and H is at least the product of the domination numbers of G and H. Recently Gaar, Krenn, Margulies and Wiegele used the graph class G of all graphs with nG vertices and domination number kG and reformulated Vizing's conjecture as the problem that for all graph classes G and H the Vizing polynomial is sum-of-squares (SOS) modulo the Vizing ideal. By solving semidefinite programs (SDPs) and clever guessing they derived SOS-certificates for some values of kG, nG, kH, and nH. In this paper, we consider their approach for kG=kH=1. For this case we are able to derive the unique reduced Gröbner basis of the Vizing ideal. Based on this, we deduce the minimum degree (nG+nH−1)/2 of an SOS-certificate for Vizing's conjecture, which is the first result of this kind. Furthermore, we present a method to find certificates for graph classes G and H with nG+nH−1=d for general d, which is again based on solving SDPs, but does not depend on guessing and depends on much smaller SDPs. We implement our new method in SageMath and give new SOS-certificates for all graph classes G and H with kG=kH=1 and nG+nH≤15.

We present BiqBin, an exact solver for linearly constrained binary quadratic problems. Our approach is based on an exact penalty method to first efficiently transform the original problem into an instance of Max-Cut, and then to solve the Max-Cut problem by a branch-and-bound algorithm. All the main ingredients are carefully developed using new semidefinite programming relaxations obtained by strengthening the existing relaxations with a set of hypermetric inequalities, applying the bundle method as the bounding routine and using new strategies for exploring the branch-and-bound tree. Furthermore, an efficient C implementation of a sequential and a parallel branch-and-bound algorithm is presented. The latter is based on a load coordinator-worker scheme using MPI for multi-node parallelization and is evaluated on a high-performance computer. The new solver is benchmarked against BiqCrunch, GUROBI, and SCIP on four families of (linearly constrained) binary quadratic problems. Numerical results demonstrate that BiqBin is a highly competitive solver. The serial version outperforms the other three solvers on the majority of the benchmark instances. We also evaluate the parallel solver and show that it has good scaling properties. The general audience can use it as an on-line service available at http://www.biqbin.eu .