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 minimum sum-of-squares clustering problem (MSSC) consists of partitioning n observations into k clusters in order to minimize the sum of squared distances from the points to the centroid of their cluster. In this paper, we propose an exact algorithm for the MSSC problem based on the branch-and-bound technique. The lower bound is computed by using a cutting-plane procedure in which valid inequalities are iteratively added to the Peng–Wei semidefinite programming (SDP) relaxation. The upper bound is computed with the constrained version of k-means in which the initial centroids are extracted from the solution of the SDP relaxation. In the branch-and-bound procedure, we incorporate instance-level must-link and cannot-link constraints to express knowledge about which data points should or should not be grouped together. We manage to reduce the size of the problem at each level, preserving the structure of the SDP problem itself. To the best of our knowledge, the obtained results show that the approach allows us to successfully solve, for the first time, real-world instances up to 4,000 data points.

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 .

Vizing's conjecture (open since 1968) relates the product of the domination numbers of two graphs to the domination number of their Cartesian product graph. In this paper, we formulate Vizing's conjecture as a Positivstellensatz existence question. In particular, we select classes of graphs according to their number of vertices and their domination number and encode the conjecture as an ideal/polynomial pair such that the polynomial is non-negative on the variety associated with the ideal if and only if the conjecture is true for this graph class. Using semidefinite programming we obtain numeric sum-of-squares certificates, which we then manage to transform into symbolic certificates confirming non-negativity of our polynomials. Specifically, we obtain exact low-degree sparse sum-of-squares certificates for particular classes of graphs. The obtained certificates allow generalizations for larger graph classes. Besides computational verification of these more general certificates, we also present theoretical proofs as well as conjectures and questions for further investigations.

Vizing's conjecture (open since 1968) relates the sizes of dominating sets in two graphs to the size of a dominating set in their Cartesian product graph. In this paper, we formulate Vizing's conjecture itself as a Positivstellensatz existence question. In particular, we encode the conjecture as an ideal/polynomial pair such that the polynomial is nonnegative if and only if the conjecture is true. We demonstrate how to use semidefinite optimization techniques to computationally obtain numeric sum-of-squares certificates, and then show how to transform these numeric certificates into symbolic certificates approving nonnegativity of our polynomial.

We present algorithm MIQCR-CB that is an advancement of MIQCR. MIQCR is a method for solving mixed-integer quadratic programs and works in two phases: the first phase determines an equivalent quadratic formulation with a convex objective function by solving a semidefinite problem (SDP); in the second phase, the equivalent formulation is solved by a standard solver. Because the reformulation relies on the solution of a large-scale semidefinite program, it is not tractable by existing semidefinite solvers even for medium-sized problems. To surmount this difficulty, we present in MIQCR-CB a subgradient algorithm within a Lagrangian duality framework for solving (SDP) that substantially speeds up the first phase. Moreover, this algorithm leads to a reformulated problem of smaller size than the one obtained by the original MIQCR method, which results in a shorter time for solving the second phase. We present extensive computational results to show the efficiency of our algorithm. First, we apply MIQCR-CB to the k-cluster problem that can be formulated by a binary quadratic program. As an illustration of the efficiency of our new algorithm, for instances of size 80 and of density 25%, MIQCR-CB is on average 78 times faster for phase 1 and 24 times faster for phase 2 than the original MIQCR. We also compare MIQCR-CB with QCR and with BiqCrunch, two methods devoted to binary quadratic programming. We show that MIQCR-CB is able to solve most of the 225 considered instances within three hours of CPU time. We also present experiments on two classes of general integer instances where we compare MIQCR-CB with MIQCR, Couenne, and Cplex12.6. We demonstrate the significant improvement over the original MIQCR approach. Finally, we show that MIQCR-CB is able to solve almost all of the considered instances, whereas Couenne and Cplex12.6 are not able to solve half of them.

The quadratic linear ordering problem naturally generalizes various optimization problems such as bipartite crossing minimization or the betweenness problem, which includes linear arrangement. These problems have important applications, e.g., in automatic graph drawing and computational biology. We present a new polyhedral approach to the quadratic linear ordering problem that is based on a linearization of the quadratic objective function. Our main result is a reformulation of the 3-dicycle inequalities using quadratic terms. After linearization, the resulting constraints are shown to be face-inducing for the polytope corresponding to the unconstrained quadratic problem. We use this result both within a branch-and-cut algorithm and within a branch-and-bound algorithm based on semidefinite programming. Experimental results for bipartite crossing minimization show that this approach clearly outperforms other methods.