Renata Sotirov In the quadratic minimum spanning tree problem (QMSTP) one wants to find the minimizer of a quadratic function over all possible spanning trees of a graph. We give two formulations of the QMSTP as mixed-integer semidefinite programs exploiting the algebraic connectivity of a graph. Based on these formulations, we derive a doubly nonnegative relaxation for … Read more
We consider the complex cut polytope: the convex hull of Hermitian rank 1 matrices \(xx^{\mathrm{H}}\), where the elements of \(x \in \mathbb{C}^n\) are \(m\)th unit roots. These polytopes have applications in \({\text{MAX-3-CUT}}\), digital communication technology, angular synchronization and more generally, complex quadratic programming. For \({m=2}\), the complex cut polytope corresponds to the well-known cut polytope. … Read more
A physical limitation in quantum circuit design is the fact that gates in a quantum system can only act on qubits that are physically adjacent in the architecture. To overcome this problem, SWAP gates need to be inserted to make the circuit physically realizable. The nearest neighbour compliance problem (NNCP) asks for an optimal embedding … Read more
It is well-known that by adding integrality constraints to the semidefinite programming (SDP) relaxation of the max-cut problem, the resulting integer semidefinite program is an exact formulation of the problem. In this paper we show similar results for a wide variety of discrete optimization problems for which SDP relaxations have been derived. Based on a … Read more
We consider semidefinite programming (SDP) approaches for solving the maximum satisfiabilityproblem (MAX-SAT) and the weighted partial MAX-SAT. It is widely known that SDP is well-suitedto approximate the (MAX-)2-SAT. Our work shows the potential of SDP also for other satisfiabilityproblems, by being competitive with some of the best solvers in the yearly MAX-SAT competition.Our solver combines … Read more
The graph partition problem (GPP) aims at clustering the vertex set of a graph into a fixed number of disjoint subsets of given sizes such that the sum of weights of edges joining different sets is minimized. This paper investigates the quality of doubly nonnegative (DNN) relaxations, i.e., relaxations having matrix variables that are both … Read more
In this paper we study the well-known Chvátal-Gomory (CG) procedure for the class of integer semidefinite programs (ISDPs). We prove several results regarding the hierarchy of relaxations obtained by iterating this procedure. We also study different formulations of the elementary closure of spectrahedra. A polyhedral description of the elementary closure for a specific type of … Read more
This paper is an in-depth analysis of the generalized $\vartheta$-number of a graph. The generalized $\vartheta$-number, $\vartheta_k(G)$, serves as a bound for both the $k$-multichromatic number of a graph and the maximum $k$-colorable subgraph problem. We present various properties of $\vartheta_k(G)$, such as that the series $(\vartheta_k(G))_k$ is increasing and bounded above by the order … Read more
We study the Quadratic Cycle Cover Problem (QCCP), which aims to find a node-disjoint cycle cover in a directed graph with minimum interaction cost between successive arcs. We derive several semidefinite programming (SDP) relaxations and use facial reduction to make these strictly feasible. We investigate a nontrivial relationship between the transformation matrix used in the … Read more
The maximum $k$-colorable subgraph (M$k$CS) problem is to find an induced $k$-colorable subgraph with maximum cardinality in a given graph. This paper is an in-depth analysis of the M$k$CS problem that considers various semidefinite programming relaxations including their theoretical and numerical comparisons. To simplify these relaxations we exploit the symmetry arising from permuting the colors, … Read more