## Lower Bounds for the Bandwidth Problem

The Bandwidth Problem asks for a simultaneous permutation of the rows and columns of the adjacency matrix of a graph such that all nonzero entries are as close as possible to the main diagonal. This work focuses on investigating novel approaches to obtain lower bounds for the bandwidth problem. In particular, we use vertex partitions … Read more

## The Quadratic Cycle Cover Problem: special cases and efficient bounds

The quadratic cycle cover problem is the problem of finding a set of node-disjoint cycles visiting all the nodes such that the total sum of interaction costs between incident arcs is minimized. In this paper we study the linearization problem for the quadratic cycle cover problem and related lower bounds. In particular, we derive various … Read more

## A polynomial time algorithm for the linearization problem of the QSPP and its applications

Given an instance of the quadratic shortest path problem (QSPP) on a digraph \$G\$, the linearization problem for the QSPP asks whether there exists an instance of the linear shortest path problem on \$G\$ such that the associated costs for both problems are equal for every \$s\$-\$t\$ path in \$G\$. We prove here that the … Read more

## On Solving the Quadratic Shortest Path Problem

The quadratic shortest path problem is the problem of finding a path in a directed graph such that the sum of interaction costs over all pairs of arcs on the path is minimized. We derive several semidefinite programming relaxations for the quadratic shortest path problem with a matrix variable of order \$m+1\$, where \$m\$ is … Read more

## Special cases of the quadratic shortest path problem

The quadratic shortest path problem (QSPP) is the problem of finding a path in a digraph such that the sum of weights of arcs and the sum of interaction costs over all pairs of arcs on the path is minimized. We first consider a variant of the QSPP known as the adjacent QSPP. It was … Read more

## Graph bisection revisited

The graph bisection problem is the problem of partitioning the vertex set of a graph into two sets of given sizes such that the sum of weights of edges joining these two sets is optimized. We present a semidefinite programming relaxation for the graph bisection problem with a matrix variable of order \$n\$ – the … Read more

## The min-cut and vertex separator problem

We consider graph three-partitions with the objective of minimizing the number of edges between the first two partition sets while keeping the size of the third block small. We review most of the existing relaxations for this min-cut problem and focus on a new class of semidefinite relaxations, based on matrices of order \$2n\$ which … Read more

## New bounds for the max-hBccut and chromatic number of a graph

We consider several semidefinite programming relaxations for the max-\$k\$-cut problem, with increasing complexity. The optimal solution of the weakest presented semidefinite programming relaxation has a closed form expression that includes the largest Laplacian eigenvalue of the graph under consideration. This is the first known eigenvalue bound for the max-\$k\$-cut when \$k>2\$ that is applicable to … Read more

## Semidefinite programming and eigenvalue bounds for the graph partition problem

The graph partition problem is the problem of partitioning the vertex set of a graph into a fixed number of sets of given sizes such that the total weight of edges joining different sets is optimized. In this paper we simplify a known matrix-lifting semidefinite programming relaxation of the graph partition problem for several classes … Read more

## A new semidenite programming relaxation for the quadratic assignment problem and its computational perspectives

Recent progress in solving quadratic assignment problems (QAPs) from the QAPLIB test set has come from mixed integer linear or quadratic programming models that are solved in a branch-and-bound framework. Semidenite programming bounds for QAP have also been studied in some detail, but their computational impact has been limited so far, mostly due to the … Read more