Adam N. Letchford It is well known that the standard (linear) knapsack problem can be solved exactly by dynamic programming in O(nc) time, where n is the number of items and c is the capacity of the knapsack. The quadratic knapsack problem, on the other hand, is NP-hard in the strong sense, which makes it unlikely that it … Read more
The Steiner Traveling Salesman Problem (STSP) is a variant of the Traveling Salesman Problem (TSP) that is particularly suitable when dealing with sparse networks, such as road networks. The standard integer programming formulation of the STSP has an exponential number of constraints, just like the standard formulation of the TSP. On the other hand, there … Read more
A wide range of problems arising in practical applications can be formulated as Mixed-Integer Nonlinear Programs (MINLPs). For the case in which the objective and constraint functions are convex, some quite effective exact and heuristic algorithms are available. When non-convexities are present, however, things become much more difficult, since then even the continuous relaxation is … Read more
Pisinger et al. introduced the concept of `aggressive reduction’ for large-scale combinatorial optimisation problems. The idea is to spend much time and effort in reducing the size of the instance, in the hope that the reduced instance will then be small enough to be solved by an exact algorithm. We present an aggressive reduction scheme … Read more
This paper introduces a fundamental family of unbounded convex sets that arises in the context of non-convex mixed-integer quadratic programming. It is shown that any mixed-integer quadratic program with linear constraints can be reduced to the minimisation of a linear function over a set in the family. Some fundamental properties of the convex sets are … Read more
In 1996, Laurent and Poljak introduced an extremely general class of cutting planes for the max-cut problem, called gap inequalities. We prove several results about them, including the following: (i) there must exist non-dominated gap inequalities with gap larger than 1, unless NP = co-NP; (ii) there must exist non-dominated gap inequalities with exponentially large … Read more
Quadratic Convex Reformulation (QCR) is a technique that was originally proposed for quadratic 0-1 programs, and then extended to various other problems. It is used to convert non-convex instances into convex ones, in such a way that the bound obtained by solving the continuous relaxation of the reformulated instance is as strong as possible. In … Read more
We compute a complete linear description of the bipartite subgraph polytope, for up to seven nodes, and a conjectured complete description for eight nodes. We then show how these descriptions were used to compute the integrality ratio of various relaxations of the max-cut problem, again for up to eight nodes. CitationL. Galli & A.N. Letchford … Read more
We consider the positive semidefinite (psd) matrices with binary entries, along with the corresponding integer polytopes. We begin by establishing some basic properties of these matrices and polytopes. Then, we show that several families of integer polytopes in the literature — the cut, boolean quadric, multicut and clique partitioning polytopes — are faces of binary … Read more
In recent years, there has been an increased literature on so-called Generalized Network Design Problems, such as the Generalized Minimum Spanning Tree Problem and the Generalized Traveling Salesman Problem. In such problems, the node set of a graph is partitioned into clusters, and the feasible solutions must contain one node from each cluster. Up to … Read more