An Exact Approach for Solving Pickup-and-Delivery Traveling Salesman Problems with Neighborhoods

This paper studies a variant of the traveling salesman problem called the pickup-and-delivery traveling salesman problem with neighborhoods that combines traditional pickup and delivery requirements with the flexibility of visiting the customers at locations within compact neighborhoods of arbitrary shape. We derive two optimality conditions for the problem, a local condition that verifies whether a … Read more

Solving Graph Partitioning on Sparse Graphs: Cuts, Projections, and Extended Formulations

This paper explores new solution approaches for the graph partitioning problem. While the classic formulations for graph partitioning are compact, they either suffer from a poor relaxation, symmetry, or contain a cubic number of constraints regardless of the graph density. These shortcomings often result in poor branch-and-bound performance. We approach this problem from perspective of … Read more

Integer Programming Methods for Solving Binary Interdiction Games

This paper studies a general class of interdiction problems in which the solution space of both the leader and follower are characterized by two discrete sets denoted the leader’s strategy set and the follower’s structure set. In this setting, the interaction between any strategy-structure pair is assumed to be binary, in the sense that the … Read more

Worst-case analysis of clique MIPs

The usual integer programming formulation for the maximum clique problem has several undesirable properties, including a weak LP relaxation, a quadratic number of constraints and nonzeros when applied to sparse graphs, and poor guarantees on the number of branch-and-bound nodes needed to solve it. With this as motivation, we propose new mixed integer programs (MIPs) … Read more

Integer Programming Formulations for Minimum Spanning Tree Interdiction

We consider a two-player interdiction problem staged over a graph where the leader’s objective is to minimize the cost of removing edges from the graph so that the follower’s objective, i.e., the weight of a minimum spanning tree in the residual graph, is increased up to a predefined level $r$. Standard approaches for graph interdiction … Read more

Why is maximum clique often easy in practice?

To this day, the maximum clique problem remains a computationally challenging problem. Indeed, despite researchers’ best efforts, there exist unsolved benchmark instances with one thousand vertices. However, relatively simple algorithms solve real-life instances with millions of vertices in a few seconds. Why is this the case? Why is the problem apparently so easy in many … Read more