Lower Bounds for the Quadratic Minimum Spanning Tree Problem Based on Reduced Cost Computation

The Minimum Spanning Tree Problem (MSTP) is one of the most known combinatorial optimization problems. It concerns the determination of a minimum edge-cost subgraph spanning all the vertices of a given connected graph. The Quadratic Minimum Spanning Tree Problem (QMSTP) is a variant of the MST whose cost considers also the interaction between every pair … Read more

A Tight Lower Bound for the Adjacent Quadratic Assignment Problem

In this paper we address the Adjacent Quadratic Assignment Problem (AQAP) which is a variant of the QAP where the cost coefficient matrix has a particular structure. Motivated by strong lower bounds obtained by applying Reformulation Linearization Technique (RLT) to the classical QAP, we propose two special RLT representations for the problem. The first is … Read more

A Short Proof that the Extension Complexity of the Correlation Polytope Grows Exponentially

We establish that the extension complexity of the nXn correlation polytope is at least 1.5^n by a short proof that is self-contained except for using the fact that every face of a polyhedron is the intersection of all facets it is contained in. The main innovative aspect of the proof is a simple combinatorial argument … Read more

Equivalence of an Approximate Linear Programming Bound with the Held-Karp Bound for the Traveling Salesman Problem

We consider two linear relaxations of the asymmetric traveling salesman problem (TSP), the Held-Karp relaxation of the TSP’s arc-based formulation, and a particular approximate linear programming (ALP) relaxation obtained by restricting the dual of the TSP’s shortest path formulation. We show that the two formulations produce equal lower bounds for the TSP’s optimal cost regardless … Read more

Optimal Toll Design: A Lower Bound Framework for the Asymmetric Traveling Salesman Problem

We propose a framework of lower bounds for the asymmetric traveling salesman problem (TSP) based on approximating the dynamic programming formulation with diff erent basis vector sets. We discuss how several well-known TSP lower bounds correspond to intuitive basis vector choices and give an economic interpretation wherein the salesman must pay tolls as he travels between … Read more

A New Relaxation Framework for Quadratic Assignment Problems based on Matrix Splitting

Quadratic assignment problems (QAPs) are among the hardest discrete optimization problems. Recent study shows that even obtaining a strong lower bound for QAPs is a computational challenge. In this paper, we first discuss how to construct new simple convex relaxations of QAPs based on various matrix splitting schemes. Then we introduce the so-called symmetric mappings … Read more

Estimating Bounds for Quadratic Assignment Problems Associated with Hamming and Manhattan Distance Matrices based on Semidefinite Programming

Quadratic assignment problems (QAPs) with a Hamming distance matrix of a hypercube or a Manhattan distance matrix of rectangular grids arise frequently from communications and facility locations and are known to be among the hardest discrete optimization problems. In this paper we consider the issue of how to obtain lower bounds for those two classes … Read more

Lower bounds for the earliness-tardiness scheduling problem on single and parallel machines

This paper addresses the parallel machine scheduling problem in which the jobs have distinct due dates with earliness and tardiness costs. New lower bounds are proposed for the problem, they can be classed into two families. First, two assignment-based lower bounds for the one-machine problem are generalized for the parallel machine case. Second, a time-indexed … Read more

Batched Bin Packing

We introduce and study the batched bin packing problem (BBPP), a bin packing problem in which items become available for packing incrementally, one batch at a time. A batched algorithm must pack a batch before the next batch becomes known. A batch may contain several items; the special case when each batch consists of merely … Read more


Given an undirected graph G with nonnegative edges costs and nonnegative vertex penalties, the prize collecting Steiner problem in graphs (PCSPG) seeks a tree of G with minimum weight. The weight of a tree is the sum of its edge costs plus the sum of the penalties of those vertices not spanned by the tree. … Read more