The linearization problem of a binary quadratic problem and its applications

We provide several applications of the linearization problem of a binary quadratic problem. We propose a new lower bounding strategy, called the linearization-based scheme, that is based on a simple certificate for a quadratic function to be non-negative on the feasible set. Each linearization-based bound requires a set of linearizable matrices as an input. We prove that the Generalized Gilmore-Lawler bounding scheme for binary quadratic problems provides linearization-based bounds. Moreover, we show that the bound obtained from the first level reformulation linearization technique is also a type of linearization-based bound, which enables us to provide a comparison among mentioned bounds. However, the strongest linearization-based bound is the one that uses the full characterization of the set of linearizable matrices. Finally, we present a polynomial-time algorithm for the linearization problem of the quadratic shortest path problem on directed acyclic graphs. Our algorithm gives a complete characterization of the set of linearizable matrices for the quadratic shortest path problem

An LP-based Characterization of Solvable QAP Instances with Chess-board and Graded Structures

The quadratic assignment problem (QAP) is perhaps the most widely studied nonlinear combinatorial optimization problem. It has many applications in various fields, yet has proven to be extremely difficult to solve. This difficulty has motivated researchers to identify special objective function structures that permit an optimal solution to be found efficiently. Previous work has shown that certain such structures can be explained in terms of a mixed 0-1 linear reformulation of the QAP known as the level-1 reformulation-linearization-technique (RLT) form. Specifically, the objective function structures were shown to ensure that a binary optimal extreme point solution exists to the continuous relaxation. This paper extends that work by considering classes of solvable cases in which the objective function coefficients have special chess-board and graded structures, and similarly characterizing them in terms of the level-1 RLT form. As part of this characterization, we develop a new relaxed version of the level-1 RLT form, the structure of which can be readily exploited to study the special instances under consideration

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 consecutive 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 sufficient conditions for the quadratic cost matrix to be linearizable, and use these conditions to compute bounds. We also show how to use a sufficient condition for linearizability within an iterative bounding procedure. In each step, our algorithm computes the best equivalent representation of the quadratic cost matrix and its optimal linearizable matrix with respect to the given sufficient condition for linearizability. Further, we show that the classical Gilmore-Lawler type bound belongs to the family of linearization based bounds, and therefore apply the above mentioned iterative reformulation technique. We also prove that the linearization vectors resulting from this iterative approach satisfy the constant value property. The best among here introduced bounds outperform existing lower bounds when taking both quality and efficiency into account

Characterizing Linearizable QAPs by the Level-1 Reformulation-Linearization Technique

The quadratic assignment problem (QAP) is an extremely challenging NP-hard combinatorial optimization program. Due to its difficulty, a research emphasis has been to identify special cases that are polynomially solvable. Included within this emphasis are instances which are linearizable; that is, which can be rewritten as a linear assignment problem having the property that the objective function value is preserved at all feasible solutions. Various known sufficient conditions for identifying linearizable instances have been explained in terms of the continuous relaxation of a weakened version of the level-1 reformulation-linearization-technique (RLT) form that does not enforce nonnegativity on a subset of the variables. Also, conditions that are both necessary and sufficient have been given in terms of decompositions of the objective coefficients. The main contribution of this paper is the identification of a relationship between polyhedral theory and linearizability that promotes a novel, yet strikingly simple, necessary and sufficient condition for identifying linearizable instances; specifically, an instance of the QAP is linearizable if and only if the continuous relaxation of the same weakened RLT form is bounded. In addition to providing a novel perspective on the QAP being linearizable, a consequence of this study is that every linearizable instance has an optimal solution to the (polynomially-sized) continuous relaxation of the level-1 RLT form that is binary. The converse, however, is not true so that the continuous relaxation can yield binary optimal solutions to instances of the QAP that are not linearizable. Another consequence follows from our defining a maximal linearly independent set of equations in the lifted RLT variable space; we answer a recent open question that the theoretically best possible linearization-based bound cannot improve upon the level-1 RLT form

Linear Programming Methods for Identifying Solvable Cases of the Quadratic Assignment Problem

This research effort is concerned with identifying and characterizing families of polynomially solvable instances of the celebrated NP-hard quadratic assignment problem (qap). The approach is novel in that it uses polyhedral methods based on an equivalent mixed 0-1 linear reformulation of the problem. The continuous relaxation of this mixed 0-1 form yields a feasible region having extreme points that are both binary and fractional. The solvable instances of concern essentially possess objective function structures that ensure a binary extreme point must be optimal, so that the linear program solves the qap. The ultimate contribution of this work is the unification and subsumption of a variety of known solvable instances of the qap, and the development of a theoretical framework for identifying richer families of solvable instances. The qap was introduced over 50 years ago in the context of facility layout and location. The underlying mathematical structure, from which the problem draws its name, consists of the minimization of a quadratic function of binary variables over an assignment polytope. Since its inception, this structure has received considerable attention from various researchers, both practitioners and theoreticians alike, due to the diversity of practical applications and the resistance to exact solution procedures. Unfortunately, the combinatorial explosion of feasible solutions to the qap, in terms of the number of binary variables, creates a significant gap between the sizes of the motivating applications and the instances that can be solved by state-of-the-art solution algorithms. The most successful algorithms rely on linear forms of the qap to compute bounds within enumerative schemes. The inability to solve large qap instances has motivated researchers to seek special objective function structures that permit polynomial solvability. Various, seemingly unrelated, structures are found in the literature. This research shows that many such structures can be explained in terms of the linear reformulation which results from applying the level-1 reformulation-linearization technique (RLT) to the qap. In fact, the research shows that the level-1 RLT not only serves to explain many of these instances, but also allows for simplifications and/or generalizations. One important structure centers around instances deemed to be linearizable, where a qap instance is defined to be linearizazble if it can be equivalently rewritten as a linear assignment problem that preserves the objective function value at all feasible points. A contribution of this effort is that the constraint structure of a relaxed version of the continuous relaxation of the level-1 RLT form gives rise to a necessary and sufficient condition for an instance of the qap to be linearizable. Specifically, an instance of the qap is linearizable if and only if the given relaxed level-1 RLT form has a finite optimal solution. For all such cases, an optimal solution must occur at a binary extreme point. As a consequence, all linearizable qap instances are solvable via the level-1 RLT. The converse, however is not true, as the continuous relaxation of the level-1 RLT form can have a binary optimal solution when the qap is not linearizable. Thus, the linear program available from the level-1 RLT theoretically identifies a richer family of solvable instances. Notably, and as a consequence of this study, the level-1 RLT serves as a unifying entity in that it integrates the computation of linear programming-based bounds with the identification of polynomially solvable special cases, a relationship that was previously unnoticed