The quadratic assignment problem is easy for Robinsonian matrices with Toeplitz structure

We present a new polynomially solvable case of the Quadratic Assignment Problem in Koopmans-Beckman form $QAP(A,B)$, by showing that the identity permutation is optimal when $A$ and $B$ are respectively a Robinson similarity and dissimilarity matrix and one of $A$ or $B$ is a Toeplitz matrix. A Robinson (dis)similarity matrix is a symmetric matrix whose entries (increase) decrease monotonically along rows and columns when moving away from the diagonal, and such matrices arise in the classical seriation problem.Comment: 15 pages, 2 figure

Linearizable special cases of the QAP

We consider special cases of the quadratic assignment problem (QAP) that are linearizable in the sense of Bookhold. We provide combinatorial characterizations of the linearizable instances of the weighted feedback arc set QAP, and of the linearizable instances of the traveling salesman QAP. As a by-product, this yields a new well-solvable special case of the weighted feedback arc set 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

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

The multi-stripe travelling salesman problem

In the classical Travelling Salesman Problem (TSP), the objective function sums the costs for travelling from one city to the next city along the tour. In the q-stripe TSP with q ≥ 1, the objective function sums the costs for travelling from one city to each of the next q cities along the tour. The resulting q-stripe TSP generalizes the TSP and forms a special case of the quadratic assignment problem. We analyze the computational complexity of the q-stripe TSP for various classes of specially structured distance matrices. We derive NP-hardness results as well as polyomially solvable cases. One of our main results generalizes a well-known theorem of Kalmanson from the classical TSP to the q-stripe TSP

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

Quadratic assignment problem : linearizations and polynomial time solvable cases

Cataloged from PDF version of article.The Quadratic Assignment Problem (QAP) is one of the hardest combinatorial optimization problems known. Exact solution attempts proposed for instances of size larger than 15 have been generally unsuccessful even though successful implementations have been reported on some test problems from the QAPLIB up to size 36. In this dissertation, we analyze the binary structure of the QAP and present new IP formulations. We focus on “flow-based” formulations, strengthen the formulations with valid inequalities, and report computational experience with a branch-and-cut algorithm. Next, we present new classes of instances of the QAP that can be completely or partially reduced to the Linear Assignment Problem and give procedures to check whether or not an instance is an element of one of these classes. We also identify classes of instances of the Koopmans-Beckmann form of the QAP that are solvable in polynomial time. Lastly, we present a strong lower bound based on Bender’s decomposition.Erdoğan, GüneşPh.D