Compact Linearization for Binary Quadratic Problems Comprising Linear Constraints

In this paper, the compact linearization approach originally proposed for binary quadratic programs with assignment constraints is generalized to such programs with arbitrary linear equations and inequalities that have positive coefficients and right hand sides. Quadratic constraints may exist in addition, and the technique may as well be applied if these impose the only nonlinearities, i.e., the objective function is linear. We present special cases of linear constraints (along with prominent combinatorial optimization problems where these occur) such that the associated compact linearization yields a linear programming relaxation that is provably as least as strong as the one obtained with a classical linearization method. Moreover, we show how to compute a compact linearization automatically which might be used, e.g., by general-purpose mixed-integer programming solvers

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

Nonlinear Integer Programming

Research efforts of the past fifty years have led to a development of linear integer programming as a mature discipline of mathematical optimization. Such a level of maturity has not been reached when one considers nonlinear systems subject to integrality requirements for the variables. This chapter is dedicated to this topic. The primary goal is a study of a simple version of general nonlinear integer problems, where all constraints are still linear. Our focus is on the computational complexity of the problem, which varies significantly with the type of nonlinear objective function in combination with the underlying combinatorial structure. Numerous boundary cases of complexity emerge, which sometimes surprisingly lead even to polynomial time algorithms. We also cover recent successful approaches for more general classes of problems. Though no positive theoretical efficiency results are available, nor are they likely to ever be available, these seem to be the currently most successful and interesting approaches for solving practical problems. It is our belief that the study of algorithms motivated by theoretical considerations and those motivated by our desire to solve practical instances should and do inform one another. So it is with this viewpoint that we present the subject, and it is in this direction that we hope to spark further research.Comment: 57 pages. To appear in: M. J\"unger, T. Liebling, D. Naddef, G. Nemhauser, W. Pulleyblank, G. Reinelt, G. Rinaldi, and L. Wolsey (eds.), 50 Years of Integer Programming 1958--2008: The Early Years and State-of-the-Art Surveys, Springer-Verlag, 2009, ISBN 354068274

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 Binary Programming Models in Computational Biology

In this paper we formulate four problems in computational molecular biology as 0-1 quadratic programs. These problems are all NP-hard and the current solution methods used in practice consist of heuristics or approximation algorithms tailored to each problem. Using test problems from scientific databases, we address the question, “Can a general-purpose solver obtain good answers in reasonable time?” In addition, we use the latest heuristics as incumbent solutions to address the question, “Can a general-purpose solver confirm optimality or find an improved solution in reasonable time?” Our computational experiments compare four different reformulation methods: three forms of linearization and one form of quadratic convexification

Compact Hierarchical Graph Drawings via Quadratic Layer Assignment

We propose a new mixed-integer programming formulation that very naturally expresses the layout restrictions of a layered (hierarchical) graph drawing and several associated objectives, such as a minimum total arc length, number of reversed arcs, and width, or the adaptation to a specific drawing area, as a special quadratic assignment problem. Our experiments show that it is competitive to another formulation that we slightly simplify as well