Mark W. Lewis

The unconstrained binary quadratic programming problem: a survey

In recent years the unconstrained binary quadratic program (UBQP) has grown in importance in the field of combinatorial optimization due to its application potential and its computational challenge. Research on UBQP has generated a wide range of solution techniques for this basic model that encompasses a rich collection of problem types. In this paper we survey the literature on this important model, providing an overview of the applications and solution methods.

A new modeling and solution approach for the set-partitioning problem

The set-partitioning problem (SPP) is widely known for both its application potential and its computational challenge. This NP-hard problem is known to pose considerable difficulty for classical solution methods such as those based on LP technologies. In recent years, the unconstrained binary quadratic program has proven to perform well as a unified modeling and solution framework for a variety of IP problems. In this paper we illustrate how this unified framework can be applied to SPP. Computational experience is presented, illustrating the attractiveness of the approach.

A new approach for modeling and solving set packing problems

In recent years the unconstrained quadratic binary program has emerged as a unified modeling and solution framework for a variety of combinatorial optimization problems. Experience reported in the literature with several problem classes has demonstrated that this unified approach works surprisingly well in terms of solution quality and computational times, often rivaling and sometimes surpassing special purpose methods. In this paper we report on the application of this unified framework to set packing problems with a variety of coefficient distributions. Computational experience is reported illustrating the attractiveness of the approach. In particular, our results highlight both the effectiveness and robustness of our approach across a wide array of test problems.

Using xQx to model and solve the uncapacitated task allocation problem

This paper illustrates how large instances of the unconstrained task allocation problem can be effectively modeled and efficiently solved as unconstrained quadratic binary programs. Computational experience and a comparison to the state-of-the-art commercial code (CPLEX) illustrate the attractiveness of our approach.

The Path Restoration Version of the Spare Capacity Allocation Problem with Modularity Restrictions: Models, Algorithms, and an Empirical Analysis

This investigation presents a strategy to construct a compact mathematical model of the path-restoration version of the spare capacity allocation problem. The strategy uses a node-arc formulation and combines constraints whenever multiple working paths affected by an edge failure have identical origins or destinations. Another unique feature of this model is the inclusion of modularity restrictions corresponding to the discrete capacities of the equipment used in telecommunication networks.The new model can be solved using a classical branch-and-bound algorithm with a linear-programming relaxation. A preprocessing module is developed, which generates a set of cuts that strengthens this linear programming relaxation. The overhead associated with the cuts is offset by the improved bounds produced. A new branch-and-bound algorithm is developed that exploits the modularity restrictions. In an extensive empirical analysis, a software implementation of this algorithm was found to be substantially faster than CPLEX 6.5.3. For a test suite of 50 problems, each having 50 nodes and 200 demands from a uniform distribution with a small variance, our new software obtained solutions guaranteed to be within 4% of optimality in five minutes of CPU time on a DEC AlphaStation.

A note on xQx as a modelling and solution framework for the Linear Ordering Problem

This paper expands the list of 0-1 problems that can be effectively modelled and solved as Unconstrained Quadratic Binary Programs (UQPs). UQP has been presented as a general-purpose modelling approach with application to a broad range of problem classes (Kochenberger et al., 2004). In this paper, we demonstrate that the Linear Ordering Problem (LOP) can be easily recast so that it can be treated as a UQP problem, and that large instances of the LOP can be effectively handled within this framework. Computational results are given demonstrating the viability and attractiveness of this approach.

Exact solutions to generalized vertex covering problems: a comparison of two models

The generalized vertex cover problem (GVCP) was recently introduced in the literature and modeled as a binary linear program. GVCP extends classic vertex cover problems to include both node and edge weights in the objective function. Due to reported difficulties in finding good solutions for even modest sized problems via the use of exact methods (CPLEX), heuristic solutions obtained from a customized genetic algorithm (GA) were reported by Milanovic (Comput Inf 29:1251–1265, 2010). In this paper we consider an alternative model representation for GVCP that translates the constrained linear (binary) form to an unconstrained quadratic binary program and compare the linear and quadratic models via computations carried out by CPLEX’s branch-and-cut algorithms. For problems comparable to those previously studied in the literature, our results indicate that the quadratic model efficiently yields optimal solutions for many large GVCP problems. Moreover, our quadratic model dramatically outperforms the corresponding linear model in terms of time to reach and verify optimality and in terms of the optimality gap for problems where optimality is unattained.

Guided design search in the interval-bounded sailor assignment problem

The problem of assigning Navy personnel to jobs is primarily a manual process performed by enlisted detailers, with decision support from the Enlisted Assignment Information System. In this paper, we offer an expanded interval bounded network flow model of the sailor assignment process creating teams of skilled sailors to be assigned to ships. A new integer preprocessing and solution technique, Guided Design Search (GDS), is integrated into the CPLEX solver with promising results for these difficult problems. Computational results show GDS/CPLEX speed improvements of 10-fold to optimality and for larger problems found feasible assignments when CPLEX alone could not. We show how GDS results can be used by detailers to gauge the effectiveness of alternative sailor assignments and also how it can be used to validate the objective function coefficients of the decision variables.

Computationally attractive non-linear models for combinatorial optimisation

A common approach to many combinatorial problems is to model them as 0/1 linear programs. This approach enables the use of standard linear program-based optimisation methodologies that are widely employed by the operation research community. While this methodology has worked well for many problems, it can become problematic in cases where the linear programs generated become excessively large. In such cases, linear models can lose their computational viability. In recent years, several articles have explored the computational attractiveness of non-linear alternatives to the standard linear models typically adopted to represent such problems. In many cases, comparative computational testing yields results favouring the non-linear models by a wide margin. In this article, we summarise some of these successes in an effort to encourage a broader view of model construction than the conventional wisdom, i.e. linear modelling, typically affords.

Quadratic unconstrained binary optimization problem preprocessing: Theory and empirical analysis

The Quadratic Unconstrained Binary Optimization problem QUBO has become a unifying model for representing a wide range of combinatorial optimization problems, and for linking a variety of disciplines that face these problems. A new class of quantum annealing computer that maps QUBO onto a physical qubit network structure with specific size and edge density restrictions is generating a growing interest in ways to transform the underlying QUBO structure into an equivalent graph having fewer nodes and edges. In this article, we present rules for reducing the size of the QUBO matrix by identifying variables whose value at optimality can be predetermined. We verify that the reductions improve both solution quality and time to solution and, in the case of metaheuristic methods where optimal solutions cannot be guaranteed, the quality of solutions obtained within reasonable time limits. We discuss the general QUBO structural characteristics that can take advantage of these reduction techniques and perform careful experimental design and analysis to identify and quantify the specific characteristics most affecting reduction. The rules make it possible to dramatically improve solution times on a new set of problems using both the exact Cplex solver and a tabu search metaheuristic.