LP as a Global Search Heuristic Across Different Constrainedness Regions

We study the behavior of heuristics based on the LP relaxation with respect to the underlying constraindness of the problem. Our study focuses on the Latin square (or quasigroup) completion problem [1]) as a prototype for highly combinatorial problems. We find that simple techniques based on the LP relaxation of the problem provide satisfactory guidance for underand over-constrained instances. In the critically constrained region, the performance of such simple techniques degrades, due to the inherit hardness of the problem. In this setting, we examine a technique that recomputes the LP relaxation every time a variable is set. This leads to a significant increase in performance, suggesting that carefully designed “one step at a time” LP based heuristics could provide suitable guidance even for the hardest instances. Recent years have witnessed the emergence of a new area involving hybrid solvers integrating CPand OR-based methods. OR has a long and rich history of using Linear Programming (LP) based relaxations for (Mixed) Integer Programming problems. In this approach, the LP relaxation provides bounds on overall solution quality and can be used for pruning in a branch-and-bound approach. This is particularly true in domains where we have a combination of linear constraints, well-suited for linear programming (LP) formulations, and discrete constraints, suited for constraint satisfaction problem (CSP) formulations. Nevertheless, in a purely combinatorial setting, so far it has been surprisingly difficult to integrate LP-based and CSP-based techniques. For example, despite significant LP results for Boolean satisfiability (SAT) problems (see e.g., [2–5]), practical state-of-the-art solvers do not yet incorporate LP relaxation techniques. We examine the quality of the guidance provided by the LP relaxation as a function of the structure of the problem, i.e., we attempt to characterize the performance of LP heuristics across different constraindness regions in the search space. The problem used for our study is the Latin square completion problem. A Latin square is an n by n matrix, where each cell has one of n symbols (or colors), such that each symbol occurs exactly once in each row and column. Given ? Research supported by the Intelligent Information Systems Institute, Cornell University (AFOSR grant F49620-01-1-0076). 1 The multiplication table of a quasigroup is a Latin square. a partial coloring of the n by n cells of a Latin square, determining whether there is a valid completion into a full Latin square is an NP-complete problem [6]. The underlying structure of this problem is similar to that found in a series of realworld applications, such as timetabling, experimental design, and fiber optics routing problems [7, 8]. We start by considering the LP assignment formulation [8]. In this formulation, we have n variables, some of them with pre-assigned values. Each variable, xijk (i, j, k = 1, 2 . . . , n), is a 0/1 variable that takes the value 1 if cell (i, j) is colored with color k. The objective function is to maximize the total number of colored cells in the Latin square: