Conflict graphs in implicit enumeration

Szabó, Sándor (2012) Conflict graphs in implicit enumeration. Pollack Periodica, 7 (Supple). pp. 145-156. ISSN 1788-1994

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Abstract

A zero-one linear program is a global discrete optimization problem. Namely, it is a linear programming problem with zero-one variables. The real relaxation of a zero-one program is again a continuous linear programming problem. Simply, the zero-one variables are replaced by continuous variables varying between zero and one independently of each other. If the real relaxation, as a continuous problem, solved with the simplex method happens to have a zero-one optimal solution, then this particular solution is also an optimal solution of the original zero-one programming problem. Suppose that x₁,…,x_n are all the variables of a given zero-one linear programming problem. For convenience we introduce further variables y₁,…,y_n defined by y₁=1−x₁,…, y_n=1−x_n respectively. For a unified notation the variables x₁,…,x_n and y₁,…, y_n together will be denoted by z₁,…,z_2n. These variables will play the roles the nodes of the conflict graph Г associated with the given zero-one linear programming problem. Fixing the variables z_i and z_j both to be equal to one simultaneously reduces the number of variables in the problem. If the new smaller optimization problem does not have any feasible solution, then the nodes z_i and z_j will be connected by an edge in the graph Γ. The graph Γ simply records that there is a conflict between the assignments z_i =1 and z_j =1 which explains the name conflict graph. In other words the non-directed edge between the nodes z_i and z_j codes the fact that the inequality z_i+z_j≤1holds. The conflict graph was designed to generate additional constraints the so-called valid inequalities or cuts in order to expedite the solution of the zero-one linear program via its real relaxation. This particular solution strategy is aptly named the branch and cut method. It will be shown that the conflict graph besides its intended use can also be applied in another solution technique the method of implicit enumeration. We will illustrate by examples that the conflict graph provides us with rules to prune the search tree that are not offered by the commonly applied pruning rules. In addition the computations involved can easily be organized into a highly parallel computational scheme.

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