The convex hull of a linear congruence relation in zero-one variables

Abstract

A class of optimization problems over subsets of zero-one vectors of then-dimensional unit cube given by a special linear congruence relation is considered. The general problem is formulated as a zero-one linear program, minimal and complete descriptions of the associated polytopes by linear inequalities are derived and an$\mathcal{O}(n \log n)$ time algorithm for the optimization problems is given. Since the number of inequalities that completely describe the polytope grows exponentially withn, we also give a separation algorithm that identifies violated inequalities in time$\mathcal{O}(n^2 )$. A particular variation of the bin packing problem is a special case of our problem and can thus be solved in polynomial time.