Characterisations of Totally Unimodular, Balanced and Perfect Matrices

Abstract

We consider combinatorial programming problems of the form (IP): max{cx|Ax≤e, x_j=0 or 1, ∀_j, where A is a mxn matrix of zeroes and ones, e is a column vector of m ones and c is an arbitrary (non-negative) vector of n reals. This class of problems is known as the set packing problem, see e.g. (1). It is closely related to the set partitioning problem (SPP) and to the set covering problem. In the former case, the inequality constraints Ax≤e of (IP) are replaced by equality constraints Ax=e, whereas in the latter case one requires the constraints to hold with reversed inequality, i.e. Ax≥e. With respect to the set partitioning problem (SPP), it can be shown, that by appropriately modifying the objective function, the problem (SPP) can always be transformed into the form (IP) above. This is, however, not true in general if the set covering problem is considered. It is true, however, if the matrix A has at most two +1 entries per row, i.e. if the set covering problem assumes the special form of a node-covering problem in a (finite undirected) graph. As has been noted in (2), some of the structural properties of set partitioning and set packing problems do not carry over to the (general) set covering problem.