Submodular integer cover and Its application to production planning

Abstract

Suppose there are a set of suppliers i and a set of consumers j with demands bj, and the amount of products that can be shipped from i to j is at most cij. The amount of products that a supplier i can produce is an integer multiple of its capacity ki, and every production of ki products incurs the cost of wi. The capacitated supply-demand (CSD) problem is to minimize the production cost of σi such that all the demands (or the total demand requirement specified separately) at consumers are satisfied by shipping products form the suppliers to them. To capture the core structural properties of CSD in a general frame-work, we introduce the submodular integer cover (SIC) problem, which extends the submodular set cover (SSC) problem by generalizing sub-modular constraints on subsets to those on integer vectors. Whereas it can be shown that CSD is approximable within a factor of by extending the greedy approach for SSC to CSD, we first generalize the primal-dual approach for SSC to SIC and evaluate its performance. One of the approximation ratios obtained for CSD from such an approach is the maximum number of suppliers that can ship to a single consumer; therefore, the approximability of CSD can be ensured to depend only on the network (incidence) structure and not on any numerical values of input capacities. The CSD problem also serves as a unifying framework for various types of covering problems, and any approximation bound for CSD holds for set cover generalized simultaneously into various directions. It will be seen, nevertheless, that our bound matches (or nearly matches) the best result for each generalization individually. Meanwhile, this bound being nearly tight for standard set cover, any further improvement, even if possible, is doomed to be a marginal one.