A Min-Edge Cost Flow Framework for Capacitated Covering Problems

Abstract

In this work, we introduce the Cov-MECF framework, a special case of minimum-edge cost flow in which the input graph is bipartite. We observe that several important covering (and multi-covering) problems are captured in this unifying model and introduce a new heuristic LP O for any problem in this framework. The essence of LP O harnesses as an oracle the fractional solution in deciding how to greedily modify the partial solution. We empirically establish that this heuristic returns solutions that are higher in quality than those of Wolsey's algorithm. We also apply the analogs of Leskovec et. al. 's [25] optimization to LP O and introduce a further freezing optimization to both algorithms. We observe that the former optimization generally benefits LP O more than Wolsey's algorithm, and that the additional freezing step often corrects suboptimalities while further reducing the number of subroutine calls. We tested these implementations on randomly generated testbeds, several instances from the Second DIMACS Implementation Challenge and a couple networks modeling realworld dynamics.

Hard Capacitated Set Cover and Uncapacitated Geometric Set Cover

“…Flow networks have played an important role in design of algorithms and in other areas of computer science like network design, computer vision, etc. Many of the covering problems are visualized as flow network problems [4], since there is a rich literature of network flow problems available. In capacitated set cover problem with hard capacities there is a ground set X = {1, .., n} of elements and a collection of its subset S. Each S ∈ S has a capacity k(S) and a non-negative cost w(S) associated with it.…”

“…One of the motivating applications is [4]; i.e. GMID (Glyco-molecule ID) which is a chip-based technology that is used to generate fingerprints which uniquely identify glycomolecules.…”