Conditional covering problem: Study of complexity and optimization methods

Abstract

The Conditional Covering Problem (CCP) is a facility location problem on a graph, where the set of nodes represents the demand points and the potential facility locations. The CCP minimizes the sum of the facility location costs required to cover all demand points. The key aspect of the CCP is that a facility covers all nodes within a given coverage radius, except for the node on which it is located. Our investigation of the CCP will first show that solving CCP on a general graph structure is NP-Hard, prohibiting finding exact optimal solutions in a finite amount of time. While the CCP is NP-Hard on general graphs, we will present a quadratic algorithm that will find optimal solutions to CCP on path and extended star graphs (multiple path graphs with one node in common). We will then present a polynomial time algorithm for tree graphs building off the quadratic algorithms for a star and tree. Given that CCP is NP-Hard on general graphs we next focused on determining near optimal solutions for general graphs. We applied both greedy heuristics and metaheuristics to determine near-optimal solutions. Building off our understanding of optimal solutions on tree structures, we incorporate the idea of trees into our heuristic search. We found that greedy heuristics provide near-optimal solutions in a very short period of time. We showed that simulated annealing with binary encoding provided higher quality solutions to the CCP.