On the Conditional Covering Problem on Paths

Consider a graph G, in which the set of vertices represents the set of demand points as well as the set of potential facility sites. The classical covering location problem seeks to minimize the total location cost required to cover all vertices of G. The conditional covering problem (CCP) has the same objective but with an additional constraint: each selected facility site must be covered by at least one other selected site. The CCP arises in several military and security applications that are becoming increasing important. It is strongly NP-hard. In several special cases, polynomial results are known. In this paper, an O(nlog n)-time algorithm is presented for the CCP on a path. The presented algorithm improves the previous upper bound from O(n).