Reachability in choice networks

Abstract

In this paper, we investigate the problem of determining s−t reachability in choice networks. In the traditional s−t reachability problem, we are given a weighted network tuple G=〈V,E,c,s,t〉, with the goal of checking if there exists a path from s to t in G. In an optional choice network, we are given a choice set S⊆E×E, in addition to the network tuple G. In the s−t reachability problem in choice networks (OCRD), the goal is to find whether there exists a path from vertex s to vertex t, with the caveat that at most one edge from each edge-pair (x,y)∈S is used in the path. OCRD finds applications in a number of domains, including routing in wireless networks and sensor placement. We analyze the computational complexities of the OCRD problem and its variants from a number of algorithmic perspectives. We show that the problem is NP-complete in directed acyclic graphs with bounded pathwidth. Additionally, we show that its optimization version is NPO PB-complete. Additionally, we show that the problem is fixed-parameter tractable in the cardinality of the choice set S. In particular, we show that the problem can be solved in time O∗(1.42|S|). We also consider weighted versions of the OCRD problem and detail their computational complexities; in particular, the optimization version of the WOCRD problem is NPO-complete. While similar results have been obtained for related problems, our results improve on those results by providing stronger results or by providing results for more limited graph types.