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 $S\subseteq E\times E$, in addition to the network tuple $G$. In the $s-t$ reachability problem in choice networks (OCR${}_D$), 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 $(x,y)\in S$ is used in the path. OCR${}_D$ finds applications in a number of domains, including routing in wireless networks and sensor placement. We analyze the computational complexities of the OCR${}_D$ 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 $O^{\ast }(1.42^{\left|S\right|})$. We also consider weighted versions of the OCR${}_D$ problem and detail their computational complexities; in particular, the optimization version of the $WOC{R}_D$ 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.