Analyzing the Reachability Problem in Choice Networks

Piotr Wojciechowski, K. Subramani, Alvaro Velasquez

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 $$\mathbf{G}= \langle V, E, \mathbf{c}, s, t\rangle $$ , 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 arc from each arc-pair $$(e_i,e_j) \in S$$ 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 and 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. We also consider weighted versions of the $$OCR_{D}$$ problem and detail their computational complexities; in particular, the optimization version of the $$WOCR_{D}$$ problem is NPO-complete.