Formal Language Constrained Path Problems

Given an alphabet , a (directed) graph G whose edges are weighted and -labeled, and a formal language L , we consider the problem of finding a shortest (simple) path p in G complying with the additional constraint that l(p) 2 L. Here l(p) denotes the unique word given by concatenating the -labels of the edges along the path p. The main contributions of this paper include the following: 1. We show that the problem of finding a shortest path between source and destination for a traveler whose mode choice is specified as a context free language is solvable efficiently in polynomial time. When the mode choice is specified as a regular language we provide algorithms with improved space and time bounds. 2. In contrast, we show that the problem of finding simple paths between a source and a given destination isNP-hard, even when restricted to fixed simple regular languages and very simple graphs (complete grids) 3. For the class of treewidth bounded graphs, we show that (i) the problem of finding a regular language constrained simple path between source and destination is solvable in polynomial time and (ii) the extension to finding context free language constrained simple paths isNP-complete. Our results extend the previous results in [MW95, SJB97, Ya90]. Several additional extensions and applications of our results in the context of transportation problems are presented. For instance, as a corollary of our results, we obtain a polynomial time algorithm for the BEST k-SIMILAR PATH problem studied in [SJB97]. The previous best algorithm was given by [SJB97] and takes exponential time in the worst case.