Computing and Listing st-Paths in Public Transportation Networks

Given a set of directed paths (called lines) l, a public transportation network is a directed graph g_l=(v_l,a_l)g_l=(v_l,a_l) which contains exactly the vertices and arcs of every line l\in ll\in l. An st-route is a pair (\pi ,\gamma )(\pi ,\gamma ) where \gamma =\langle l_1,\ldots ,l_h \rangle \gamma =\langle l_1,\ldots ,l_h \rangle is a line sequence and \pi \pi is an st-path in g_lg_l which is the concatenation of subpaths of the lines l_1,\ldots ,l_hl_1,\ldots ,l_h, in this order. Given a threshold \beta \beta , we present an algorithm for listing all st-paths \pi \pi for which a route (\pi ,\gamma )(\pi ,\gamma ) with |\gamma | \le \beta |\gamma | \le \beta exists, and we show that the running time of this algorithm is polynomial with respect to the input and the output size. We also present an algorithm for listing all line sequences \gamma \gamma with |\gamma |\le \beta |\gamma |\le \beta for which a route (\pi ,\gamma )(\pi ,\gamma ) exists, and show how to speed it up using preprocessing. Moreover, we show that for the problem of finding an st-route (\pi ,\gamma )(\pi ,\gamma ) that minimizes the number of different lines in \gamma \gamma , even computing an o(\log |v|)o(\log |v|)-approximation is np-hard.