Computing and Listing st-Paths in Subway Networks

Given a set of directed paths called linesL, a public transportation network is a directed graph $$G_L=V_L,A_L$$ which contains exactly the vertices and arcs of every line $$l\in L$$. An st-route is a pair $$\pi ,\gamma $$ where $$\gamma =\langle l_1,\ldots ,l_h \rangle $$ is a line sequence and $$\pi $$ is an st-path ini?ź$$G_L$$ which is the concatenation of subpaths of the lines $$l_1,\ldots ,l_h$$, in this order. Given a thresholdi?ź$$\beta $$, we present an algorithm for listing all st-pathsi?ź$$\pi $$ for which a route $$\pi ,\gamma $$ with $$|\gamma | \le \beta $$ 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 sequencesi?ź$$\gamma $$ with $$|\gamma |\le \beta $$ for which a route $$\pi ,\gamma $$ exists, and show how to speed it up using preprocessing. Moreover, we show that for the problem of finding an st-route $$\pi ,\gamma $$ that minimizes the number of different lines ini?ź$$\gamma $$, even computing an $$o\log |V|$$-approximation is NP-hard.