Shortest Two Disjoint Paths in Polynomial Time

Given an undirected graph and two pairs of vertices $(s_i,t_i)$ for $i\in\{1,2\}$ we show that there is a polynomial time Monte Carlo algorithm that finds disjoint paths of smallest total length joining $s_i$ and $t_i$ for $i\in\{1,2\}$ respectively, or concludes that there most likely are no such paths at all. Our algorithm applies to both the vertex- and edge-disjoint versions of the problem. Our algorithm is algebraic and uses permanents over the polynomial rings $Z_2[x]$ and $Z_4[x]$ in combination with Mulmuley, Vazirani and Vazirani's isolation lemma to detect a solution. We develop a fast algorithm for permanents over these rings by modifying Valiant's 1979 algorithm for the permanent over $Z_{2^l}$