Using a geometric lens to find k disjoint shortest paths
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Given an undirected n-vertex graph and k pairs of terminal vertices (s_1,t_1), …, (s_k,t_k), the k-Disjoint Shortest Paths (k-DSP)-problem asks whether there are k pairwise vertex-disjoint paths P_1,…, P_k such that P_i is a shortest s_i-t_i-path for each i ∈ [k]. Recently, Lochet [arXiv 2019] provided an algorithm that solves k-DSP in n^O(k^4^k) time, answering a 20-year old question about the computational complexity of k-DSP for constant k. On the one hand, we present an improved O(k n^12k · k! + k + 1)-time algorithm based on a novel geometric view on this problem. For the special case k=2, we show that the running time can be further reduced to O(n^2m) by small modifications of the algorithm and a further refined analysis. On the other hand, we show that k-DSP is W[1]-hard with respect to k, showing that the dependency of the degree of the polynomial running time on the parameter k is presumably unavoidable.
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