Parameterized Complexity of Finding Dissimilar Shortest Paths

Abstract

We consider the problem of finding ``dissimilar'' $k$ shortest paths from $s$ to $t$ in an edge-weighted directed graph $D$, where the dissimilarity is measured by the minimum pairwise Hamming distances between these paths. More formally, given an edge-weighted directed graph $D=(V,A)$, two specified vertices $s,t\in V$, and integers $d,k$, the goal of Dissimilar Shortest Paths is to decide whether $D$ has $k$ shortest paths $P_1,\dots ,P_k$ from $s$ to $t$ such that $|A(P_i)△A(P_j)|\ge d$ for distinct $P_i$ and $P_j$. We design a deterministic algorithm to solve Dissimilar Shortest Paths with running time $2^{O(3^{k}d{k}^2)}{n}^{O(1)}$, that is, Dissimilar Shortest Paths is fixed-parameter tractable parameterized by $k+d$. To complement this positive result, we show that Dissimilar Shortest Paths is W[1]-hard when parameterized by only $k$ and paraNP-hard parameterized by $d$.

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