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 $|A(P_i) \mathbin{\triangle} A(P_j)| \ge d$ for distinct $P_i$ and $P_j$. We design a deterministic algorithm to solve Dissimilar Shortest Paths with running time $2^{O(3^kdk^2)}n^{O(1)}$, 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 $d$.

翻译：我们研究在边加权有向图 $D$ 中寻找从 $s$ 到 $t$ 的"差异" $k$ 条最短路径问题，其中差异度通过路径间的最小成对汉明距离衡量。更形式化地说，给定边加权有向图 $D = (V, A)$、两个指定顶点 $s, t \in V$ 以及整数 $d, k$，差异最短路径问题的目标是判断 $D$ 中是否存在 $k$ 条从 $s$ 到 $t$ 的最短路径 $P_1, \dots, P_k$，使得对任意不同的 $P_i$ 与 $P_j$ 满足 $|A(P_i) \mathbin{\triangle} A(P_j)| \ge d$。我们设计了一个运行时间为 $2^{O(3^kdk^2)}n^{O(1)}$ 的确定性算法求解差异最短路径问题，即该问题在参数 $k + d$ 下是固定参数可解的。为补充这一正面结果，我们证明当仅以 $k$ 为参数时差异最短路径问题是 W[1]-难问题，而以 $d$ 为参数时是 paraNP-难问题。