Given a graph $G=(V,E)$ and a set $T=\{ (s_i, t_i) : 1\leq i\leq k \}\subseteq V\times V$ of $k$ pairs, the $k$-vertex-disjoint-paths (resp. $k$-edge-disjoint-paths) problem asks to determine whether there exist~$k$ pairwise vertex-disjoint (resp. edge-disjoint) paths $P_1, P_2, ..., P_k$ in $G$ such that, for each $1\leq i\leq k$, $P_i$ connects $s_i$ to $t_i$. Both the edge-disjoint and vertex-disjoint versions in undirected graphs are famously known to be FPT (parameterized by $k$) due to the Graph Minor Theory of Robertson and Seymour. Eilam-Tzoreff [DAM `98] introduced a variant, known as the $k$-disjoint-shortest-paths problem, where each individual path is further required to be a shortest path connecting its pair. They showed that the $k$-disjoint-shortest-paths problem is NP-complete on both directed and undirected graphs; this holds even if the graphs are planar and have unit edge lengths. We focus on four versions of the problem, corresponding to considering edge/vertex disjointness, and to considering directed/undirected graphs. Building on the reduction of Chitnis [SIDMA `23] for $k$-edge-disjoint-paths on planar DAGs, we obtain the following inapproximability lower bound for each of the four versions of $k$-disjoint-shortest-paths on $n$-vertex graphs: - Under Gap-ETH, there exists a constant $\delta>0$ such that for any constant $0<\epsilon\leq \frac{1}{2}$ and any computable function $f$, there is no $(\frac{1}{2}+\epsilon)$-approx in $f(k)\cdot n^{\delta\cdot k}$ time. We further strengthen our results as follows: Directed: Inapprox lower bound for edge-disjoint (resp. vertex-disjoint) paths holds even if the input graph is a planar (resp. 1-planar) DAG with max in-degree and max out-degree at most $2$. Undirected: Inapprox lower bound for edge-disjoint (resp. vertex-disjoint) paths hold even if the input graph is planar (resp. 1-planar) and has max degree $4$.

翻译：给定图 $G=(V,E)$ 及 $k$ 个点对构成的集合 $T=\{ (s_i, t_i) : 1\leq i\leq k \}\subseteq V\times V$，$k$-点不相交路径（相应地，$k$-边不相交路径）问题要求判断在 $G$ 中是否存在 $k$ 条两两点不相交（相应地，边不相交）的路径 $P_1, P_2, ..., P_k$，使得对每个 $1\leq i\leq k$，$P_i$ 连接 $s_i$ 与 $t_i$。由于 Robertson 与 Seymour 的图子式理论，无向图中的边不相交与点不相交版本均被证明是参数 $k$ 下的 FPT 问题。Eilam-Tzoreff [DAM `98] 引入了一个变体，称为 $k$-不相交最短路径问题，其中每条路径还需是其对应点对间的最短路径。他们证明了 $k$-不相交最短路径问题在有向图与无向图上均是 NP 完全的；即使图是平面图且边权为单位长度，该结论依然成立。我们关注该问题的四个版本，分别对应于考虑边/点不相交性，以及考虑有向/无向图。基于 Chitnis [SIDMA `23] 针对平面有向无环图上 $k$-边不相交路径问题的归约，我们对 $n$ 个顶点图上的 $k$-不相交最短路径问题的四个版本分别得到以下不可近似性下界：- 在 Gap-ETH 假设下，存在常数 $\delta>0$，使得对任意常数 $0<\epsilon\leq \frac{1}{2}$ 及任意可计算函数 $f$，不存在在 $f(k)\cdot n^{\delta\cdot k}$ 时间内达到 $(\frac{1}{2}+\epsilon)$-近似的算法。我们进一步强化结果如下：有向图：边不相交（相应地，点不相交）路径的不可近似下界即使输入图是最大入度与最大出度至多为 $2$ 的平面（相应地，1-平面）有向无环图时依然成立。无向图：边不相交（相应地，点不相交）路径的不可近似下界即使输入图是平面（相应地，1-平面）且最大度为 $4$ 时依然成立。