We examine the possibility of approximating Maximum Vertex-Disjoint Shortest Paths. In this problem, the input is an edge-weighted (directed or undirected) $n$-vertex graph $G$ along with $k$ terminal pairs $(s_1,t_1),(s_2,t_2),\ldots,(s_k,t_k)$. The task is to connect as many terminal pairs as possible by pairwise vertex-disjoint paths such that each path is a shortest path between the respective terminals. Our work is anchored in the recent breakthrough by Lochet [SODA '21], which demonstrates the polynomial-time solvability of the problem for a fixed value of $k$. Lochet's result implies the existence of a polynomial-time $ck$-approximation for Maximum Vertex-Disjoint Shortest Paths, where $c \leq 1$ is a constant. Our first result suggests that this approximation algorithm is, in a sense, the best we can hope for. More precisely, assuming the gap-ETH, we exclude the existence of an $o(k)$-approximations within $f(k) \cdot $poly($n$) time for any function $f$ that only depends on $k$. Our second result demonstrates the infeasibility of achieving an approximation ratio of $n^{\frac{1}{2}-\varepsilon}$ in polynomial time, unless P = NP. It is not difficult to show that a greedy algorithm selecting a path with the minimum number of arcs results in a $\lceil\sqrt{\ell}\rceil$-approximation, where $\ell$ is the number of edges in all the paths of an optimal solution. Since $\ell \leq n$, this underscores the tightness of the $n^{\frac{1}{2}-\varepsilon}$-inapproximability bound. Additionally, we establish that Maximum Vertex-Disjoint Shortest Paths is fixed-parameter tractable when parameterized by $\ell$ but does not admit a polynomial kernel. Our hardness results hold for undirected graphs with unit weights, while our positive results extend to scenarios where the input graph is directed and features arbitrary (non-negative) edge weights.

翻译：我们研究了最大顶点不相交最短路径问题的近似可能性。在该问题中，输入为一个边加权（有向或无向）的 $n$ 顶点图 $G$，以及 $k$ 个终端对 $(s_1,t_1),(s_2,t_2),\ldots,(s_k,t_k)$。目标是通过成对顶点不相交的路径连接尽可能多的终端对，且每条路径均为对应终端之间的最短路径。我们的工作基于 Lochet [SODA '21] 的最新突破，该工作证明了问题在固定 $k$ 下的多项式时间可解性。Lochet 的结果意味着存在一种多项式时间的 $ck$-近似算法用于最大顶点不相交最短路径问题，其中 $c \leq 1$ 为常数。我们的第一个结果表明，这种近似算法在某种意义上是我们所能期待的最佳结果。更精确地，假设 gap-ETH 成立，我们排除了在 $f(k) \cdot \text{poly}(n)$ 时间内存在 $o(k)$-近似算法的可能性，其中 $f$ 为仅依赖于 $k$ 的任意函数。我们的第二个结果表明，除非 P = NP，否则在多项式时间内无法达到 $n^{\frac{1}{2}-\varepsilon}$ 的近似比。不难证明，选择具有最少弧数的路径的贪心算法可实现 $\lceil\sqrt{\ell}\rceil$-近似，其中 $\ell$ 为最优解中所有路径的边数。由于 $\ell \leq n$，这凸显了 $n^{\frac{1}{2}-\varepsilon}$ 不可近似性界的紧性。此外，我们证明最大顶点不相交最短路径问题在以 $\ell$ 为参数时是固定参数可解的，但不存在多项式核。我们的困难结果对单位权重的无向图成立，而正面结果则扩展到输入图为有向图且具有任意（非负）边权重的场景。