Tight Approximation and Kernelization Bounds for Vertex-Disjoint Shortest Paths

TL;DR

This paper investigates the approximability and kernelization bounds for the problem of finding maximum vertex-disjoint shortest paths, establishing tight inapproximability results and exploring algorithmic complexities under various assumptions.

Contribution

It provides the first tight bounds on approximation ratios for the problem, showing the limits of polynomial-time algorithms and kernelization, and extends results to directed graphs with arbitrary weights.

Findings

Excludes $o(k)$-approximation algorithms assuming gap-ETH.

Shows inapproximability within $n^{1/2- ext{epsilon}}$ unless P=NP.

Establishes exponential-time algorithms and kernelization hardness results.

Abstract

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 $f(k)$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 $\sqrt{\ell}$-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 the problem can be solved in $2^{O(\ell)}$poly($n$) time, but does not admit a polynomial kernel in $\ell$. Moreover, it cannot be solved in $2^{o(\ell)}$poly($n$) time unless the ETH fails. 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.