# Tight Approximation and Kernelization Bounds for Vertex-Disjoint Shortest Paths

## TL;DR

This paper studies the limits of approximating maximum vertex-disjoint shortest paths in graphs and provides tight bounds for approximation algorithms.

## Contribution

The paper establishes tight approximation and kernelization bounds for the Maximum Vertex-Disjoint Shortest Paths problem.

## Key findings

- Assuming gap-ETH, no o(k)-approximation exists in f(k)poly(n) time.
- An m^(1/2 - ε)-approximation is impossible in polynomial time unless P = NP.
- A simple √ℓ-approximation algorithm is provided, matching the lower bound.

## 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 \documentclass[12pt]{minimal}
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\begin{document}$$(s_1,t_1),(s_2,t_2),\ldots ,(s_k,t_k)$$\end{document}(s1,t1),(s2,t2),…,(sk,tk). 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 \documentclass[12pt]{minimal}
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\begin{document}$$c \le 1$$\end{document}c≤1 is a constant. (One can guess 1/c terminal pairs to connect in \documentclass[12pt]{minimal}
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\begin{document}$$k^{O({1}/{c})}$$\end{document}kO(1/c) time and then utilize Lochet’s algorithm to compute the solution in \documentclass[12pt]{minimal}
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\begin{document}$$n^{f({1}/{c})}$$\end{document}nf(1/c) time.) 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)-approximation within \documentclass[12pt]{minimal}
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\begin{document}$$f(k){{\,\textrm{poly}\,}}(n)$$\end{document}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 \documentclass[12pt]{minimal}
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\begin{document}$$m^{{1}/{2}-\varepsilon }$$\end{document}m1/2-ε in polynomial time, unless P \documentclass[12pt]{minimal}
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\begin{document}$$=$$\end{document}= NP. We also show that this bound is tight by providing a simple \documentclass[12pt]{minimal}
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\begin{document}$$\sqrt{\ell }$$\end{document}ℓ-approximation algorithm, where \documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}ℓ is the number of edges in all paths of an optimal solution. Additionally, we establish that Maximum Vertex-Disjoint Shortest Paths can be solved in \documentclass[12pt]{minimal}
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\begin{document}$$2^{O(\ell )} {{\,\textrm{poly}\,}}(n)$$\end{document}2O(ℓ)poly(n) time, but does not admit a polynomial kernel in \documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}ℓ. Moreover, it cannot be solved in \documentclass[12pt]{minimal}
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\begin{document}$$2^{o(\ell )}{{\,\textrm{poly}\,}}(n)$$\end{document}2o(ℓ)poly(n) time under ETH. 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.