Year: 2026
Type: article
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 $$(s_1,t_1),(s_2,t_2),\ldots ,(s_k,t_k)$$ ( s 1 , t 1 ) , ( s 2 , t 2 ) , … , ( 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 $$c \le 1$$ c ≤ 1 is a constant. (One can guess 1/ c terminal pairs to connect in $$k^{O({1}/{c})}$$ k O ( 1 / c ) time and then utilize Lochet’s algorithm to compute the solution in $$n^{f({1}/{c})}$$ n f ( 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 $$f(k){{\,\textrm{poly}\,}}(n)$$ 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 $$m^{{1}/{2}-\varepsilon }$$ m 1 / 2 - ε in polynomial time, unless P $$=$$ = NP. We also show that this bound is tight by providing a simple $$\sqrt{\ell }$$ ℓ -approximation algorithm, where $$\ell $$ ℓ
Source: Theory of Computing Systems
Institutions: University of Bergen, Technische Universität Berlin
Language: English
Subfield: Computational Theory and Mathematics
Field: Computer Science
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,190
Awards: G5355697469, G5399623649
OpenAlex ID: W7128005051