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In this problem, the input is an edge-weighted (directed or undirected)\n n<\/jats:italic>\n -vertex graph\n G<\/jats:italic>\n along with\n k<\/jats:italic>\n \u00a0terminal pairs\n \n \n $$(s_1,t_1),(s_2,t_2),\\ldots ,(s_k,t_k)$$<\/jats:tex-math>\n \n \n \n (<\/mml:mo>\n \n s<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n t<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n ,<\/mml:mo>\n \n (<\/mml:mo>\n \n s<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n t<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n ,<\/mml:mo>\n \u2026<\/mml:mo>\n ,<\/mml:mo>\n \n (<\/mml:mo>\n \n s<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n ,<\/mml:mo>\n \n t<\/mml:mi>\n k<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . 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 \u201921], which demonstrates the polynomial-time solvability of the problem for a fixed value of\n k<\/jats:italic>\n . Lochet\u2019s result implies the existence of a polynomial-time\n ck<\/jats:italic>\n -approximation for\n Maximum Vertex-Disjoint Shortest Paths<\/jats:sc>\n , where\n \n \n $$c \\le 1$$<\/jats:tex-math>\n \n \n c<\/mml:mi>\n \u2264<\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is a constant. (One can guess 1\/\n c<\/jats:italic>\n terminal pairs to connect in\n \n \n $$k^{O({1}\/{c})}$$<\/jats:tex-math>\n \n \n k<\/mml:mi>\n \n O<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n c<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n \u00a0time and then utilize Lochet\u2019s algorithm to compute the solution in\n \n \n $$n^{f({1}\/{c})}$$<\/jats:tex-math>\n \n \n n<\/mml:mi>\n \n f<\/mml:mi>\n (<\/mml:mo>\n 1<\/mml:mn>\n \/<\/mml:mo>\n c<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n \u00a0time.) 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\u00a0\n o<\/jats:italic>\n (\n k<\/jats:italic>\n )-approximation within\n \n \n $$f(k){{\\,\\textrm{poly}\\,}}(n)$$<\/jats:tex-math>\n \n \n f<\/mml:mi>\n (<\/mml:mo>\n k<\/mml:mi>\n )<\/mml:mo>\n \n \n poly<\/mml:mtext>\n \n <\/mml:mrow>\n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n \u00a0time for any function\n f<\/jats:italic>\n that only depends on\n k<\/jats:italic>\n . Our second result demonstrates the infeasibility of achieving an approximation ratio of\n \n \n $$m^{{1}\/{2}-\\varepsilon }$$<\/jats:tex-math>\n \n \n m<\/mml:mi>\n \n 1<\/mml:mn>\n \/<\/mml:mo>\n 2<\/mml:mn>\n -<\/mml:mo>\n \u03b5<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n in polynomial time, unless P\n \n \n $$=$$<\/jats:tex-math>\n \n =<\/mml:mo>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n NP. We also show that this bound is tight by providing a simple\n \n \n $$\\sqrt{\\ell }$$<\/jats:tex-math>\n \n \n \u2113<\/mml:mi>\n <\/mml:msqrt>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n -approximation algorithm, where\n \n \n $$\\ell $$<\/jats:tex-math>\n \n \u2113<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n is the number of edges in all paths of an optimal solution. Additionally, we establish that\n Maximum Vertex-Disjoint Shortest Paths<\/jats:sc>\n \u00a0can be solved in\n \n \n $$2^{O(\\ell )} {{\\,\\textrm{poly}\\,}}(n)$$<\/jats:tex-math>\n \n \n \n 2<\/mml:mn>\n \n O<\/mml:mi>\n (<\/mml:mo>\n \u2113<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n \n \n poly<\/mml:mtext>\n \n <\/mml:mrow>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n time, but does not admit a polynomial kernel in\n \n \n $$\\ell $$<\/jats:tex-math>\n \n \u2113<\/mml:mi>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\n . Moreover, it cannot be solved in\n \n \n $$2^{o(\\ell )}{{\\,\\textrm{poly}\\,}}(n)$$<\/jats:tex-math>\n \n \n \n 2<\/mml:mn>\n \n o<\/mml:mi>\n (<\/mml:mo>\n \u2113<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:msup>\n \n \n poly<\/mml:mtext>\n \n <\/mml:mrow>\n \n (<\/mml:mo>\n n<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n <\/mml:mrow>\n <\/mml:math>\n <\/jats:alternatives>\n <\/jats:inline-formula>\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.\n <\/jats:p>","DOI":"10.1007\/s00224-025-10252-9","type":"journal-article","created":{"date-parts":[[2026,2,5]],"date-time":"2026-02-05T09:02:50Z","timestamp":1770282170000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Tight Approximation and Kernelization Bounds for Vertex-Disjoint Shortest Paths"],"prefix":"10.1007","volume":"70","author":[{"given":"Matthias","family":"Bentert","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fedor V.","family":"Fomin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Petr A.","family":"Golovach","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,2,5]]},"reference":[{"key":"10252_CR1","doi-asserted-by":"crossref","unstructured":"Bentert, M., Fomin, F.V., Golovach, P.A.: Tight approximation and kernelization bounds for vertex-disjoint shortest paths. 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