{"paper":{"title":"On the Complexity of the Minimum-($k,\\rho$)-Shortcut Problem","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The minimum (k,ρ)-shortcut problem is NP-hard for k≥2 and ρ≥k+2 in both directed and undirected graphs.","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Alexander Leonhardt, Conrad Schecker, Julian Christoph Brinkmann, Tatiana Rocha Avila","submitted_at":"2026-05-13T12:59:40Z","abstract_excerpt":"We consider the Minimum-$(k,\\rho)$-$\\mathrm{Shortcut}$ problem ($\\min(k,\\rho)\\text{-}\\mathrm{Shortcut}$), where the goal is to find the smallest set of shortcut edges such that every vertex in a given graph can reach its $\\rho$ closest vertices using paths of at most $k$ edges. This is a fundamental graph optimization problem used to accelerate parallel shortest path algorithms.\n  It is well-known that the problem is trivially solvable for the cases $k=1$ and $k\\geq\\rho$. While recent work by Leonhardt, Meyer, and Penschuck (ESA 2024) showed that in undirected graphs $\\min(k,\\rho)\\text{-}\\math"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We present a simpler and more direct reduction from the Hitting Set problem which establishes that min(k,ρ)-Shortcut is NP-hard for k≥2 and ρ≥k+2 in both directed and undirected graphs.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The constructed reduction from Hitting Set instances to shortcut instances is polynomial-time computable and correctly preserves yes/no answers.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The Minimum-(k,ρ)-Shortcut problem is NP-hard for k≥2 and ρ≥k+2 in directed and undirected graphs, while undirected graphs with ρ=k+1 are solvable in polynomial time.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The minimum (k,ρ)-shortcut problem is NP-hard for k≥2 and ρ≥k+2 in both directed and undirected graphs.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"1ad621fb960803ae732d7255b2b39d171d1664803d5e5a4d9a74ef2c4606b07c"},"source":{"id":"2605.13474","kind":"arxiv","version":1},"verdict":{"id":"7cbfe22a-d94f-459e-a795-a7e22c9c92eb","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-14T18:25:24.753965Z","strongest_claim":"We present a simpler and more direct reduction from the Hitting Set problem which establishes that min(k,ρ)-Shortcut is NP-hard for k≥2 and ρ≥k+2 in both directed and undirected graphs.","one_line_summary":"The Minimum-(k,ρ)-Shortcut problem is NP-hard for k≥2 and ρ≥k+2 in directed and undirected graphs, while undirected graphs with ρ=k+1 are solvable in polynomial time.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The constructed reduction from Hitting Set instances to shortcut instances is polynomial-time computable and correctly preserves yes/no answers.","pith_extraction_headline":"The minimum (k,ρ)-shortcut problem is NP-hard for k≥2 and ρ≥k+2 in both directed and undirected graphs."},"references":{"count":19,"sample":[{"doi":"","year":2003,"title":"2003 , url =","work_id":"a41c2cb3-4021-4193-ab24-7ce4f9df62cc","ref_index":1,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2024,"title":"2024 , doi =","work_id":"bfb9204f-16dc-48d3-bcb8-13ce5e5de21b","ref_index":2,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"10.1016/0890-5401(90)90043-h","year":1990,"title":"The Monadic Second-Order Logic of Graphs","work_id":"69e8ab4c-158e-499e-8f23-ebc09d518761","ref_index":3,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2003,"title":"2003 , doi =","work_id":"360be629-9503-44d9-b930-2a794a6ea8c9","ref_index":4,"cited_arxiv_id":"","is_internal_anchor":false},{"doi":"","year":2018,"title":"Berndt and Sun Kim and Alexandru Zaharescu , title =","work_id":"ec284562-dc13-4a7a-9a2c-4b49eef9610c","ref_index":5,"cited_arxiv_id":"","is_internal_anchor":false}],"resolved_work":19,"snapshot_sha256":"e459083ee19430afee34b6fe036ea2a374aa8a43efff1f3117237d6b9b034a42","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}