{
  "schema_version": 1,
  "problem_number": "OWR-1111-001",
  "title": "Blocking Shortest Paths between Three Terminals Is NP-Hard: A Reduction from Three-Terminal Cut",
  "author": {
    "name": "Alper Ferudun",
    "affiliation": "Mercury Software GmbH",
    "url": "https://github.com/AlperTheKing"
  },
  "abstract": "In the problem Blocking Shortest Paths one is given an undirected graph with edge costs and some pairs of vertices, and asks for an edge set of minimum cost whose deletion increases the distance of each pair by at least 1; that is, the set must meet every shortest path between the two vertices of each pair. Krause showed that the problem can be solved in polynomial time when all pairs have a vertex in common and that it is NP-hard for two disjoint pairs, and left open the remaining case of three pairs: the three pairs of three given vertices (Oberwolfach Report 50/2005; thesis, Braunschweig 2006). We show that this case is NP-hard, with edge costs and with unit costs. The proof is a short reduction from the minimum cut problem with three terminals: from a graph H with edge set D we build a bipartite graph with three terminals in which the minimum cost of a blocking set is 2 w(D) plus the minimum weight of a three-terminal cut of H, for every positive integral weight w on D. So the question of record OWR-1111-001 of the UnsolvedMath collection, to determine the complexity of the problem for three vertices, with or without edge costs, has the answer: NP-hard in both cases, and the decision problem is NP-complete. The construction was tested on all connected graphs H with at most six vertices. The hardness of the three-terminal cut problem is the theorem of Dahlhaus, Johnson, Papadimitriou, Seymour and Yannakakis. No proof or statement of the result was found in the literature accessible to us; a search that finds nothing is not a proof of novelty. This is an unrefereed note.",
  "result_type": "COMPLETE_PROOF",
  "categories": [
    "cs.CC",
    "cs.DM",
    "math.CO"
  ],
  "keywords": [
    "blocking shortest paths",
    "increasing distances in graphs",
    "edge deletion",
    "multiway cut",
    "three-terminal cut",
    "NP-hardness",
    "combinatorial optimization",
    "shortest paths",
    "interdiction",
    "Oberwolfach Reports problems",
    "UnsolvedMath",
    "OWR-1111-001",
    "cs.CC",
    "cs.DM",
    "math.CO",
    "open mathematics"
  ],
  "manuscript_version_date": "2026-10-13",
  "publication_date": "2026-10-13",
  "publication_date_kind": "first public online release",
  "version": "1.0",
  "date_modified": "2026-10-13",
  "presentation_revision_only": false,
  "doi_archived_file_version": "1.0",
  "status": "unrefereed preprint",
  "canonical_url": "https://eulersolve.org/papers/owr-1111-001/",
  "pdf_url": "https://eulersolve.org/papers/owr-1111-001/paper.pdf?v=dae8f586d056",
  "doi": "10.5281/zenodo.23358810",
  "zenodo_record_url": "https://zenodo.org/records/23358810",
  "license": "https://creativecommons.org/licenses/by/4.0/",
  "scope_caveat": "Answers open problem (1) of S. Krause (Oberwolfach Report 50/2005) for undirected graphs: the triangle instances of Blocking Shortest Paths (the three pairs of three given vertices) are NP-hard with positive integer edge costs and with unit costs, and the decision problem is NP-complete, by a reduction from the three-terminal cut problem. The hardness of three-terminal cut is cited from Dahlhaus, Johnson, Papadimitriou, Seymour and Yannakakis (STOC 1992 version read; the SIAM 1994 full version was not accessible). Planar instances and approximability are not addressed. Novelty is not certified.",
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  "ai_use_disclosure": "AI-assisted tools supported research, computation, proof development, and manuscript preparation. The author remains responsible for all claims and the final text."
}