{"paper":{"title":"Discrete Convexity and Polynomial Solvability in Minimum 0-Extension Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.MG","authors_text":"Hiroshi Hirai","submitted_at":"2014-05-20T06:10:35Z","abstract_excerpt":"For a graph G and a set V containing the vertex set of G, a 0-extension of G is a metric d on V such that d extends the shortest path metric of G and for all x in V there exists a vertex s in G with d(x, s) = 0. The minimum 0-extension problem 0-Ext[G] on G is: given a set V containing V(G) and a nonnegative cost function c defined on the set of all pairs of V, find a 0-extension d of G with \\sum c(xy)d(x, y) minimum. The 0-extension problem generalizes a number of basic combinatorial optimization problems, such as minimum (s,t)-cut problem and multiway cut problem. Karzanov proved the polynom"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1405.4960","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","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"}