{"paper":{"title":"On k-Submodular Relaxation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CC","cs.DS"],"primary_cat":"math.OC","authors_text":"Hiroshi Hirai, Yuni Iwamasa","submitted_at":"2015-04-29T12:27:44Z","abstract_excerpt":"$k$-submodular functions, introduced by Huber and Kolmogorov, are functions defined on $\\{0, 1, 2, \\dots, k\\}^n$ satisfying certain submodular-type inequalities. $k$-submodular functions typically arise as relaxations of NP-hard problems, and the relaxations by $k$-submodular functions play key roles in design of efficient, approximation, or fixed-parameter tractable algorithms. Motivated by this, we consider the following problem: Given a function $f : \\{1, 2, \\dots, k\\}^n \\rightarrow \\mathbb{R} \\cup \\{\\infty\\}$, determine whether $f$ is extended to a $k$-submodular function $g : \\{0, 1, 2, \\"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.07830","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"}