{"paper":{"title":"Optimal Matroid Partitioning Problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Hanna Sumita, Kazuhisa Makino, Kei Kimura, Yasushi Kawase","submitted_at":"2017-10-03T01:00:19Z","abstract_excerpt":"This paper studies optimal matroid partitioning problems for various objective functions. In the problem, we are given a finite set $E$ and $k$ weighted matroids $(E, \\mathcal{I}_i, w_i)$, $i = 1, \\dots, k$, and our task is to find a minimum partition $(I_1,\\dots,I_k)$ of $E$ such that $I_i \\in \\mathcal{I}_i$ for all $i$. For each objective function, we give a polynomial-time algorithm or prove NP-hardness. In particular, for the case when the given weighted matroids are identical and the objective function is the sum of the maximum weight in each set (i.e., $\\sum_{i=1}^k\\max_{e\\in I_i}w_i(e)$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1710.00950","kind":"arxiv","version":1},"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"}