{"paper":{"title":"Approximations and Hardness of Packing Partially Ordered Items","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"cs.DS","authors_text":"Baruch Schieber, Guy Kortsarz, Hadas Shachnai, Ilan Doron-Arad, Joseph Naor","submitted_at":"2024-03-03T17:17:08Z","abstract_excerpt":"Motivated by applications in production planning and storage allocation in hierarchical databases, we initiate the study of covering partially ordered items (CPO). Given a capacity $k \\in \\mathbb{Z}^+$, and a directed graph $G=(V,E)$ where each vertex has a size in $\\{0,1, \\ldots,k\\}$, we seek a collection of subsets of vertices $S_1, \\ldots, S_m$ that cover all the vertices, such that for any $1 \\leq j \\leq m$, the total size of vertices in $S_j$ is bounded by $k$, and there are no edges from $V \\setminus S_j$ to $S_j$. The objective is to minimize the number of subsets $m$. CPO is closely re"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2403.01568","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2403.01568/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}