{"paper":{"title":"The Complexity of Maximum $k$-Order Bounded Component Set Problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"cs.DS","authors_text":"Shijin Rajakrishnan, Sounaka Mishra","submitted_at":"2017-11-29T09:52:09Z","abstract_excerpt":"Given a graph $G=(V, E)$ and a positive integer $k$, in Maximum $k$-Order Bounded Component Set (Max-$k$-OBCS), it is required to find a vertex set $S \\subseteq V$ of maximum size such that each component in the induced graph $G[S]$ has at most $k$ vertices. We prove that for constant $k$, Max-$k$-OBCS is hard to approximate within a factor of $n^{1 -\\epsilon}$, for any $\\epsilon > 0$, unless $\\mathsf{P} = \\mathsf{NP}$. This is an improvement on the previous lower bound of $\\sqrt{n}$ for Max-2-OBCS due to Orlovich et al. We provide lower bounds on the approximability when $k$ is not a constant"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.02870","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"}