{"paper":{"title":"Multi-Colored Spanning Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CG","authors_text":"Csaba D. T\\'oth, Hugo A. Akitaya, Maarten L\\\"offler","submitted_at":"2016-08-25T09:05:30Z","abstract_excerpt":"We study a problem proposed by Hurtado et al. (2016) motivated by sparse set visualization. Given $n$ points in the plane, each labeled with one or more primary colors, a \\emph{colored spanning graph} (CSG) is a graph such that for each primary color, the vertices of that color induce a connected subgraph. The \\textsc{Min-CSG} problem asks for the minimum sum of edge lengths in a colored spanning graph. We show that the problem is NP-hard for $k$ primary colors when $k\\ge 3$ and provide a $(2-\\frac{1}{3+2\\varrho})$-approximation algorithm for $k=3$ that runs in polynomial time, where $\\varrho$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.07056","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"}