2026-07-22T11:56:21Z https://drops.dagstuhl.de/oai

oai:drops-oai.dagstuhl.de:21080 2024-09-23T09:13:16Z ddc:004 open_access

On Connections Between k-Coloring and Euclidean k-Means Aman, Enver Karthik C. S. Punna, Sharath k-means k-minsum Euclidean space fine-grained complexity In the Euclidean k-means problems we are given as input a set of n points in ℝ^d and the goal is to find a set of k points C ⊆ ℝ^d, so as to minimize the sum of the squared Euclidean distances from each point in P to its closest center in C. In this paper, we formally explore connections between the k-coloring problem on graphs and the Euclidean k-means problem. Our results are as follows: - For all k ≥ 3, we provide a simple reduction from the k-coloring problem on regular graphs to the Euclidean k-means problem. Moreover, our technique extends to enable a reduction from a structured max-cut problem (which may be considered as a partial 2-coloring problem) to the Euclidean 2-means problem. Thus, we have a simple and alternate proof of the NP-hardness of Euclidean 2-means problem. - In the other direction, we mimic the O(1.7297ⁿ) time algorithm of Williams [TCS'05] for the max-cut of problem on n vertices to obtain an algorithm for the Euclidean 2-means problem with the same runtime, improving on the naive exhaustive search running in 2ⁿ⋅ poly(n,d) time. - We prove similar results and connections as above for the Euclidean k-min-sum problem. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Enver Aman and Karthik C. S. and Sharath Punna 2024 Is Part Of LIPIcs, Volume 308, 32nd Annual European Symposium on Algorithms (ESA 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.ESA.2024.9 urn:nbn:de:0030-drops-210808 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2024.9 eng https://creativecommons.org/licenses/by/4.0/legalcode