eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2023-09-04
51:1
51:12
10.4230/LIPIcs.APPROX/RANDOM.2023.51
article
NP-Hardness of Almost Coloring Almost 3-Colorable Graphs
Hecht, Yahli
1
https://orcid.org/0009-0000-0596-080X
Minzer, Dor
2
https://orcid.org/0000-0002-8093-1328
Safra, Muli
1
https://orcid.org/0000-0002-5022-7727
School of Computer Science, Tel Aviv University, Israel
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA
A graph G = (V,E) is said to be (k,δ) almost colorable if there is a subset of vertices V' ⊆ V of size at least (1-δ)|V| such that the induced subgraph of G on V' is k-colorable. We prove that for all k, there exists δ > 0 such for all ε > 0, given a graph G it is NP-hard (under randomized reductions) to distinguish between:
1) Yes case: G is (3,ε) almost colorable.
2) No case: G is not (k,δ) almost colorable. This improves upon an earlier result of Khot et al. [Irit Dinur et al., 2018], who showed a weaker result wherein in the "yes case" the graph is (4,ε) almost colorable.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol275-approx-random2023/LIPIcs.APPROX-RANDOM.2023.51/LIPIcs.APPROX-RANDOM.2023.51.pdf
PCP
Hardness of approximation