NP-Hardness of Almost Coloring Almost 3-Colorable Graphs

Yahli Hecht*, Dor Minzer*, Muli Safra*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

Abstract

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. [16], who showed a weaker result wherein in the “yes case” the graph is (4, ε) almost colorable.

Original languageEnglish
Title of host publicationApproximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023
EditorsNicole Megow, Adam Smith
PublisherSchloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
ISBN (Electronic)9783959772969
DOIs
StatePublished - Sep 2023
Event26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023 - Atlanta, United States
Duration: 11 Sep 202313 Sep 2023

Publication series

NameLeibniz International Proceedings in Informatics, LIPIcs
Volume275
ISSN (Print)1868-8969

Conference

Conference26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023
Country/TerritoryUnited States
CityAtlanta
Period11/09/2313/09/23

Funding

FundersFunder number
National Science Foundation2227876, 2239160
Bloom's Syndrome Foundation
Horizon 2020 Framework Programme835152
Iowa Science Foundation
European Research Council

    Keywords

    • PCP, Hardness of approximation

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