TY - GEN

T1 - NP-Hardness of Almost Coloring Almost 3-Colorable Graphs

AU - Hecht, Yahli

AU - Minzer, Dor

AU - Safra, Muli

N1 - Publisher Copyright:
© 2023 Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. All rights reserved.

PY - 2023/9

Y1 - 2023/9

N2 - 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.

AB - 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.

KW - PCP, Hardness of approximation

UR - http://www.scopus.com/inward/record.url?scp=85171999364&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.APPROX/RANDOM.2023.51

DO - 10.4230/LIPIcs.APPROX/RANDOM.2023.51

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AN - SCOPUS:85171999364

T3 - Leibniz International Proceedings in Informatics, LIPIcs

BT - Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques, APPROX/RANDOM 2023

A2 - Megow, Nicole

A2 - Smith, Adam

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

T2 - 26th International Conference on Approximation Algorithms for Combinatorial Optimization Problems, APPROX 2023 and the 27th International Conference on Randomization and Computation, RANDOM 2023

Y2 - 11 September 2023 through 13 September 2023

ER -