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Abstract: We give a new proof showing that it is NP-hard to color a 3-colorable graph using just four colors. This result is already known [19], but our proof is novel as it does not rely on the PCP theorem, while the one in [19] does. This highlights a qualitative difference between the known hardness result for coloring 3-colorable graphs and the factor n hardness for approximating the chromatic number of general graphs, as the latter result is known to imply (some form of) PCP theorem [3]. (Update)
Context of citations to this paper: More
...NP hard problem. Finding a 4 coloring of such a graph is also known to be NP hard (Khanna, Linial and Safra [KLS00] and Guruswami and Khanna [GK00]) Karger, Motwani and Sudan [KMS98] show, on the other hand, using semidefinite programming, that a 3 colorable graph on n...
...results are much weaker. For example, for 3 colorable graphs the best known hardness result only rules out coloring using 4 colors [20, 16]. This paper is motivated by the quest for strong (super constant) inapproximability for coloring graphs whose chromatic number is a...
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BibTeX entry: (Update)
V. Guruswami and S. Khanna. On the hardness of 4-coloring a 3-colorable graph. Proc. of Complexity 2000, to appear. http://citeseer.ist.psu.edu/article/guruswami00hardness.html More
@inproceedings{ guruswami00hardness,
author = "Venkatesan Guruswami and Sanjeev Khanna",
title = "On the Hardness of 4-Coloring a 3-Colorable Graph",
booktitle = "{IEEE} Conference on Computational Complexity",
pages = "188-197",
year = "2000",
url = "citeseer.ist.psu.edu/article/guruswami00hardness.html" }
Citations (may not include all citations):773 Reducibility among combinatorial problems (context) - Karp - 1972
403 Proof verification and hardness of approximation problems - Arora, Lund et al. - 1998
300 the hardness of approximating minimization problems (context) - Lund, Yannakakis - 1994
212 Probabilistic checking of proofs: A new characterization of .. - Arora, Safra - 1998
104 approximation and complexity classes (context) - Papadimitriou, Yannakakis - 1991
98 Zero-knowledge and the chromatic number - Feige, Kilian - 1996
94 Interactive proofs and the hardness of approximating cliques - Feige, Goldwasser et al. - 1996
64 Johnson Computers and Intractability --- A guide to the theo.. (context) - Garey - 1979
62 the hardness of approximating the chromatic number - Khanna, Linial et al. - 1993
53 The hardness of approximation: Gap Location - Petrank - 1994
51 Nearoptimal hardness results and approximation algorithms fo.. - Guruswami, Khanna et al. - 1999
44 PCP's and non-approximability -- towards tight results (context) - Bellare, Goldreich et al. - 1998
38 A still better performance guarantee for approximate graph c.. (context) - Halld'orsson - 1993
34 The complexity of near-optimal graph coloring (context) - Garey, Johnson - 1976
34 Improving the performance guarantee for approximate graph co.. (context) - Wigderson - 1983
30 Clique is hard to approximate within n (context) - astad
23 New approximation algorithms for graph coloring - Blum - 1994
20 Algorithms for Approximate Graph Coloring - Blum - 1991
19 coloring algorithm for 3-colorable graphs - Blum, Karger - 1997
13 Hardness of Approximate Hypergraph Coloring - Guruswami, astad et al. - 2000
6 Approximate graph coloring using semidefinite programming (context) - Karger, Motwani et al. - 1998
5 Improved hardness results for approximating the chromatic nu.. (context) - urer - 1995
1 Longest directed path is n -hard (context) - Khanna - 1999
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