TY - CHAP

T1 - Approximating the Achromatic Number Problem on Bipartite Graphs

AU - Kortsarz, Guy

AU - Shende, Sunil

N1 - Copyright:
Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2003

Y1 - 2003

N2 - The achromatic number of a graph is the largest number of colors needed to legally color the vertices of the graph so that adjacent vertices get different colors and for every pair of distinct colors c1,c2 there exists at least one edge whose endpoints are colored by c1,c 2. We give a greedy O(n4/5) ratio approximation for the problem of finding the achromatic number of a bipartite graph with n vertices. The previous best known ratio was n · log log n/log n [12]. We also establish the first non-constant hardness of approximation ratio for the achromatic number problem; in particular, this hardness result also gives the first such result for bipartite graphs. We show that unless NP has a randomized quasi-polynomial algorithm, it is not possible to approximate achromatic number on bipartite graph within a factor of (ln n)1/4-ε. The methods used for proving the hardness result build upon the combination of one-round, two-provers techniques and zero-knowledge techniques inspired by Feige et.al. [6].

AB - The achromatic number of a graph is the largest number of colors needed to legally color the vertices of the graph so that adjacent vertices get different colors and for every pair of distinct colors c1,c2 there exists at least one edge whose endpoints are colored by c1,c 2. We give a greedy O(n4/5) ratio approximation for the problem of finding the achromatic number of a bipartite graph with n vertices. The previous best known ratio was n · log log n/log n [12]. We also establish the first non-constant hardness of approximation ratio for the achromatic number problem; in particular, this hardness result also gives the first such result for bipartite graphs. We show that unless NP has a randomized quasi-polynomial algorithm, it is not possible to approximate achromatic number on bipartite graph within a factor of (ln n)1/4-ε. The methods used for proving the hardness result build upon the combination of one-round, two-provers techniques and zero-knowledge techniques inspired by Feige et.al. [6].

UR - http://www.scopus.com/inward/record.url?scp=0142214928&partnerID=8YFLogxK

U2 - 10.1007/978-3-540-39658-1_36

DO - 10.1007/978-3-540-39658-1_36

M3 - ???researchoutput.researchoutputtypes.contributiontobookanthology.chapter???

AN - SCOPUS:0142214928

SN - 3540200649

SN - 9783540200642

T3 - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

SP - 385

EP - 396

BT - Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

A2 - di Battista, Giuseppe

A2 - Zwick, Uri

PB - Springer Verlag

ER -