Abstract
The chromatic sum of a graph G is the smallest total among all proper colorings of G using natural numbers. It was shown that computing the chromatic sum is NP-hard. In this article we prove that a simple greedy algorithm applied to sparse graphs gives a "good" approximation of the chromatic sum. For all graphs the existence of a polynomial time algorithm that approximates the chromatic sum with a linear function error implies P = NP.
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
G. Chartrand and L. Lesniak, Graphs & Digraphs, 2nd Edition. Wadsworth and Brooks/Cole (1986).
P. Erdös, E. Kubicka and A. Schwenk, Graphs that require extra colors to attain the chromatic sum, submitted to Congr. Numerantium.
M. R. Garey, D. S. Johnson, Computers and Intractability, W. H. Freeman (San Francisco) (1979).
D. S. Johnson, Approximation algorithms for combinatorial problems, Journal of Computer and System Sciences 9 (1974) 256–278.
D. S. Johnson, Worst Case Behavior of Graph Coloring Algorithms, Proc. of 5th Southeastern Conference on Combinatorics, Graph Theory and Computing (1974) 513–524.
E. Kubicka and A. J. Schwenk, Introduction to Chromatic Sums, Proceedings of ACM 1989 Computer Science Conference (1989) 39–45.
F. T. Leighton, A Graph Coloring Algorithm for Large Scheduling Problems, Journal of Research on the NBS 84 (1979) 489–506.
A. J. Mansfield and D. J. A. Welsh, Some colouring problems and their complexity, Annals of Discrete Mathematics 13 (1982) 159–170.
B. Manvel, Coloring Large Graphs, Congressus Numerantium 33 (1981) 197–204.
D. W. Matula, G. Marble, and J. D. Isaacson, Graph coloring algorithms, in R. C. Read (ed), Graph Theory and Computing, Academic Press, New York (1972).
D. W. Matula, Bounded color functions on graphs, Networks 2 (1972) 129–44.
C. Thomassen, P. Erdös, Y. Alavi, P. J. Malde, and A. J. Schwenk, Tight Bounds on the Chromatic Sum of a Connected Graph, J. Graph Theory 13 (1989) 353–357
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Kubicka, E., Kubicki, G., Kountanis, D. (1991). Approximation algorithms for the chromatic sum. In: Sherwani, N.A., de Doncker, E., Kapenga, J.A. (eds) Computing in the 90's. Great Lakes CS 1989. Lecture Notes in Computer Science, vol 507. Springer, New York, NY. https://doi.org/10.1007/BFb0038467
Download citation
DOI: https://doi.org/10.1007/BFb0038467
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-97628-0
Online ISBN: 978-0-387-34815-5
eBook Packages: Springer Book Archive