Tight Bounds for Online Coloring of Basic Graph Classes
We resolve a number of long-standing open problems in online graph coloring. More specifically, we develop tight lower bounds on the performance of online algorithms for fundamental graph classes. An important contribution is that our bounds also hold for randomized online algorithms, for which hardly any results were known. Technically, we construct lower bounds for chordal graphs. The constructions then allow us to derive results on the performance of randomized online algorithms for the following further graph classes: trees, planar, bipartite, inductive, bounded-treewidth and disk graphs. It shows that the best competitive ratio of both deterministic and randomized online algorithms is Theta(log n), where n is the number of vertices of a graph. Furthermore, we prove that this guarantee cannot be improved if an online algorithm has a lookahead of size O(n/log n) or access to a reordering buffer of size n^(1-epsilon), for any 0 < epsilon <= 1. A consequence of our results is that, for all of the above mentioned graph classes except bipartite graphs, the natural First Fit coloring algorithm achieves an optimal performance, up to constant factors, among deterministic and randomized
online algorithms.
graph coloring
online algorithms
lower bounds
randomization
7:1-7:14
Regular Paper
Susanne
Albers
Susanne Albers
Sebastian
Schraink
Sebastian Schraink
10.4230/LIPIcs.ESA.2017.7
N. Avigdor-Elgrabli and Y. Rabani. An optimal randomized online algorithm for reordering buffer management. In Proc. 54th Annual IEEE Symposium on Foundations of Computer Science (FOCS), pages 1-10, 2013.
A. Bar-Noy, P. Cheilaris, S. Olonetsky, and S. Smorodinsky. Online conflict-free colouring for hypergraphs. Combinatorics, Probability & Computing, 19(4):493-516, 2010.
A. Bar-Noy, P. Cheilaris, and S. Smorodinsky. Deterministic conflict-free coloring for intervals: From offline to online. ACM Trans. Algorithms, 4(4):44:1-44:18, 2008.
D. Bean. Effective coloration. J. Symbolic Logic, 41(2):469-480, 1976.
S. Ben-David, A. Borodin, R. Karp, G. Tardos, and A. Wigderson. On the power of randomization in on-line algorithms. Algorithmica, 11(1):2-14, 1994.
M. P. Bianchi, H.-J. Böckenhauer, J. Hromkovic, and L. Keller. Online coloring of bipartite graphs with and without advice. Algorithmica, 70(1):92-111, 2014.
H. L. Bodlaender. A tourist guide through treewidth. Acta Cybernetica, 11(1-2):1-21, 1993.
E. Burjons, J. Hromkovic, X. Muñoz, and W. Unger. Online graph coloring with advice and randomized adversary. In Proc. 42nd International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM'16), pages 229-240. Springer LNCS 9587, 2016.
I. Caragiannis, A. V. Fishkin, C. Kaklamanis, and E. Papaioannou. A tight bound for online colouring of disk graphs. Theoretical Computer Science, 384(2):152-160, 2007.
R. G. Downey and C. McCartin. Online promise problems with online width metrics. Journal of Computer and System Sciences, 73(1):57-72, 2007.
M. Englert, D. Özmen, and M. Westermann. The power of reordering for online minimum makespan scheduling. SIAM J. Comput., 43(3):1220-1237, 2014.
T. Erlebach and J. Fiala. On-line coloring of geometric intersection graphs. Computational Geometry, 23(2):243-255, 2002.
T. Erlebach and J. Fiala. Independence and coloring problems on intersection graphs of disks. In E. Bampis, K. Jansen, and C. Kenyon, editors, Efficient Approximation and Online Algorithms: Recent Progress on Classical Combinatorial Optimization Problems and New Applications, pages 135-155. Springer LNCS 3484, 2006.
Martin Farber. Characterizations of strongly chordal graphs. Discrete Mathematics, 43(2-3):173-189, 1983.
A. Gyárfás and J. Lehel. On-line and first fit colorings of graphs. Journal of Graph Theory, 12(2):217-227, 1988.
M. M. Halldórsson. Parallel and on-line graph coloring. J. Algorithms, 23(2):265-280, 1997.
M. M. Halldórsson and M. Szegedy. Lower bounds for on-line graph coloring. Theoretical Computer Science, 130(1):163-174, 1994.
S. Irani. Coloring inductive graphs on-line. Algorithmica, 11(1):53-72, 1994. Preliminary version in FOCS'90 .
H. A. Kierstead. Coloring graphs on-line. In A. Fiat and G. J. Woeginger, editors, Online Algorithms, pages 281-305. Springer LNCS 1442, 1998.
H. A. Kierstead and W. A. Trotter. An extremal problem in recursive combinatorics. Congressus Numerantium, 33:143-153, 1981.
S. Leonardi and A. Vitaletti. Randomized lower bounds for online path coloring. In Proc. 2nd International Workshop on Randomization and Approximation Techniques in Computer Science (RANDOM'98), pages 232-247. Springer LNCS 1518, 1998.
L. Lovász, M. Saks, and W. T. Trotter. An on-line graph coloring algorithm with sublinear performance ratio. Annals of Discrete Mathematics, 43:319-325, 1989.
D. Marx. Graph colouring problems and their applications in scheduling. Periodica Polytechnica, Electrical Engineering, 48(1–2):11-16, 2004.
L. Narayanan. Channel assignment and graph multicoloring. Handbook of Wireless Networks and Mobile Computing, pages 71-94, 2004.
D. D. Sleator and R. E. Tarjan. Amortized efficiency of list update and paging rules. Commun. ACM, 28(2):202-208, 1985.
S. Vishwanathan. Randomized online graph coloring. J. Algorithms, 13(4):657-669, 1992. Preliminary version in FOCS'90 .
D. B. West. Introduction to Graph Theory, 2nd Edition. Pearson, 2001.
A. C. C. Yao. Probabilistic computations: Toward a unified measure of complexity. In Proc. 18th Annual Symposium on Foundations of Computer Science, pages 222-227, 1977.
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