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Efficient approximation algorithms for domatic partition and on-line coloring of circular arc graphs. (English) Zbl 0847.68084
Summary: We exploit the close relationship between circular arc graphs and interval graphs to design efficient approximation algorithms for NP-hard optimization problems on circular arc graphs. The problems considered here are maximum domatic partition and on-line minimum vertex coloring.
We present a heuristic for the domatic partition problem with a performance ratio of 4. For on-line coloring, we consider two different on-line models. In the first model, arcs are presented in the increasing order of their left endpoints. For this model, our heuristic guarantees a solution which is within a factor of 2 of the optimal (off-line) value, and we show that no on-line coloring algorithm can achieve a performance guarantee of less than 4/3. In the second on-line model, arcs are presented in an arbitrary order; and it is known that no on-line coloring algorithm can achieve a performance guarantee of less than 3. For this model, we present a heuristic which provides a performance guarantee of 4.

68R10 Graph theory (including graph drawing) in computer science
05C05 Trees
Full Text: DOI
[1] Berge, C., Graphs and hypergraphs, (1973), North-Holland Amsterdam · Zbl 0483.05029
[2] Bonucelli, M.A., Dominating sets and domatic number of circular arc graphs, Discrete appl. math., 12, 203-213, (1985) · Zbl 0579.05051
[3] Chaitin, C.G., Register allocation and spilling via graph coloring, (), 98-105, 6
[4] Cockayne, E.J.; Hedetniemi, S.T., Towards a theory of domination in graphs, Networks, 7, 247-261, (1977) · Zbl 0384.05051
[5] Garey, M.R.; Johnson, D.S.; Miller, G.L.; Papadimitriou, C.H., The complexity of coloring circular arcs and chords, SIAM J. algebraic discrete methods, 1, 216-227, (1980) · Zbl 0499.05058
[6] Golumbic, M.C., Algorithmic graph theory and perfect graphs, (1980), Academic Press New York · Zbl 0541.05054
[7] Gupta, U.I; Lee, D.T.; Leung, J.Y.-T., Efficient algorithms for interval graphs and circular arc graphs, Networks, 12, 459-467, (1982) · Zbl 0493.68066
[8] Hunt, H.B.; Marathe, M.V.; Radhakrishnan, V.; Ravi, S.S.; Rosenkrantz, D.J.; Stearns, R.E., A unified approach to approximation schemes for NP-and PSPACE-hard problems for geometric graphs, (), 424-435
[9] Irani, S., Coloring inductive graphs on-line, (), 470-479
[10] Kant, G.; van Leeuwen, J., The file distribution problem for processor networks, ()
[11] Kierstead, H., A polynomial time approximation algorithm for dynamic storage allocation, Discrete math., 88, 231-237, (1991) · Zbl 0761.05087
[12] Kierstead, H.; Trotter, W., An extremal problem in recursive combinatorics, (), 143-153
[13] Marathe, M.V.; Hunt, H.B.; Ravi, S.S., Geometric heuristics for unit disk graphs, (), 244-249
[14] Marathe, M.V.; Hunt, H.B.; Ravi, S.S., Efficient approximation algorithms for domatic partition and on-line coloring of circular arc graphs (extended abstract), (), 26-30
[15] Masuda, S.; Nakajima, K., An optimal algorithm for finding a maximum independent set of a circular arc graph, SIAM J. comput., 17, 41-52, (1988) · Zbl 0646.68084
[16] Patterson, D.A., Reduced instruction set computers, Comm. ACM, 28, 8-21, (1985)
[17] Slusarek, M., A coloring algorithm for interval graphs, (), 471-480 · Zbl 0755.68112
[18] Slusarek, M., Optimal on-line coloring of circular arc graphs, (1994), Private communication
[19] Rao, A.Srinivas; Rangan, C.Pandu, A linear algorithm for domatic number of interval graphs, Inform. process. lett., 33, 29-33, (1989-1990) · Zbl 0685.68062
[20] Supowit, K.J., Decomposing a set of points into chains, with applications to permutation and circle graphs, Inform. process. lett., 21, 229-252, (1985) · Zbl 0586.68034
[21] Tucker, A., Clolouring a family of circular arcs, SIAM J. appl. math., 29, 493-502, (1975) · Zbl 0312.05105
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