Greed is good: Approximating independent sets in sparse and bounded-degree graphs
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Theminimum-degree greedy algorithm, or Greedy for short, is a simple and well-studied method for finding independent sets in graphs. We show that it achieves a performance ratio of (Δ+2)/3 for approximating independent sets in graphs with degree bounded by Δ. The analysis yields a precise characterization of the size of the independent sets found by the algorithm as a function of the independence number, as well as a generalization of Turán’s bound. We also analyze the algorithm when run in combination with a known preprocessing technique, and obtain an improved \((2\bar d + 3)/5\) performance ratio on graphs with average degree \(\bar d\) , improving on the previous best \((\bar d + 1)/2\) of Hochbaum. Finally, we present an efficient parallel and distributed algorithm attaining the performance guarantees of Greedy.
- Alon, N., Feige, U., Wigderson, A., Zuckerman, D. (1995) Derandomized graph products. Computational Complexity 5: pp. 60-75 CrossRef
- S. Arora, C. Lund, R. Motwani, M. Sudan, and M. Szegedy. Proof verification and hardness of approximation problems.Proc. 33rd Ann. IEEE Symp. on Foundations of Computer Science, pages 14–23, Oct. 1992.
- Berman, P., Fujito, T. (1995) On the approximation properties of the independent set problem in degree 3 graphs. Proc. Fourth Workshop on Algorithms and Data Structures. Springer-Verlag, Berlin, pp. 449-460
- P. Berman and M. Fürer. Approximating maximum independent set in bounded degree graphs.Proc. Fifth Ann. ACM-SIAM Symp. on Discrete Algorithms, pages 365–371, Jan. 1994.
- Brooks, R. L. (1991) On coloring the nodes of a network. Math. Proc. Cambridge Philos. Soc. 37: pp. 194-197 CrossRef
- Chvátal, V. (1983) Linear Programming. Freeman, New York
- Chvátal, V., McDiarmid, C. (1992) Small transversals in hypergraphs. Combinatorica 12: pp. 19-26 CrossRef
- Erdős, P. (1970) On the graph theorem of Turán. Mat. Lapok 21: pp. 249-251
- Goldberg, A. V., Plotkin, S. A., Shannon, G. E. (1988) Parallel symmetry-breaking in sparse graphs. SIAM J. Discrete Math. 1: pp. 434-446 CrossRef
- Griggs, J. R. (1983) Lower bounds on the independence number in terms of the degrees. J. Combin. Theory Ser. B 34: pp. 22-39 CrossRef
- Halldórsson, M. M., Radhakrishnan, J. (1994) Improved approximations of independent sets in bounded-degree graphs. Springer-Verlag, Berlin
- Halldórsson, M. M., Radhakrishnan, J. (1994) Improved approximations of independent sets in bounded-degree via subgraph removal. Nordic J. Comput. 1: pp. 475-492
- Halldórsson, M. M., Yoshihara, K. (1995) Greedy approximations of independent sets in low degree graphs. Springer-Verlag, Berlin
- Hochbaum, D. S. (1983) Efficient bounds for the stable set, vertex cover, and set packing problems. Discrete Appl. Math. 6: pp. 243-254 CrossRef
- Hopcroft, J., Karp, R. (1973) Ann 5/2 algorithm for maximal matchings in bipartite graphs. SIAM J. Comput. 4: pp. 225-231 CrossRef
- D. S. Johnson. Worst case behavior of graph coloring algorithms.Proc. 5th Southeastern Conf. on Combinatorics, Graph Theory, and Computing, pages 513–527. Congressus Numerantium, X, Utilitas Math., Winnipeg, Manitoba, 1974.
- Karp, R., Ramachandran, V. A survey of parallel algorithms for shared-memory machines. In: Leeuwen, J. eds. (1990) Handbook of Theoretical Computer Science. Elsevier, Amsterdam, pp. 869-941
- S. Khanna, R. Motwani, M. Sudan, and U. Vazirani. On syntactic versus computational views of approximability.Proc. 35th Ann. IEEE Symp. on Foundations of Computer Science, pages 819–830, 1994.
- Kubicka, E., Kubicki, G., Kountanis, D. (1989) Approximation algorithms for the chromatic sum. Springer-Verlag, Berlin
- Lawler, E. (1976) Combinatorial Optimization: Networks and Matroids. Holt, Rinehart and Winston, New York
- Linial, N. (1992) Locality in distributive algorithms. SIAM J. Comput. 21: pp. 193-201 CrossRef
- Lovász, L. (1975) Three short proofs in graph theory. J. Combin. Theory Ser. B 19: pp. 269-271 CrossRef
- McDiarmid, C. (1984) Colouring random graphs. Ann. Oper. Res. 1: pp. 183-200 CrossRef
- Nemhauser, G. L., Trotter, L. (1975) Vertex packings Structural properties and algorithms. Math. Programming 8: pp. 232-248 CrossRef
- Papadimitriou, C. H., Yannakakis, M. (1991) Optimization, approximation, and complexity classes. J. Comput. System Sci. 43: pp. 425-440 CrossRef
- Shearer, J. B. (1983) A note on the independence number of triangle-free graphs. Discrete Math. 46: pp. 83-87 CrossRef
- Simon, H. U. (1990) On approximate solutions for combinatorial optimization problems. SIAM J. Discrete Math. 3: pp. 294-310 CrossRef
- Turán, P. (1941) On an extremal problem in graph theory. Mat. Fiz. Lapok 48: pp. 436-452
- Twentieth Century Fox.Wall Street. Motion picture, 1987.
- V. K. Wei. A lower bound on the stability number of a simple graph. Technical Memorandum No. 81-11217-9, Bell Laboratories, 1981.
- Greed is good: Approximating independent sets in sparse and bounded-degree graphs
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