On the Hardness of Approximating Minimization Problems

Journal of the ACM (JACM)

Volumn 41, Issue 5, 1994, Pages 960-981

  Lund, Carsten ^a   Yannakakis, Mihalis ^a  

^a LUCENT TECHNOLOGIES   (United States)

Author keywords

approximation algorithms; chromatic number; clique cover; dominating set; graph coloring; hitting set; independent set; NP hard; probabilistically checkable proofs; set cover

Indexed keywords

ALGORITHMS; COLOR; COMPUTATIONAL COMPLEXITY; GRAPH THEORY; OPTIMIZATION; PROBABILITY;

APPROXIMATION ALGORITHMS; CHROMATIC NUMBER; CLIQUE COVER; DOMINATING SET; GRAPH COLORING; HITTING SET; INDEPENDENT SET; MINIMIZATION PROBLEMS; PROBABILISTICALLY CHECKABLE PROOFS; SET COVER;

APPROXIMATION THEORY;

EID: 0028514351     PISSN: 00045411     EISSN: 1557735X     Source Type: Journal    
DOI: 10.1145/185675.306789     Document Type: Article

References (33)

1. ALON, N., GOLDREICH, O., HÅSTAD, J., AND PERALTA, R. 1990. Simple constructions of almost k-wise independent random variables. In Proceedings of the 31st IEEE Symposium on Foundations of Computer Science. IEEE, New York, pp. 544-553.
2. ARORA, S., LUND, C, MOTWANI, R., SUDAN, M. AND SZEGEDY, M. 1992. Proof verification and intractability of approximation problems. In Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science. IEEE, New York, pp. 14-23.
3. ARORA, S., AND SAFRA, S. 1992. Approximating clique is NP-complete. In Proceedings of the 33rd IEEE Symposium on Foundations of Computer Science. IEEE, New York, pp. 2-13.
4. BELLARE, M. 1993. Interactive proofs and approximation: Reductions from two provers in one round. In Proceedings of the 2nd Israel Symposium on Theory and Computing Systems. Springer-Verlag, Berlin, pp. 266-274.
5. BELLARE, M., GOLDWASSER, S., LUND, C., AND RUSSELL., A. 1993. Efficient probabilistically checkable proofs and applications to approximation. In Proceedings of the 25th ACM Symposium on Theory of Computing (San Diego, Calif., May 16-18). ACM, New York, pp. 294-304.
6. BELLARE, M., AND ROGOWAY, P. 1993. The complexity of approximating a nonlinear program. In Complexity in Numerical Optimization. P. Pardalos, Ed., World-Scientific, Singapore, pp. 16-32.
7. BERMAN, P., AND SCHNITGER, G. 1989. On the complexity of approximating the independent set problem. In Proceedings of the 7th Symposium on Theoretical Aspects of Computer Science. Lecture Notes in Computer Science, vol. 349. Springer-Verlag, New York, pp. 256-267.
8. BLUM, A. 1990. Some tools for approximate 3-coloring. In Proceedings of the 31st IEEE Symposium on Foundations of Computer Science. IEEE, New York, pp. 554-562.
9. CHVÁTAL, V. 1979. A greedy heuristic for the set covering problem. Math. Op. Res. 4, 233-235.
10. DOBSON, G. 1982. Worst-case analysis of greedy heuristics for integer programming with nonnegative data. Math. Op. Res. 7, 515-531.
11. FEIGE, U., GOLDWASSER, S., LOVÁSZ, L., SAFRA, S., AND SZEGEDY, M. 1991. Approximating clique is almost NP-complete. In Proceedings of the 32nd IEEE Symposium on Foundations of Computer Science. IEEE, New York, pp. 2-12.
12. FEIGE, U., AND LOVÁSZ, L. 1992. Two-prover one-round proof systems: Their power and their problems. In Proceedings of the 24th ACM Symposium on the Theory of Computing (Victoria, B.C., Canada, May 4-6). ACM, New York, pp. 733-744.
13. FISHER, M. L., AND WOLSEY, L. A. 1982. On the greedy heuristic for continuous covering and packing problems. SIAM J. Alg. Disc. Meth. 3, 584-591.
14. GAREY, M. R., AND JOHNSON, D. S. 1976. The complexity of near-optimal graph coloring. J. ACM 23, 1 (Jan.), 43-49.
15. GROTSCHEL, M., LOVÁSZ, L., AND SCHRIJVER, A. 1981. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica 1, 169-197.
16. HALLDÓRSSON, M. M. 1990. A still better performance guarantee for approximate graph coloring. Tech. Rep. 90-44. DIMACS.
17. JOHNSON, D. S. 1974a. Approximation algorithms for combinatorial problems. J. Comput. Syst. Sci., 9, 256-278.
18. JOHNSON, D. S. 1974b. Worst case behavior of graph coloring algorithms. In Proceedings of the 5th Southeastern Conf. on Combinatorics, Graph Theory and Computing. Utilitas Mathematica Publishers, Winnipeg, Ontario, Canada.
19. KARP, R. M. 1972. Reducibility among combinatorial problems. In Complexity of Computer Computations. R. E. Miller and J. W. Thatcher, eds., Advances in Computing Research, Plenum Press, New York.
20. KHANNA, S., LINIAL, N., AND SAFRA, S. 1993. On the hardness of approximating the chromatic number. In Proceedings of the 2nd Israel Symposium on Theory and Computing Systems. Springer-Verlag, New York, pp. 250-260.
21. KOLAITIS, P. G., AND THAKUR, M. N. 1991. Approximation properties of np minimization classes. In Proceedings of the 6th Conference on Structure in Complexity Theory, pp. 353-366.
22. KOU, L. T., STOCKMEYER, L. J., AND WONG, C. K. 1978. Covering edges by cliques with regard to keyword conflicts and intersection graphs. Commun. ACM 21, 2 (Feb.), 135-139.
23. LAPIDOT, D., AND SHAMIR, A. 1991. Fully parallelized multiprover protocols for NEXPTIME. In Proceedings of the 32nd Annual IEEE Symposium on Foundations of Computer Science. IEEE, New York, pp. 13-18.
24. LINIAL, N” AND VAZIRANI, U. 1989. Graph products and chromatic numbers. In Proceedings of the 30th IEEE Symposium on Foundations of Computer Science. IEEE, New York, pp. 124-133.
25. LOVÁSZ, L. 1975. On the ratio of optimal integral and fractional covers. Disc. Math. 13, 383-390.
26. LUND, C., AND YANNAKAKIS, M. 1993. The approximation of maximum subgraph problems. In Proceedings of the 20th International Colloquium on Automata, Languages and Programming. Springer-Verlag, Berlin, pp. 40-51.
27. ORLIN, J. 1977. Contentment in graph theory: Covering graphs with cliques. In Proceedings of the Koninklijke Nederlandse. Vol. 80 Akademie van Wetenschappen, Amsterdam, The Netherlands, pp. 406-424.
28. PAZ, A., AND MORAN, S. 1981. Nondeterministic polynomial optimization problems and their approximations. Theoret. Comput. Sci. 15, 251-277.
29. PAPADIMITRIOU, C., AND YANNAKAKIS, M. 1988. Optimization, approximation and complexity classes. In Proceedings of the 20th ACM Symposium on the Theory of Computing (Chicago, Ill., May 2-4). ACM, New York, pp. 229-234.
30. SIMON, H. U. 1990. On approximate solutions for combinatorial optimization problems. SIAM J. Algebraic Discrete Methods 3, 294-310.
31. WIGDERSON, A. 1983. Improving the performance guarantee for approximate graph coloring. J. ACM 30, 4 (Oct.), 729-735.
32. WOLSEY, L. A. 1982. An analysis of the greedy algorithm for the submodulat set covering problem. Combinatorica 2, 385-393.
33. YANNAKAKIS, M. 1979. The effect of a connectivity requirement on the complexity of maximum subgraph problems. J. ACM 26, 4 (Oct.), 618-630.