Combinatorial Optimization pp 361-396 | Cite as
Approximation Algorithms
Chapter
Abstract
In this chapter we introduce the important concept of approximation algorithms. So far we have dealt mostly with polynomially solvable problems. In the remaining chapters we shall indicate some strategies to cope with NP-hard combinatorial optimization problems. Here approximation algorithms must be mentioned in the first place.
Keywords
Approximation Algorithm Chromatic Number Vertex Cover Truth Assignment Perfect Graph
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
Preview
Unable to display preview. Download preview PDF.
References
General Literature
- Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., Protasi, M. [1999]: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer, Berlin 1999MATHGoogle Scholar
- Garey, M.R., and Johnson, D.S. [1979]: Computers and Intractability; A Guide to the Theory of NP-Completeness. Freeman, San Francisco 1979, Chapter 4Google Scholar
- Hochbaum, D.S. [1996]: Approximation Algorithms for NP-Hard Problems. PWS, Boston, 1996Google Scholar
- Horowitz, E., and Sahni, S. [1978]: Fundamentals of Computer Algorithms. Computer Science Press, Potomac 1978, Chapter 12MATHGoogle Scholar
- Shmoys, D.B. [1995]: Computing near-optimal solutions to combinatorial optimization problems. In: Combinatorial Optimization; DIMACS Series in Discrete Mathematics and Theoretical Computer Science 20 (W. Cook, L. Lovász, P. Seymour, eds.), AMS, Providence 1995Google Scholar
- Papadimitriou, C.H. [1994]: Computational Complexity, Addison-Wesley, Reading 1994, Chapter 13MATHGoogle Scholar
- Vazirani, V.V. [2001]: Approximation Algorithms. Springer, Berlin, 2001Google Scholar
Cited References
- Ajtai, M. [1994]: Recursive construction for 3-regular expanders. Combinatorica 14 (1994), 379–416MathSciNetMATHCrossRefGoogle Scholar
- Appel, K., and Haken, W. [1977]: Every planar map is four colorable; Part I; Discharging. Illinois Journal of Mathematics 21 (1977), 429–490MathSciNetMATHGoogle Scholar
- Appel, K., Haken, W., and Koch, J. [1977]: Every planar map is four colorable; Part II; Reducibility. Illinois Journal of Mathematics 21 (1977), 491–567MathSciNetMATHGoogle Scholar
- Arora, S. [1994]: Probabilistic checking of proofs and the hardness of approximation problems, Ph.D. thesis, U.C. Berkeley, 1994Google Scholar
- Arora, S., Lund, C, Motwani, R., Sudan, M., and Szegedy, M. [1998]: Proof verification and hardness of approximation problems. Journal of the ACM 45 (1998), 501–555MathSciNetMATHCrossRefGoogle Scholar
- Arora, S., and Safra, S. [1998]: Probabilistic checking of proofs. Journal of the ACM 45 (1998), 70–122MathSciNetMATHCrossRefGoogle Scholar
- Asano, T., and Williamson, D.P. [2000]: Improved approximation algorithms for MAX SAT. Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms (2000), 96-105Google Scholar
- Bar-Yehuda, R., and Even, S. [1981]: A linear-time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms 2 (1981), 198–203MathSciNetMATHCrossRefGoogle Scholar
- Becker, A., and Geiger, D. [1994]: Approximation algorithms for the loop cutset problem. Proceedings of the 10th Conference on Uncertainty in Artificial Intelligence (1994), 60-68Google Scholar
- Bellare, M., and Sudan, M. [1994]: Improved non-approximability results. Proceedings of the 26th Annual ACM Symposium on the Theory of Computing (1994), 184-193Google Scholar
- Bellare, M., Goldreich, O., and Sudan, M. [1998]: Free bits, PCPs and nonapproximability — towards tight results. SIAM Journal on Computing 27 (1998), 804–915MathSciNetMATHCrossRefGoogle Scholar
- Berge, C. [1961]: Färbung von Graphen, deren sämtliche bzw. deren ungerade Kreise starr sind. Wissenschaftliche Zeitschrift, Martin Luther Universität Halle-Wittenberg, Mathematisch-Naturwissenschaftliche Reihe (1961), 114-115Google Scholar
- Berge, C. [1962]: Sur une conjecture relative au problème des codes optimaux. Communication, 13ème assemblée générale de l’URSI, Tokyo 1962Google Scholar
- Berman, P., and Fujito, T. [1999]: On approximation properties of the independent set problem for low degree graphs. Theory of Computing Systems 32 (1999), 115–132MathSciNetMATHCrossRefGoogle Scholar
- Boppana, R., and Halldorsson, M.M. [1992]: Approximating maximum independent set by excluding subgraphs. BIT 32 (1992), 180–196MathSciNetMATHCrossRefGoogle Scholar
- Brooks, R.L. [1941]: On colouring the nodes of a network. Proceedings of the Cambridge Philosophical Society 37 (1941), 194–197MathSciNetCrossRefGoogle Scholar
- Chen, J., Friesen, D.K., and Zheng, H. [1999]: Tight bound on Johnson’s algorithm for maximum satisfiability. Journal of Computer and System Sciences 58 (1999), 622–640MathSciNetMATHCrossRefGoogle Scholar
- Chvátal, V. [1975]: On certain polytopes associated with graphs. Journal of Combinatorial Theory B 18 (1975), 138–154MATHCrossRefGoogle Scholar
- Chvátal, V. [1979]: A greedy heuristic for the set cover problem. Mathematics of Operations Research 4 (1979), 233–235MathSciNetMATHCrossRefGoogle Scholar
- Clementi, A.E.F., and Trevisan, L. [1999]: Improved non-approximability results for minimum vertex cover with density constraints. Theoretical Computer Science 225 (1999), 113–128MathSciNetMATHCrossRefGoogle Scholar
- Erdős, P. [1967]: On bipartite subgraphs of graphs. Mat. Lapok. 18 (1967), 283–288Google Scholar
- Feige, U. [1998]: A threshold of In n for the approximating set cover. Journal of the ACM 45 (1998), 634–652MathSciNetMATHCrossRefGoogle Scholar
- Feige, U., and Goemans, M.X. [1995]: Approximating the value of two prover proof systems, with applications to MAX 2SAT and MAX DICUT. Proceedings of the 3rd Israel Symposium on Theory of Computing and Systems (1995), 182-189Google Scholar
- Feige, U., Goldwasser, S., Lovász, L., Safra, S., and Szegedy, M. [1996]: Interactive proofs and the hardness of approximating cliques. Journal of the ACM 43 (1996), 268–292MathSciNetMATHCrossRefGoogle Scholar
- Fernández-Baca, D., and Lagergren, J. [1998]: On the approximability of the Steiner tree problem in phylogeny. Discrete Applied Mathematics 88 (1998), 129–145MathSciNetMATHCrossRefGoogle Scholar
- Fulkerson, D.R. [1972]: Anti-blocking polyhedra. Journal of Combinatorial Theory B 12 (1972), 50–71MathSciNetMATHCrossRefGoogle Scholar
- Fürer, M., and Raghavachari, B. [1994]: Approximating the minimum-degree Steiner tree to within one of optimal. Journal of Algorithms 17 (1994), 409–423MathSciNetCrossRefGoogle Scholar
- Garey, M.R., and Johnson, D.S. [1976]: The complexity of near-optimal graph coloring. Journal of the ACM 23 (1976), 43–49MathSciNetMATHCrossRefGoogle Scholar
- Garey, M.R., Johnson, D.S., and Stockmeyer, L. [1976]: Some simplified NP-complete graph problems. Theoretical Computer Science 1 (1976), 237–267MathSciNetMATHCrossRefGoogle Scholar
- Goemans, M.X., and Williamson, D.P. [1994]: New 3/4-approximation algorithms for the maximum satisfiability problem. SIAM Journal on Discrete Mathematics 7 (1994), 656–666MathSciNetMATHCrossRefGoogle Scholar
- Goemans, M.X., and Williamson, D.P. [1995]: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming Journal of the ACM 42 (1995), 1115–1145MathSciNetMATHCrossRefGoogle Scholar
- Grötschel, M., Lovász, L., and Schrijver, A. [1988]: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin 1988MATHCrossRefGoogle Scholar
- Halldórsson, M.M., and Radhakrishnan, J. [1997]: Greed is good: approximating independent sets in sparse and bounded degree graphs. Algorithmica 18 (1997), 145–163MathSciNetMATHCrossRefGoogle Scholar
- Håstad, J. [1996]: Clique is hard to approximate within n 1−ε. Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science (1996), 627-636Google Scholar
- Håstad, J. [1997]: Getting optimal in-approximability results. Proceedings of the 29th Annual ACM Symposium on the Theory of Computing (1997), 1-10Google Scholar
- Heawood, P.J. [1890]: Map colour theorem. Quarterly Journal of Pure Mathematics 24 (1890), 332–338Google Scholar
- Hochbaum, D.S. [1982]: Approximation algorithms for the set covering and vertex cover problems. SIAM Journal on Computing 11 (1982), 555–556MathSciNetMATHCrossRefGoogle Scholar
- Hochbaum, D.S., and Shmoys, D.B. [1985]: A best possible heuristic for the k-center problem. Mathematics of Operations Research 10 (1985), 180–184MathSciNetMATHCrossRefGoogle Scholar
- Holyer, I. [1981]: The NP-completeness of edge-coloring. SIAM Journal on Computing 10 (1981), 718–720MathSciNetMATHCrossRefGoogle Scholar
- Hougardy, S., Prömel, HJ., and Steger, A. [1994]: Probabilistically checkable proofs and their consequences for approximation algorithms. Discrete Mathematics 136 (1994), 175–223MathSciNetMATHCrossRefGoogle Scholar
- Hsu, W.L., and Nemhauser, G.L. [1979]: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1 (1979), 209–216MathSciNetMATHCrossRefGoogle Scholar
- Johnson, D.S. [1974]: Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences 9 (1974), 256–278MathSciNetMATHCrossRefGoogle Scholar
- Khanna, S., Linial, N., and Safra, S. [2000]: On the hardness of approximating the chromatic number. Combinatorica 20 (2000), 393–415MathSciNetMATHCrossRefGoogle Scholar
- König, D. [1916]: Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Mathematische Annalen 77 (1916), 453–465MathSciNetMATHCrossRefGoogle Scholar
- Lieberherr, K., and Specker, E. [1981]: Complexity of partial satisfaction. Journal of the ACM 28 (1981), 411–421MathSciNetMATHCrossRefGoogle Scholar
- Lovász, L. [1972]: Normal hypergraphs and the perfect graph conjecture. Discrete Mathematics 2 (1972), 253–267MathSciNetMATHCrossRefGoogle Scholar
- Lovász, L. [1975]: On the ratio of optimal integral and fractional covers. Discrete Mathematics 13 (1975), 383–390MathSciNetMATHCrossRefGoogle Scholar
- Lovász, L. [1979]: On the Shannon capacity of a graph. IEEE Transactions on Information Theory 25 (1979), 1–7MATHCrossRefGoogle Scholar
- Lovász, L. [1979]: Graph theory and integer programming. In: Discrete Optimization I; Annals of Discrete Mathematics 4 (P.L. Hammer, E.L. Johnson, B.H. Korte, eds.), North-Holland, Amsterdam 1979, pp. 141–158Google Scholar
- Mahajan, S., and Ramesh, H. [1999]: Derandomizing approximation algorithms based on semidefinite programming. SIAM Journal on Computing 28 (1999), 1641–1663MathSciNetMATHCrossRefGoogle Scholar
- Papadimitriou, C.H., and Steiglitz, K. [1982]: Combinatorial Optimization; Algorithms and Complexity. Prentice-Hall, Englewood Cliffs 1982, pp. 406–408MATHGoogle Scholar
- Papadimitriou, C.H., and Yannakakis, M. [1991]: Optimization, approximation, and complexity classes. Journal of Computer and System Sciences 43 (1991), 425–440MathSciNetMATHCrossRefGoogle Scholar
- Papadimitriou, C.H., and Yannakakis, M. [1993]: The traveling salesman problem with distances one and two. Mathematics of Operations Research 18 (1993), 1–12MathSciNetMATHCrossRefGoogle Scholar
- Raghavan, P., and Thompson, CD. [1987]: Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica 7 (1987), 365–374MathSciNetMATHCrossRefGoogle Scholar
- Raz, R., and Safra, S. [1997]: A sub constant error probability low degree test, and a sub constant error probability PCP characterization of NP. Proceedings of the 29th Annual ACM Symposium on the Theory of Computing (1997), 475-484Google Scholar
- Robertson, N., Sanders, D.P., Seymour, P., and Thomas, R. [1997]: The four colour theorem. Journal of Combinatorial Theory B 70 (1997), 2–44MathSciNetMATHCrossRefGoogle Scholar
- Robertson, N., Sanders, D.P., Seymour, P., and Thomas, R. [1996]: Efficiently four-coloring planar graphs. Proceedings of the 28th Annual ACM Symposium on the Theory of Computing (1996), 571-575Google Scholar
- Slavík, P. [1997]: A tight analysis of the greedy algorithm for set cover. Journal of Algorithms 25 (1997), 237–254MathSciNetMATHCrossRefGoogle Scholar
- Stockmeyer, L.J. [1973]: Planar 3-colorability is polynomial complete. ACM SIGACT News 5 (1973), 19–25CrossRefGoogle Scholar
- Vizing, V.G. [1964]: On an estimate of the chromatic class of a p-graph. Diskret. Analiz 3 (1964), 23–30 [in Russian]MathSciNetGoogle Scholar
- Yannakakis, M. [1994]: On the approximation of maximum satisfiability. Journal of Algorithms 17 (1994), 475–502MathSciNetMATHCrossRefGoogle Scholar
Copyright information
© Springer-Verlag Berlin Heidelberg 2002