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.
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General Literature
Asano, T., Iwama, K., Takada, H., and Yamashita, Y. [2000]: Designing high-quality approximation algorithms for combinatorial optimization problems. IEICE Transactions on Communications/Electronics/Information and Systems E83-D (2000), 462–478
Ausiello, G., Crescenzi, P., Gambosi, G., Kann, V., Marchetti-Spaccamela, A., and Protasi, M. [1999]: Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer, Berlin 1999
Garey, M.R., and Johnson, D.S. [1979]: Computers and Intractability; A Guide to the Theory of NP-Completeness. Freeman, San Francisco 1979, Chapter 4
Hochbaum, D.S. [1996]: Approximation Algorithms for NP-Hard Problems. PWS, Boston, 1996
Horowitz, E., and Sahni, S. [1978]: Fundamentals of Computer Algorithms. Computer Science Press, Potomac 1978, Chapter 12
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 1995
Papadimitriou, C.H. [1994]: Computational Complexity, Addison-Wesley, Reading 1994, Chapter 13
Vazirani, V.V. [2001]: Approximation Algorithms. Springer, Berlin, 2001
Cited References
Ajtai, M. [1994]: Recursive construction for 3-regular expanders. Combinatorica 14 (1994), 379–416
Alizadeh, F. [1995]: Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM Journal on Optimization 5 (1995), 13–51
Appel, K., and Haken, W. [1977]: Every planar map is four colorable; Part I; Discharging. Illinois Journal of Mathematics 21 (1977), 429–490
Appel, K., Haken, W., and Koch, J. [1977]: Every planar map is four colorable; Part II; Reducibility. Illinois Journal of Mathematics 21 (1977), 491–567
Arora, S. [1994]: Probabilistic checking of proofs and the hardness of approximation problems, Ph.D. thesis, U.C. Berkeley, 1994
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), 501–555
Arora, S., and Safra, S. [1998]: Probabilistic checking of proofs. Journal of the ACM 45 (1998), 70–122
Asano, T. [2006]: An improved analysis of Goemans and Williamson’s LP-relaxation for MAX SAT. Theoretical Computer Science 354 (2006), 339–353
Bar-Yehuda, R., and Even, S. [1981]: A linear-time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms 2 (1981), 198–203
Becker, A., and Geiger, D. [1996]: Optimization of Pearl’s method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artificial Intelligence Journal 83 (1996), 1–22
Bellare, M., and Sudan, M. [1994]: Improved non-approximability results. Proceedings of the 26th Annual ACM Symposium on the Theory of Computing (1994), 184–193
Bellare, M., Goldreich, O., and Sudan, M. [1998]: Free bits, PCPs and nonapproximability — towards tight results. SIAM Journal on Computing 27 (1998), 804–915
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), 114–115
Berge, C. [1962]: Sur une conjecture relative au problème des codes optimaux. Communication, 13ème assemblée générale de l’URSI, Tokyo 1962
Berman, P., and Fujito, T. [1999]: On approximation properties of the independent set problem for low degree graphs. Theory of Computing Systems 32 (1999), 115–132
Brooks, R.L. [1941]: On colouring the nodes of a network. Proceedings of the Cambridge Philosophical Society 37 (1941), 194–197
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), 622–640
ChlebÃk, M. and ChlebÃková, J. [2006]: Complexity of approximating bounded variants of optimization problems. Theoretical Computer Science 354 (2006), 320–338
Chudnovsky, M., Cornuéjols, G., Liu, X., Seymour, P., and Vusković, K. [2005]: Recognizing Berge graphs. Combinatorica 25 (2005), 143–186
Chudnovsky, M., Robertson, N., Seymour, P., and Thomas, R. [2006]: The strong perfect graph theorem. Annals of Mathematics 164 (2006), 51–229
Chvátal, V. [1975]: On certain polytopes associated with graphs. Journal of Combinatorial Theory B 18 (1975), 138–154
Chvátal, V. [1979]: A greedy heuristic for the set cover problem. Mathematics of Operations Research 4 (1979), 233–235
Clementi, A.E.F., and Trevisan, L. [1999]: Improved non-approximability results for minimum vertex cover with density constraints. Theoretical Computer Science 225 (1999), 113–128
Deza, M.M., and Laurent, M. [1997]: Geometry of Cuts and Metrics. Springer, Berlin 1997
Dinur, I. [2007]: The PCP theorem by gap amplification. Journal of the ACM 54 (2007), Article 12
Dinur, I., and Safra, S. [2002]: On the hardness of approximating minimum vertex cover. Annals of Mathematics 162 (2005), 439–485
Erdős, P. [1967]: On bipartite subgraphs of graphs. Mat. Lapok. 18 (1967), 283–288
Feige, U. [1998]: A threshold of ln n for the approximating set cover. Journal of the ACM 45 (1998), 634–652
Feige, U. [2004]: Approximating maximum clique by removing subgraphs. SIAM Journal on Discrete Mathematics 18 (2004), 219–225
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), 182–189
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), 268–292
Fernández-Baca, D., and Lagergren, J. [1998]: On the approximability of the Steiner tree problem in phylogeny. Discrete Applied Mathematics 88 (1998), 129–145
Fulkerson, D.R. [1972]: Anti-blocking polyhedra. Journal of Combinatorial Theory B 12 (1972), 50–71
Fürer, M., and Raghavachari, B. [1994]: Approximating the minimum-degree Steiner tree to within one of optimal. Journal of Algorithms 17 (1994), 409–423
Garey, M.R., and Johnson, D.S. [1976]: The complexity of near-optimal graph coloring. Journal of the ACM 23 (1976), 43–49
Garey, M.R., Johnson, D.S., and Stockmeyer, L. [1976]: Some simplified NP-complete graph problems. Theoretical Computer Science 1 (1976), 237–267
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), 656–666
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), 1115–1145
Grötschel, M., Lovász, L., and Schrijver, A. [1988]: Geometric Algorithms and Combinatorial Optimization. Springer, Berlin 1988
Halldórsson, M.M., and Radhakrishnan, J. [1997]: Greed is good: approximating independent sets in sparse and bounded degree graphs. Algorithmica 18 (1997), 145–163
Håstad, J. [2001]: Some optimal inapproximability results. Journal of the ACM 48 (2001), 798–859
Heawood, P.J. [1890]: Map colour theorem. Quarterly Journal of Pure Mathematics 24 (1890), 332–338
Hochbaum, D.S. [1982]: Approximation algorithms for the set covering and vertex cover problems. SIAM Journal on Computing 11 (1982), 555–556
Hochbaum, D.S., and Shmoys, D.B. [1985]: A best possible heuristic for the k-center problem. Mathematics of Operations Research 10 (1985), 180–184
Holyer, I. [1981]: The NP-completeness of edge-coloring. SIAM Journal on Computing 10 (1981), 718–720
Hougardy, S., Prömel, H.J., and Steger, A. [1994]: Probabilistically checkable proofs and their consequences for approximation algorithms. Discrete Mathematics 136 (1994), 175–223
Hsu, W.L., and Nemhauser, G.L. [1979]: Easy and hard bottleneck location problems. Discrete Applied Mathematics 1 (1979), 209–216
Johnson, D.S. [1974]: Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences 9 (1974), 256–278
Khanna, S., Linial, N., and Safra, S. [2000]: On the hardness of approximating the chromatic number. Combinatorica 20 (2000), 393–415
Knuth, D.E. [1969]: The Art of Computer Programming; Vol. 2. Seminumerical Algorithms. Addison-Wesley, Reading 1969 (third edition: 1997)
König, D. [1916]: Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Mathematische Annalen 77 (1916), 453–465
Lieberherr, K., and Specker, E. [1981]: Complexity of partial satisfaction. Journal of the ACM 28 (1981), 411–421
Lovász, L. [1972]: Normal hypergraphs and the perfect graph conjecture. Discrete Mathematics 2 (1972), 253–267
Lovász, L. [1975]: On the ratio of optimal integral and fractional covers. Discrete Mathematics 13 (1975), 383–390
Lovász, L. [1979a]: On the Shannon capacity of a graph. IEEE Transactions on Information Theory 25 (1979), 1–7
Lovász, L. [1979b]: 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. 141–158
Lovász, L. [2003]: Semidefinite programs and combinatorial optimization. In: Recent Advances in Algorithms and Combinatorics (B.A. Reed, C.L. Sales, eds.), Springer, New York (2003), pp. 137–194
Mahajan, S., and Ramesh, H. [1999]: Derandomizing approximation algorithms based on semidefinite programming. SIAM Journal on Computing 28 (1999), 1641–1663
Papadimitriou, C.H., and Steiglitz, K. [1982]: Combinatorial Optimization; Algorithms and Complexity. Prentice-Hall, Englewood Cliffs 1982, pp. 406–408
Papadimitriou, C.H., and Yannakakis, M. [1991]: Optimization, approximation, and complexity classes. Journal of Computer and System Sciences 43 (1991), 425–440
Papadimitriou, C.H., and Yannakakis, M. [1993]: The traveling salesman problem with distances one and two. Mathematics of Operations Research 18 (1993), 1–12
Raghavan, P., and Thompson, C.D. [1987]: Randomized rounding: a technique for provably good algorithms and algorithmic proofs. Combinatorica 7 (1987), 365–374
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 Theory of Computing (1997), 475–484
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), 571–575
Robertson, N., Sanders, D.P., Seymour, P., and Thomas, R. [1997]: The four colour theorem. Journal of Combinatorial Theory B 70 (1997), 2–44
Sanders, P., and Steurer, D. [2005]: An asymptotic approximation scheme for multigraph edge coloring. Proceedings of the 16th Annual ACM-SIAM Symposium on Discrete Algorithms (2005), 897–906
Singh, M. and Lau, L.C. [2007]: Approximating minimum bounded degree spanning trees to within one of optimal. Proceedings of the 39th Annual ACM Symposium on Theory of Computing (2007), 661–670
SlavÃk, P. [1997]: A tight analysis of the greedy algorithm for set cover. Journal of Algorithms 25 (1997), 237–254
Stockmeyer, L.J. [1973]: Planar 3-colorability is polynomial complete. ACM SIGACT News 5 (1973), 19–25
Trevisan, L. [2004]: On local versus global satisfiability. SIAM Journal on Discrete Mathematics 17 (2004), 541–547
Vizing, V.G. [1964]: On an estimate of the chromatic class of a p-graph. Diskret. Analiz 3 (1964), 23–30 [in Russian]
Wigderson, A. [1983]: Improving the performance guarantee for approximate graph coloring. Journal of the ACM 30 (1983), 729–735
Yannakakis, M. [1994]: On the approximation of maximum satisfiability. Journal of Algorithms 17 (1994), 475–502
Zuckerman, D. [2006]: Linear degree extractors and the inapproximability of Max Clique and Chromatic Number. Proceedings of the 38th Annual ACM Symposium on Theory of Computing (2006), 681–690
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(2008). Approximation Algorithms. In: Combinatorial Optimization. Algorithms and Combinatorics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-71844-4_16
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DOI: https://doi.org/10.1007/978-3-540-71844-4_16
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