The design of approximation algorithms

The Design of Approximation Algorithms

Volumn 9780521195270, Issue , 2011, Pages 1-504

  Williamson, David P ^a   Shmoys, David B ^a,b  

^a Social Dynamics Laboratory   (United States)

^b Institute for Computational Sustainability   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

APPROXIMATION ALGORITHMS; DESIGN; DYNAMIC PROGRAMMING; MARKETING; OPERATIONS RESEARCH; OPTIMAL SYSTEMS;

ALGORITHMIC TECHNIQUES; DISCRETE OPTIMIZATION PROBLEMS; FACILITY LOCATIONS; HEURISTIC SOLUTIONS; LINEAR AND SEMIDEFINITE PROGRAMMING; LOCAL SEARCH ALGORITHM; NEAR-OPTIMAL SOLUTIONS; OPTIMIZATION PROBLEMS;

OPTIMIZATION;

EID: 84924016148     PISSN: None     EISSN: None     Source Type: Book    
DOI: 10.1017/CBO9780511921735     Document Type: Book

References (296)

1. A. A. Ageev and M. I. Sviridenko. An 0.828-approximation algorithm for the uncapacitated facility location problem. Discrete Applied Mathematics, 93:149-156, 1999.
2. A. A. Ageev and M. I. Sviridenko. Approximation algorithms for maximum coverage and max cut with given sizes of parts. In G. Cornuéjols, R. E. Burkard, and G. J. Woeginger, editors, Lecture Notes in Computer Science, vol. 1610, Integer Programming and Combinatorial Optimization, pages 17-30. Springer-Verlag, Berlin, Germany, 1999.
3. A. A. Ageev and M. I. Sviridenko. Pipage rounding: a new method of constructing algorithms with proven performance guarantee. Journal of Combinatorial Optimization, 8:307-328, 2004.
4. A. Agrawal, P. Klein, and R. Ravi. When trees collide: an approximation algorithm for the generalized Steiner problem on networks. SIAM Journal on Computing, 24:440-456, 1995.
5. F. Alizadeh. Interior point methods in semidefinite programming with applications to combinatorial optimization. SIAM Journal on Optimization, 5:13-51, 1995.
6. N. Alon, Y. Azar, G. J. Woeginger, and T. Yadid. Approximation schemes for scheduling. In Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 493-500, 1997.
7. N. Alon, R. M. Karp, D. Peleg, and D. West. A graph-theoretic game and its application to the k-server problem. SIAM Journal on Computing, 24:78-100, 1995.
8. R. Andersen and U. Feige. Interchanging distance and capacity in probabilistic mappings. CoRR, abs/0907.3631, 2009. Available at http://arxiv.org/abs/0907.3631. Accessed June 4, 2010.
9. D. L. Applegate, R. E. Bixby, V. Chvátal, and W. J. Cook. The Traveling Salesman Problem: A Computational Study. Princeton University Press, Princeton, NJ, USA, 2006.
10. S. Arnborg and A. Proskurowski. Linear time algorithms for NP-hard problems restricted to partial k-trees. Discrete Applied Mathematics, 23:11-24, 1989.
11. S. Arora. Polynomial time approximation schemes for Euclidean traveling salesman and other geometric problems. Journal of the ACM, 45:753-782, 1998.
12. S. Arora. Approximation schemes for NP-hard geometric optimization problems: a survey. Mathematical Programming, 97:43-69, 2003.
13. S. Arora, L. Babai, J. Stern, and Z. Sweedyk. The hardness of approximate optima in lattices, codes, and systems of linear equations. Journal of Computer and System Sciences, 54:317-331, 1997.
14. S. Arora and B. Barak. Computational Complexity: A Modern Approach. Cambridge University Press, New York, NY, USA, 2009.
15. S. Arora, E. Chlamtac, and M. Charikar. New approximation guarantee for chromatic number. In Proceedings of the 38th Annual ACM Symposium on the Theory of Computing, pages 215-224, 2006.
16. S. Arora, D. Karger, and M. Karpinski. Polynomial time approximation schemes for dense instances of NP-hard problems. Journal of Computer and System Sciences, 58:193-210, 1999.
17. S. Arora, J. R. Lee, and A. Naor. Euclidean distortion and the sparsest cut. Journal of the American Mathematical Society, 21:1-21, 2008.
18. S. Arora and C. Lund. Hardness of approximations. In D. S. Hochbaum, editor, Approximation Algorithms for NP-Hard Problems, chapter 10. PWS Publishing Company, Boston, MA, USA, 1997.
19. S. Arora, C. Lund, R. Motwani, M. Sudan, andM. Szegedy. Proof verification and the hardness of approximation problems. Journal of the ACM, 45:501-555, 1998.
20. S. Arora, P. Raghavan, and S. Rao. Approximation schemes for Euclidean k-medians and related problems. In Proceedings of the 30th Annual ACM Symposium on the Theory of Computing, pages 106-113, 1998.
21. S. Arora, S. Rao, and U. Vazirani. Geometry, flows, and graph-partitioning algorithms. Communications of the ACM, 51:96-105, 2008.
22. S. Arora, S. Rao, and U. Vazirani. Expander flows, geometric embeddings and graph partitioning. Journal of the ACM, 56, 2009. Article 5.
23. S. Arora and S. Safra. Probabilistic checking of proofs: a new characterization of NP. Journal of the ACM, 45:70-122, 1998.
24. V. Arya, N. Garg, R. Khandekar, A. Meyerson, K. Munagala, and V. Pandit. Local search heuristics for k-median and facility location problems. SIAM Journal on Computing, 33:544- 562, 2004.
25. A. Asadpour, M. X. Goemans, A. Madry, S. Oveis Gharan, and A. Saberi. An O(logn/log log n)-approximation algorithm for the asymmetric traveling salesman problem. In Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms, pages 379-389, 2010.
26. Y. Aumann and Y. Rabani. An O(log k) approximate min-cut max-flow theorem and approximation algorithm. SIAM Journal on Computing, 27:291-301, 1998.
27. G. Ausiello, P. Crescenzi, G. Gambosi, V. Kann, A. Marchetti-Spaccamela, and M. Protasi. Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties. Springer-Verlag, Berlin, Germany, 1999.
28. B. Awerbuch and Y. Azar. Buy-at-bulk network design. In Proceedings of the 38th Annual IEEE Symposium on Foundations of Computer Science, pages 542-547, 1997.
29. V. Bafna, P. Berman, and T. Fujito. A 2-approximation algorithm for the undirected feedback vertex set problem. SIAM Journal on Discrete Mathematics, 12:289-297, 1999.
30. B. S. Baker. Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM, 41:153-180, 1994.
31. E. Balas. The prize collecting traveling salesman problem. Networks, 19:621-636, 1989.
32. N. Bansal, R. Khandekar, and V. Nagarajan. Additive guarantees for degree-bounded directed network design. SIAM Journal on Computing, 39:1413-1431, 2009.
33. R. Bar-Yehuda. One for the price of two: a unified approach for approximating covering problems. Algorithmica, 27:131-144, 2000.
34. R. Bar-Yehuda, K. Bendel, A. Freund, and D. Rawitz. Local ratio: a unified framework for approximation algorithms. In Memorium: Shimon Even 1935-2004. ACMComputing Surveys, 36:422-463, 2004.
35. R. Bar-Yehuda and S. Even. A linear time approximation algorithm for the weighted vertex cover problem. Journal of Algorithms, 2:198-203, 1981.
36. R. Bar-Yehuda and S. Even. A local-ratio theorem for approximating the weighted vertex cover problem. Annals of Discrete Mathematics, 25:27-46, 1985.
37. R. Bar-Yehuda, D. Geiger, J. Naor, and R. M. Roth. Approximation algorithms for the feedback vertex set problem with applications to constraint satisfaction and Bayesian inference. SIAM Journal on Computing, 27:942-959, 1998.
38. R. Bar-Yehuda and D. Rawitz. On the equivalence between the primal-dual schema and the local ratio technique. SIAM Journal on Discrete Mathematics, 19:762-797, 2005.
39. Y. Bartal. Probabilistic approximation of metric spaces and its algorithmic applications. In Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, pages 184-193, 1996.
40. Y. Bartal. On approximating arbitrary metrics by tree metrics. In Proceedings of the 30th Annual ACM Symposium on the Theory of Computing, pages 161-168, 1998.
41. A. Becker and D. Geiger. Optimization of Pearl's method of conditioning and greedy-like approximation algorithms for the vertex feedback set problem. Artificial Intelligence, 83:167- 188, 1996.
42. M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCPs, and nonapproximability - towards tight results. SIAM Journal on Computing, 27:804-915, 1998.
43. M. Bellare, S. Goldwasser, C. Lund, andA. Russell. Efficient probabilistically checkable proofs and applications to approximation. In Proceedings of the 25th Annual ACM Symposium on the Theory of Computing, pages 294-304, 1993.
44. M. Bern and P. Plassmann. The Steiner problem with edge lengths 1 and 2. Information Processing Letters, 32:171-176, 1989.
45. D. P. Bertsekas and J. N. Tsitsiklis. Introduction to Probability. Athena Scientific, Nashua, NH, USA, second edition, 2008.
46. D. Bertsimas and C.-P. Teo. From valid inequalities to heuristics: a unified view of primal-dual approximation algorithms in covering problems. Operations Research, 46:503-514, 1998.
47. D. Bertsimas and J. N. Tsitsiklis. Introduction to Linear Optimization. Athena Scientific, Belmont, MA, USA, 1997.
48. D. Bienstock, M. X. Goemans, D. Simchi-Levi, and D. Williamson. A note on the prize collecting traveling salesman problem. Mathematical Programming, 59:413-420, 1993.
49. G. Birkhoff. Tres observaciones sobre el algebra lineal [in Spanish]. Revista Facultad de Ciencias Exactas, Puras y Aplicadas Universidad Nacional de Tucumán, Serie A (Matemáticas y Física Téorica), 5:147-151, 1946.
50. R. G. Bland, D. Goldfarb, and M. J. Todd. The ellipsoid method: a survey. Operations Research, 29:1039-1091, 1981.
51. A. Blum, R. Ravi, and S. Vempala. A constant-factor approximation algorithm for the k-MST problem. Journal of Computer and System Sciences, 58:101-108, 1999.
52. H. L. Bodlaender. Planar graphs with bounded treewidth. Technical Report RUU-CS-88-14, Utrecht University, Department of Computer Science, Netherlands, 1988.
53. A. Borchers and D.-Z. Du. The k-Steiner ratio in graphs. SIAM Journal on Computing, 26:857- 869, 1997.
54. C. Borell. The Brunn-Minkowski inequality in Gauss space. Inventiones Mathematicae, 30:207-216, 1975.
55. J. Bourgain. On Lipschitz embedding of finite metric spaces in Hilbert space. Israel Journal of Mathematics, 52:46-52, 1985.
56. S. C. Boyd and W. R. Pulleyblank. Optimizing over the subtour polytope of the travelling salesman problem. Mathematical Programming, 49:163-187, 1991.
57. Y. Boykov, O. Veksler, and R. Zabih. Fast approximate energy minimization via graph cuts. IEEE Transactions on Pattern Analysis and Machine Intelligence, 23:1222-1239, 2001.
58. J. Byrka and K. Aardal. An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem. SIAM Journal on Computing, 39:2212-2231, 2010.
59. J. Byrka, F. Grandoni, T. Rothvoß, and L. Sanità. An improved LP-based approximation for Steiner tree. In Proceedings of the 42nd Annual ACM Symposium on the Theory of Computing, pages 583-592, 2010.
60. G. Călinescu, H. Karloff, and Y. Rabani. An improved approximation algorithm for multiway cut. Journal of Computer and System Sciences, 60:564-574, 2000.
61. T. Carnes and D. Shmoys. Primal-dual schema for capacitated covering problems. In A. Lodi, A. Panconesi, and G. Rinaldi, editors, Lecture Notes in Computer Science, vol. 5035, Integer Programming and Combinatorial Optimization, pages 288-302. Springer, New York, NY, USA, 2008.
62. R. D. Carr, L. K. Fleischer, V. J. Leung, and C. A. Phillips. Strengthening integrality gaps for capacitated network design and covering problems. In Proceedings of the 11th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 106-115, 2000.
63. D. Chakrabarty, J. Könemann, and D. Pritchard. Integrality gap of the hypergraphic relaxation of Steiner trees:Ashort proof of a 1.55 upper bound. Operations Research Letters, 38:567-570, 2010.
64. M. Charikar and S. Guha. Improved combinatorial algorithms for facility location problems. SIAM Journal on Computing, 34:803-824, 2005.
65. M. Charikar, K. Makarychev, and Y. Makarychev. Near-optimal algorithms for unique games. In Proceedings of the 38th Annual ACM Symposium on the Theory of Computing, pages 205-214, 2006.
66. M. Charikar and B. Raghavachari. The finite capacity dial-a-ride problem. In Proceedings of the 39th Annual IEEE Symposium on Foundations of Computer Science, pages 458-467, 1998.
67. M. Charikar and A. Wirth. Maximizing quadratic programs: extending Grothendieck's inequality. In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pages 54-60, 2004.
68. S. Chawla, R. Krauthgamer, R. Kumar, Y. Rabani, and D. Sivakumar. On the hardness of approximating multicut and sparsest-cut. Computational Complexity, 15:94-114, 2006.
69. C. Chekuri and M. Bender. An efficient approximation algorithm for minimizing makespan on uniformly related machines. Journal of Algorithms, 41:212-224, 2001.
70. C. Chekuri, S. Guha, and J. Naor. The Steiner k-cut problem. SIAM Journal on Discrete Mathematics, 20:261-271, 2006.
71. H. Chernoff. A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations. Annals of Mathematical Statistics, 23:493-507, 1952.
72. E. Chlamtac, K. Makarychev, andY. Makarychev. Howto play unique games using embeddings. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pages 687-696, 2006.
73. N. Christofides. Worst-case analysis of a new heuristic for the travelling salesman problem. Report 388, Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh, PA, USA, 1976.
74. F. A. Chudak, M. X. Goemans, D. S. Hochbaum, and D. P. Williamson. A primal-dual interpretation of two 2-approximation algorithms for the feedback vertex set problem in undirected graphs. Operations Research Letters, 22:111-118, 1998.
75. F. A. Chudak, T. Roughgarden, and D. P. Williamson. Approximate k-MSTs and k-Steiner trees via the primal-dual method and Lagrangean relaxation. Mathematical Programming, 100:411-421, 2004.
76. F. A. Chudak and D. B. Shmoys. Approximation algorithms for precedence-constrainted scheduling problems on parallel machines that run at different speeds. Journal of Algorithms, 30:323-343, 1999.
77. F. A. Chudak and D. B. Shmoys. Improved approximation algorithms for the uncapacitated facility location problem. SIAM Journal on Computing, 33:1-25, 2003.
78. V. Chvátal. A greedy heuristic for the set-covering problem. Mathematics of Operations Research, 4:233-235, 1979.
79. V. Chvátal. Linear Programming. W. H. Freeman, New York, NY, USA, 1983.
80. W. Cook and P. Seymour. Tour merging via branch-decomposition. INFORMS Journal on Computing, 15:233-248, 2003.
81. W. J. Cook, W. H. Cunningham, W. R. Pulleyblank, and A. Schrijver. Combinatorial Optimization. Wiley & Sons, New York, NY, USA, 1998.
82. T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein. Introduction to Algorithms. MIT Press, Cambridge, MA, USA, third edition, 2009.
83. G. Cornuejols, M. L. Fisher, and G. L. Nemhauser. Location of bank accounts to optimize float: an analytic study of exact and approximate algorithms. Management Science, 23:789- 810, 1977.
84. G. Cornuéjols, J. Fonlupt, and D. Naddef. The traveling salesman problem on a graph and some related integer polyhedra. Mathematical Programming, 33:1-27, 1985.
85. W. H. Cunningham. Minimum cuts, modular functions, and matroid polyhedra. Networks, 15:205-215, 1985.
86. E. Dahlhaus, D. S. Johnson, C. H. Papadimitriou, P. D. Seymour, and M. Yannakakis. The complexity of multiterminal cuts. SIAM Journal on Computing, 23:864-894, 1994.
87. M. M. Deza and M. Laurent. Geometry of Cuts and Metrics. Springer, Berlin, Germany, 1997.
88. E. W. Dijkstra. A note on two problems in connexion with graphs. Numerische Mathematik, 1:269-271, 1959.
89. I. Dinur. The PCP theorem by gap amplification. Journal of the ACM, 54, 2007. Article 12.
90. I. Dinur, E. Mossel, andO. Regev. Conditonal hardness for approximate coloring. SIAM Journal on Computing, 39:843-873, 2009.
91. I. Dinur and S. Safra. The importance of being biased. In Proceedings of the 34th Annual ACM Symposium on the Theory of Computing, pages 33-42, 2002.
92. Y. Dodis and S. Khanna. Designing networks with bounded pairwise distance. In Proceedings of the 31st Annual ACM Symposium on the Theory of Computing, pages 750-759, 1999.
93. R. Durrett. The Essentials of Probability. Duxbury Press, Belmont, CA, USA, 1994.
94. R. Durrett. Elementary Probability for Applications. Cambridge University Press, New York, NY, USA, 2009.
95. J. Edmonds. Optimum branchings. Journal of Research of the National Bureau of Standards B, 71B:233-240, 1967.
96. J. Edmonds. Matroids and the greedy algorithm. Mathematical Programming, 1:127-136, 1971.
97. K. Edwards. The complexity of colouring problems on dense graphs. Theoretical Computer Science, 43:337-343, 1986.
98. F. Eisenbrand, F. Grandoni, T. Rothvoß, and G. Schäfer. Connected facility location via random facility sampling and core detouring. Journal of Computer and Systems Sciences, 76:709-726, 2010.
99. P. Erdős. Gráfok páros körüljárású részgráfjairól [On bipartite subgraphs of graphs, in Hungarian]. Matematikai Lapok, 18:283-288, 1967.
100. P. Erdős and L. Pósa. On the maximal number of disjoint circuits of a graph. Publicationes Mathematicae Debrecen, 9:3-12, 1962.
101. P. Erdős and J. L. Selfridge. On a combinatorial game. Journal of Combinatorial Theory B, 14:293-301, 1973.
102. G. Even, J. Naor, S. Rao, and B. Schieber. Fast approximate graph partitioning algorithms. SIAM Journal on Computing, 28:2187-2214, 1999.
103. G. Even, J. Naor, S. Rao, and B. Schieber. Divide-and-conquer approximation algorithms via spreading metrics. Journal of the ACM, 47:585-616, 2000.
104. S. Even, A. Itai, and A. Shamir. On the complexity of timetable and multicommodity flow problems. SIAM Journal on Computing, 5:691-703, 1976.
105. J. Fakcharoenphol, S. Rao, and K. Talwar. Algorithms column: Approximating metrics by tree metrics. SIGACT News, 35:60-70, 2004.
106. J. Fakcharoenphol, S. Rao, and K. Talwar. A tight bound on approximating arbitrary metrics by tree metrics. Journal of Computer and System Sciences, 69:485-497, 2004.
107. U. Feige. A threshold of ln n for approximating set cover. Journal of the ACM, 45:634-652, 1998.
108. U. Feige, S. Goldwasser, L. Lovász, S. Safra, and M. Szegedy. Interactive proofs and the hardness of approximating cliques. Journal of the ACM, 43:268-292, 1996.
109. U. Feige and G. Schechtman. On the optimality of the random hyperplane rounding technique for MAX CUT. Random Structures and Algorithms, 20:403-440, 2002.
110. W. Fernandez de laVega. MAX-CUT has a randomized approximation scheme in dense graphs. Random Structures and Algorithms, 8:187-198, 1996.
111. W. Fernandez de la Vega and G. L. Lueker. Bin packing can be solved within 1 + ∊ in linear time. Combinatorica, 1:349-355, 1981.
112. M. C. Ferris, O. L. Mangasarian, and S. J. Wright. Linear Programming with MATLAB. Society for Industrial and Applied Mathematics and the Mathematical Programming Society, Philadelphia, PA, USA, 2007.
113. G. Finn and E. Horowitz. Alinear time approximation algorithm for multiprocessor scheduling. BIT, 19:312-320, 1979.
114. M. L. Fisher, G. L. Nemhauser, and L. A. Wolsey. An analysis of approximations formaximizing submodular set functions - II. Mathematical Programming Study, 8:73-87, 1978.
115. L. Fleischer, J. Könemann, S. Leonardi, and G. Schäfer. Simple cost sharing schemes for multicommodity rent-or-buy and stochastic Steiner tree. In Proceedings of the 38th Annual ACM Symposium on the Theory of Computing, pages 663-670, 2006.
116. D. Fotakis. A primal-dual algorithm for online non-uniform facility location. Journal of Discrete Algorithms, 5:141-148, 2007.
117. T. Fujito. Approximating node-deletion problems for matroidal properties. Journal of Algorithms, 31:211-227, 1999.
118. M. Fürer and B. Raghavachari. Approximating the minimum degree spanning tree to within one from the optimal degree. In Proceedings of the 3rd Annual ACM-SIAM Symposium on Discrete Algorithms, pages 317-324, 1992.
119. M. Fürer and B. Raghavachari. Approximating the minimum-degree Steiner tree to within one of optimal. Journal of Algorithms, 17:409-423, 1994.
120. H. N. Gabow, M. X. Goemans, É. Tardos, and D. P. Williamson. Approximating the smallest k-edge connected spanning subgraph by LP-rounding. Networks, 53:345-357, 2009.
121. D. Gale. Optimal assignments in an ordered set: An application of matroid theory. Journal of Combinatorial Theory, 4:176-180, 1968.
122. M. R. Garey and D. S. Johnson. "Strong" NP-completeness results: motivation, examples, and implications. Journal of the ACM, 25:499-508, 1978.
123. M. R. Garey and D. S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman and Company, New York, NY, USA, 1979.
124. M. R. Garey, D. S. Johnson, and L. Stockmeyer. Some simplified NP-complete graph problems. Theoretical Computer Science, 1:237-267, 1976.
125. N. Garg. A 3-approximation for the minimum tree spanning k vertices. In Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science, pages 302-309, 1996.
126. N. Garg, V. V. Vazirani, and M. Yannakakis. Approximate max-flow min-(multi)cut theorems and their applications. SIAM Journal on Computing, 25:235-251, 1996.
127. N. Garg, V. V. Vazirani, and M. Yannakakis. Primal-dual approximation algorithms for integral flow and multicut in trees. Algorithmica, 18:3-20, 1997.
128. G. Gens and E. Levner. Complexity of approximation algorithms for combinatorial problems: a survey. SIGACT News, 12:52-65, 1980.
129. G. V. Gens and E. V. Levner. On approximation algorithms for universal scheduling problems [in Russian]. Izvestiya Akademii Nauk SSSR, Tehnicheskaya Kibernetika, 6:38-43, 1978.
130. E. N. Gilbert and H. O. Pollak. Steiner minimal trees. SIAM Journal on Applied Mathematics, 16:1-29, 1968.
131. M. Goemans and J. Kleinberg. An improved approximation ratio for the minimum latency problem. Mathematical Programming, 82:111-124, 1998.
132. M. X. Goemans. A supermodular relaxation for scheduling with release dates. In W. H. Cunningham, S. T. McCormick, and M. Queyranne, editors, Lecture Notes in Computer Science, vol. 1084, Integer Programming and Combinatorial Optimization, pages 288-300. Springer-Verlag, Berlin, Germany, 1996.
133. M. X. Goemans. Improved approximation algorithms for scheduling with release dates. In Proceedings of the 8th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 591- 598, 1997.
134. M. X. Goemans. Minimum bounded-degree spanning trees. In Proceedings of the 47th Annual IEEE Symposium on Foundations of Computer Science, pages 273-282, 2006.
135. M. X. Goemans, A. V. Goldberg, S. Plotkin, D. B. Shmoys, É. Tardos, and D. P. Williamson. Improved approximation algorithms for network design problems. In Proceedings of the 5th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 223-232, 1994.
136. M. X. Goemans, N. J. A. Harvey, K. Jain, and M. Singh. A randomized rounding algorithm for the asymmetric traveling salesman problem. CoRR, abs/0909.0941, 2009. Available at http://arxiv.org/abs/0909.0941. Accessed June 10, 2010.
137. M. X. Goemans and D. P. Williamson. New 3/4-approximation algorithms for the maximum satisfiability problem. SIAM Journal on Discrete Mathematics, 7:656-666, 1994.
138. M. X. Goemans and D. P. Williamson. A general approximation technique for constrained forest problems. SIAM Journal on Computing, 24:296-317, 1995.
139. M. X. Goemans and D. P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM, 42:1115- 1145, 1995.
140. M. X. Goemans and D. P. Williamson. The primal-dual method for approximation algorithms and its application to network design problems. In D. S. Hochbaum, editor, Approximation Algorithms for NP-Hard Problems, chapter 4. PWS Publishing Company, Boston, MA, USA, 1997.
141. T. F. Gonzalez. Clustering to minimize the maximum intercluster distance. Theoretical Computer Science, 38:293-306, 1985.
142. R. L. Graham. Bounds for certain multiprocessor anomalies. Bell System Technical Journal, 45:1563-1581, 1966.
143. R. L. Graham. Bounds on multiprocessing timing anomalies. SIAM Journal on Applied Mathematics, 17:416-429, 1969.
144. M. Grötschel, L. Lovász, and A. Schrijver. The ellipsoid method and its consequences in combinatorial optimization. Combinatorica, 1:169-197, 1981.
145. M. Grötschel, L. Lovász, and A. Schrijver. Geometric Algorithms and Combinatorial Optimization. Springer-Verlag, Berlin, Germany, 1988.
146. S. Guha and S. Khuller. Greedy strikes back: improved facility location algorithms. Journal of Algorithms, 31:228-248, 1999.
147. A. Gupta. Steiner points in tree metrics don't (really) help. In Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 220-227, 2001.
148. A. Gupta, A. Kumar, M. Pál, and T. Roughgarden. Approximation via cost-sharing: simpler and better approximation algorithms for network design. Journal of the ACM, 54, 2007. Article 11.
149. A. Gupta, A. Kumar, and T. Roughgarden. Simpler and better approximation algorithms for network design. In Proceedings of the 35th Annual ACMSymposium on the Theory of Computing, pages 365-372, 2003.
150. A. Gupta and K. Talwar. Approximating unique games. In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 99-106, 2006.
151. A. Gupta and K. Tangwongsan. Simpler analyses of local search algorithms for facility location CoRR, abs/0809.2554, 2008. Available at http://arxiv.org/abs/0809.2554. Accessed November 19, 2010.
152. V. Guruswami and S. Khanna. On the hardness of 4-coloring a 3-colorable graph. SIAM Journal on Discrete Mathematics, 18:30-40, 2004.
153. V. Guruswami, S. Khanna, R. Rajaraman, B. Shepherd, and M. Yannakakis. Near-optimal hardness results and approximation algorithms for edge-disjoint paths and related problems. Journal of Computer and System Sciences, 67:473-496, 2003.
154. M. Hajiaghayi and K. Jain. The prize-collecting generalized Steiner tree problem via a new approach of primal-dual schema. In Proceedings of the 17th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 631-640, 2006.
155. L. A. Hall, A. S. Schulz, D. B. Shmoys, and J. Wein. Scheduling tominimize average completion time: off-line and on-line approximation algorithms. Mathematics of Operations Research, 22:513-544, 1997.
156. P. Harsha, M. Charikar, M. Andrews, S. Arora, S. Khot, D. Moshkovitz, L. Zhang, A. Aazami, D. Desai, I. Gorodezky, G. Jagannathan, A. S. Kulikov, D. J. Mir, A. Newman, A. Nikolov, D. Pritchard, and G. Spencer. Limits of approximation algorithms: PCPs and unique games (DIMACS tutorial lecture notes). CoRR, abs/1002.3864, 2010. Available at http://arxiv.org/abs/1002.3864. Accessed June 2, 2010.
157. R. Hassin. Approximation schemes for the restricted shortest path problem. Mathematics of Operations Research, 17:36-42, 1992.
158. J. Håstad. Clique is hard to approximate within n1-∊. Acta Mathematica, 182:105-142, 1999.
159. J. Håstad. Some optimal inapproximability results. Journal of the ACM, 48:798-859, 2001.
160. D. S. Hochbaum. Approximation algorithms for the set covering and vertex cover problems. SIAM Journal on Computing, 11:555-556, 1982.
161. D. S. Hochbaum. Heuristics for the fixed cost median problem. Mathematical Programming, 22:148-162, 1982.
162. D. S. Hochbaum, editor. Approximation Algorithms for NP-Hard Problems. PWS Publishing Company, Boston, MA, USA, 1997.
163. D. S. Hochbaum and D. B. Shmoys. A best possible heuristic for the k-center problem. Mathematics of Operations Research, 10:180-184, 1985.
164. D. S. Hochbaum and D. B. Shmoys. A unified approach to approximation algorithms for bottleneck problems. Journal of the ACM, 33:533-550, 1986.
165. D. S. Hochbaum and D. B. Shmoys. Using dual approximation algorithms for scheduling problems: theoretical and practical results. Journal of the ACM, 34:144-162, 1987.
166. W. Hoeffding. Probability inequalities for sums of bounded random variables. Journal of the American Statistical Association, 58:13-30, 1963.
167. A. J. Hoffman. Some recent applications of the theory of linear inequalities to extremal combinatorial analysis. In R. Bellman and M. Hall, Jr., editors. Proceedings of Symposia in AppliedMathematics, vol. X, Combinatorial Analysis, pages 113-127. American Mathematical Society, Providence, RI, USA, 1960.
168. A. J. Hoffman. On simple combinatorial optimization problems. Discrete Mathematics, 106/107:285-289, 1992.
169. K. Hogstedt, D. Kimelman, V. T. Rajan, T. Roth, and M. Wegman. Graph cutting algorithms for distributed applications partitioning. ACM SIGMETRICS Performance Evaluation Review, 28:27-29, 2001.
170. I. Holyer. The NP-completeness of edge coloring. SIAM Journal on Computing, 10:718-720, 1981.
171. R. A. Horn and C. R. Johnson. Matrix Analysis. Cambridge University Press, New York, NY, USA, 1985.
172. W.-L. Hsu and G. L. Nemhauser. Easy and hard bottleneck location problems. Discrete Applied Mathematics, 1:209-215, 1979.
173. O. H. Ibarra and C. E. Kim. Fast approximation algorithms for the knapsack and sum of subset problems. Journal of the ACM, 22:463-468, 1975.
174. J. R. Jackson. Scheduling a production line to minimize maximum tardiness. Research Report 43, Management Science Research Project, University of California at Los Angeles, USA, 1955.
175. K. Jain. A factor 2 approximation algorithm for the generalized Steiner network problem. Combinatorica, 21:39-60, 2001.
176. K. Jain, M. Mahdian, E. Markakis, A. Saberi, and V. V. Vazirani. Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. Journal of the ACM, 50:795- 824, 2003.
177. K. Jain and V. V. Vazirani. Approximation algorithms for metric facility location and kmedian problems using the primal-dual schema and Lagrangian relaxation. Journal of the ACM, 48:274-296, 2001.
178. D. S. Johnson. Near-Optimal Bin Packing Algorithms. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, June 1973.
179. D. S. Johnson. Approximation algorithms for combinatorial problems. Journal of Computer and System Sciences, 9:256-278, 1974.
180. M. Jünger andW. Pulleyblank. Newprimal and dualmatching heuristics. Algorithmica, 13:357- 380, 1995.
181. N. Kahale. On reducing the cut ratio to the multicut problem. Techical Report 93-78, DIMACS, Rutgers, Piscataway, NJ, USA, 1993.
182. D. Karger, R. Motwani, and M. Sudan. Approximate graph coloring by semidefinite programming. Journal of the ACM, 45:246-265, 1998.
183. D. R. Karger. Global min-cuts in RNC, and other ramifications of a simple min-cut algorithm. In Proceedings of the 4th Annual ACM-SIAM Symposium onDiscrete Algorithms, pages 21-30, 1993.
184. D. R. Karger. Minimum cuts in near-linear time. Journal of the ACM, 47:46-76, 2000.
185. D. R. Karger, P. Klein, C. Stein, M. Thorup, and N. E. Young. Rounding algorithms for a geometric embedding of minimum multiway cut. Mathematics of Operations Research, 29:436-461, 2004.
186. O. Kariv and S. L. Hakimi. An algorithmic approach to network location problems. II: the p-medians. SIAM Journal on Applied Mathematics, 37:539-560, 1979.
187. N. Karmarkar and R. M. Karp. An efficient approximation scheme for the one-dimensional bin-packing problem. In Proceedings of the 23rd Annual IEEE Symposium on Foundations of Computer Science, pages 312-320, 1982.
188. J. O. Kephart, G. B. Sorkin, W. C. Arnold, D. M. Chess, G. J. Tesauro, and S. R. White. Biologically inspired defenses against computer viruses. In Proceedings of the International Joint Conference on Artificial Intelligence, pages 985-996, 1995.
189. L. G. Khachiyan. A polynomial algorithm in linear programming [in Russian]. Doklady Akademii Nauk SSSR, 244:1093-1096, 1979.
190. S. Khanna, N. Linial, and S. Safra. On the hardness of approximating the chromatic number. Combinatorica, 20:393-415, 2000.
191. S. Khot. Lecture notes from Fall 2004, Georgia Tech CS 8002: PCPs and the hardness of approximation, lecture 3: hardness of set cover. Available at http://www.cs.nyu.edu/~khot/pcp-lecnotes/lec3.ps. Accessed June 2, 2010.
192. S. Khot. On the power of unique 2-prover 1-round games. In Proceedings of the 34th Annual ACM Symposium on the Theory of Computing, pages 767-775, 2002.
193. S. Khot, G. Kindler, E. Mossel, and R. O'Donnell. Optimal inapproximability results for MAX-CUT and other 2-variable CSPs? SIAM Journal on Computing, 37:319-357, 2007.
194. S. Khot and O. Regev. Vertex cover might be hard to approximate to with 2-∊. Journal of Computer and System Sciences, 74:335-349, 2008.
195. H. Kise, T. Ibaraki, and H. Mine. Performance analysis of six approximation algorithms for the one-machine maximum lateness scheduling problem with ready times. Journal of the Operations Research Society of Japan, 22:205-224, 1979.
196. P. Klein and R. Ravi. Anearly best-possible approximation algorithm for node-weighted Steiner trees. Journal of Algorithms, 19:104-115, 1995.
197. J. Kleinberg and É. Tardos. Approximation algorithms for classification problemswith pairwise relationships: metric labeling and Markov random fields. Journal of the ACM, 49:616-639, 2002.
198. J. Kleinberg and É. Tardos. Algorithm Design. Pearson Education, Boston, MA, USA, 2006.
199. J. M. Kleinberg. Approximation Algorithms for Disjoint Paths Problems. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, May 1996.
200. D. E. Knuth. The Art of Computer Programming, vol. 2, Seminumerical Algorithms. Addison- Wesley, Reading, MA, USA, third edition, 1998.
201. D. Kőnig. Grafokés alkalmazásuk a determinánsokés a halmazok elméletére [in Hungarian]. Mathematikai és Természettudományi Értesitő, 34:104-119, 1916.
202. G. Konjevod, R. Ravi, and F. Sibel Salman. On approximating planar metrics by tree metrics. Information Processing Letters, 80:213-219, 2001.
203. B. Korte and J. Vygen. Combinatorial Optimization. Springer, Berlin, Germany, fourth edition, 2007.
204. G. Kortsarz, R. Krauthgamer, and J. R. Lee. Hardness of approximation for vertex-connectivity network design problems. SIAM Journal on Computing, 33:704-720, 2004.
205. M. R. Korupolu, C. G. Plaxton, and R. Rajaraman. Analysis of a local search heuristic for facility location problems. Journal of Algorithms, 37:146-188, 2000.
206. A. A. Kuehn and M. J. Hamburger. A heuristic program for locating warehouses. Management Science, 9:643-666, 1963.
207. Y. Lando and Z. Nutov. Inapproximability of survivable networks. Theoretical Computer Science, 410:2122-2125, 2009.
208. L. C. Lau, R. Ravi, and M. Singh. IterativeMethods in CombinatorialOptimization. Cambridge University Press, New York, NY, USA, 2011.
209. L. C. Lau and M. Singh. Iterative rounding and relaxation. To appear in RIMS Kôkyûroku Bessatsu, 2008. Available at http://www.cse.cuhk.edu/~chi/papers/relaxation.pdf. Accessed November 19, 2010.
210. E. Lawler, J. Lenstra, A. Rinnooy Kan, and D. Shmoys. The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization. John Wiley & Sons, Chichester, UK, 1985.
211. E. L. Lawler. Fast approximation algorithms for knapsack problems. Mathematics ofOperations Research, 4:339-356, 1979.
212. J. R. Lee. Distance scales, embeddings, and metrics of negative type. Unpublished manuscript, 2005. Available at http://www.cs.washington.edu/homes/jrl/papers/soda05-full.pdf. Accessed November 19, 2010.
213. T. Leighton and S. Rao. An approximate max-flow min-cut theorem for uniform multicommodity flow problems with applications to approximation algorithms. In Proceedings of the 29th Annual IEEE Symposium on Foundations of Computer Science, pages 422-431, 1988.
214. T. Leighton and S. Rao. Multicommodity max-flowmin-cut theorems and their use in designing approximation algorithms. Journal of the ACM, 46:787-832, 1999.
215. J. K. Lenstra, D. B. Shmoys, and É. Tardos. Approximation algorithms for scheduling unrelated parallel machines. Mathematical Programming, 46:259-271, 1990.
216. K. J. Lieberherr and E. Specker. Complexity of partial satisfaction. Journal of the ACM, 28:411-421, 1981.
217. N. Linial, E. London, and Y. Rabinovich. The geometry of graphs and some of its algorithmic applications. Combinatorica, 15:215-245, 1995.
218. L. Lovász. On the ratio of optimal integral and fractional covers. Discrete Mathematics, 13:383-390, 1975.
219. L. Lovász. On the Shannon capacity of a graph. IEEE Transactions on Information Theory, IT-25:1-7, 1979.
220. C. Lund andM. Yannakakis. On the hardness of approximatingminimization problems. Journal of the ACM, 41:960-981, 1994.
221. N. Maeda, H. Nagamochi, and T. Ibaraki. Approximate algorithms for multiway objective point split problems of graphs [in Japanese]. Surikaisekikenkyusho Kôkyûroku, 833:98-109, 1993.
222. S. Mahajan and H. Ramesh. Derandomizing approximation algorithms based on semidefinite programming. SIAM Journal on Computing, 28:1641-1663, 1999.
223. C. Mathieu andW. Schudy. Yet another algorithm for densemax cut: go greedy. In Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 176-182, 2008.
224. A. Megretski. Relaxations of quadratic programs in operator theory and system analysis. In A. A. Borichev and N. K. Nikolski, editors, Systems, Approximation, Singular Integral Operators, and Related Topics: InternationalWorkshop on Operator Theory and Applications, IWOTA 2000, pages 365-392. Birkhäuser, Basel, Switzerland, 2001.
225. J. S. B. Mitchell. Guillotine subdivisions approximate polygonal subdivisions: a simple polynomial-time approximation scheme for geometric TSP, k-MST, and related problems. SIAM Journal on Computing, 28:1298-1309, 1999.
226. M. Mitzenmacher and E. Upfal. Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, New York, NY, USA, 2005.
227. E. Mossel, R. O'Donnell, andK. Oleszkiewicz. Noise stability of functions with lowinfluences: Invariance and optimality. Annals of Mathematics, 171:295-341, 2010.
228. R. Motwani and P. Raghavan. Randomized Algorithms. Cambridge University Press, NewYork, NY, USA, 1995.
229. C. Nagarajan and D. P. Williamson. Offline and online facility leasing. In A. Lodi, A. Panconesi, and G. Rinaldi, editors, Lecture Notes in Computer Science, vol. 5035, Integer Programming and Combinatorial Optimization, pages 303-315. Springer, Berlin, Germany, 2008.
230. V. Nagarajan, R. Ravi, and M. Singh. Simpler analysis of LP extreme points for traveling salesman and survivable network design problems. Operations Research Letters, 38:156-160, 2010.
231. G. L. Nemhauser and L. E. Trotter, Jr. Vertex packings: Structural properties and algorithms. Mathematical Programming, 8:232-248, 1975.
232. G. L. Nemhauser and L. A. Wolsey. Integer and Combinatorial Optimization. Wiley, New York, NY, USA, 1988.
233. G. L. Nemhauser, L. A. Wolsey, andM. L. Fisher. An analysis of approximations for maximizing submodular set functions - I. Mathematical Programming, 14:265-294, 1978.
234. A. Nemirovski, C. Roos, and T. Terlaky. On maximization of quadratic form over intersection of ellipsoids with common center. Mathematical Programming, 86:463-473, 1999.
235. Y. Nesterov. Semidefinite relaxation and nonconvex quadratic optimization. Optimization Methods and Software, 9:141-160, 1998.
236. Y. Nesterov and A. Nemirovskii. Interior-Point Polynomial Algorithms in Convex Programming. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1994.
237. C. H. Norton. Problems in Discrete Optimization. PhD thesis, Massachusetts Institute of Technology, Cambridge, MA, USA, September 1993.
238. C. H. Papadimitriou and K. Steiglitz. CombinatorialOptimization: Algorithms and Complexity. Prentice-Hall, Englewood Cliffs, NJ, USA, 1982. Reprinted by Dover Publications, Mineola, NY, USA, 1998.
239. C. H. Papadimitriou and S. Vempala. On the approximability of the traveling salesman problem. Combinatorica, 26:101-120, 2006.
240. C. H. Papadimitriou and M. Yannakakis. Optimization, approximation, and complexity classes. Journal of Computer and System Sciences, 43:425-440, 1991.
241. C. Phillips, C. Stein, and J. Wein. Minimizing average completion time in the presence of release dates. Mathematical Programming, 82:199-223, 1998.
242. H. J. Prömel and A. Steger. The Steiner Tree Problem: A Tour through Graphs, Algorithms, and Complexity. Vieweg, Braunschweig, Germany, 2002.
243. M. Queyranne. Structure of a simple scheduling polyhedron. Mathematical Programming, 58:263-285, 1993.
244. H. Räcke. Optimal hierarchical decompositions for congestion minimization in networks. In Proceedings of the 40th Annual ACMSymposium on the Theory of Computing, pages 255-264, 2008.
245. J. Radhakrishnan and M. Sudan. On Dinur's proof of the PCP theorem. Bulletin of the American Mathematical Society, 44:19-61, 2007.
246. R. Rado. Note on independence functions. Proceedings of the London Mathematical Society, 3-7:300-320, 1957.
247. P. Raghavan and C. D. Thompson, Randomized rounding: a technique for provably good algorithms and algorithmic proofs, Combinatorica, 7: 365-374, 1987.
248. P. Raghavendra. Optimal algorithms and inapproximability results for every CSP? In Proceedings of the 40th Annual ACM Symposium on the Theory of Computing, pages 245-254, 2008.
249. P. Raghavendra and D. Steurer. How to round any CSP. In Proceedings of the 50th Annual IEEE Symposium on Foundations of Computer Science, pages 586-594, 2009.
250. R. Raz. A parallel repetition theorem. SIAM Journal on Computing, 27:763-803, 1998.
251. A. Rényi. Probability Theory. North-Holland, Amsterdam, Netherlands, 1970.
252. N. Robertson and P. D. Seymour. Graph minors. II. Algorithmic aspects of tree-width. Journal of Algorithms, 7:309-322, 1986.
253. N. Robertson and P. D. Seymour. Graph minors. X. Obstructions to tree-decomposition. Journal of Combinatorial Theory B, 52:153-190, 1991.
254. G. Robins and A. Zelikovsky. Tighter bounds for graph Steiner tree approximation. SIAM Journal on Discrete Mathematics, 19:122-134, 2005.
255. D. J. Rosenkrantz, R. E. Stearns, and P. M. Lewis II. An analysis of several heuristics for the traveling salesman problem. SIAM Journal on Computing, 6:563-581, 1977.
256. S. Ross. A First Course in Probability. Prentice Hall, Englewood Cliffs, NJ, USA, eighth edition, 2009.
257. S. Sahni and T. Gonzalez. P-complete approximation problems. Journal of the ACM, 23:555- 565, 1976.
258. S. K. Sahni. Algorithms for scheduling independent tasks. Journal of the ACM, 23:116-127, 1976.
259. H. Saran and V. V. Vazirani. Finding k cuts within twice the optimal. SIAM Journal on Computing, 24:101-108, 1995.
260. F. Schalekamp and D. B. Shmoys. Universal and a priori TSP. Operations Research Letters, 36:1-3, 2008.
261. A. S. Schulz and M. Skutella. Scheduling unrelated machines by randomized rounding. SIAM Journal on Discrete Mathematics, 15:450-469, 2002.
262. D. B. Shmoys. Cut problems and their application to divide-and-conquer. In D. S. Hochbaum, editor, Approximation Algorithms for NP-Hard Problems, chapter 5. PWS Publishing Company, Boston, MA, USA, 1997.
263. D. B. Shmoys and É. Tardos. An approximation algorithm for the generalized assignment problem. Mathematical Programming, 62:461-474, 1993.
264. D. B. Shmoys, É. Tardos, and K. Aardal. Approximation algorithms for facility location problems. In Proceedings of the 29th Annual ACM Symposium on the Theory of Computing, pages 265-274, 1997.
265. D. B. Shmoys, J. Wein, and D. P. Williamson. Scheduling parallel machines on-line. SIAM Journal on Computing, 24:1313-1331, 1995.
266. D. B. Shmoys and D. P. Williamson. Analyzing the Held-Karp TSP bound: a monotonicity property with application. Information Processing Letters, 35:281-285, 1990.
267. N. Z. Shor. Cut-off method with space extension in convex programming problems [in Russian]. Kibernetika, 13:94-95, 1977.
268. M. Singh. Iterative Methods in Combinatorial Optimization. PhD thesis, Carnegie Mellon University, Pittsburgh, PA, USA, May 2008.
269. M. Singh and L. Lau. Approximating minimum bounded degree spanning trees to within one of optimal. In Proceedings of the 39th Annual ACM Symposium on the Theory of Computing, pages 661-670, 2007.
270. W. E. Smith. Various optimizers for single-stage production. Naval Research Logistics Quarterly, 3:59-66, 1956.
271. J. Spencer. Ten Lectures on the Probabilistic Method. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 1987.
272. D. Steurer and N. K. Vishnoi. Connections between unique games and multicut. Report TR09-125, Electronic Colloquium on Computational Complexity, Potsdam, Germany, 2009. Available at http://eccc.hpi-web.de/report/2009/125/. Accessed November 19, 2010.
273. G. Strang. Linear Algebra and Its Applications. Brooks/Cole, Stanford, CT, USA, fourth edition, 2005.
274. G. Strang. Introduction to Linear Algebra. Wellesley-Cambridge Press, Wellesley, MA, USA, fourth edition, 2009.
275. V. N. Sudakov and B. S. Tsirel'son. Extremal properties of semi-spaces for spherically symmetric measures [in Russian]. In V. N. Sudakov, editor, Zapiski Nauchnykh Seminarov LOMI, vol. 41, Problems of the Theory of Probability Distributions. Part II, pages 14-24. Nauka, Leningrad, Russia, 1974.
276. O. Svensson. Santa Claus schedules jobs on unrelated machines. CoRR, abs/1011.1168, 2010. Available at http://arxiv.org/abs/1011.1168. Accessed November 4, 2010.
277. C. Swamy. Correlation clustering: maximizing agreements via semidefinite programming. In Proceedings of the 15th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 519- 520, 2004.
278. A. Tamir. An O(pn2) algorithm for the p-median and related problems in tree graphs. Operations Research Letters, 19:59-64, 1996.
279. L. Trevisan. Positive linear programming, parallel approximation, and PCP's. In J. Diaz and M. Serna, editors, Lecture Notes in Computer Science, vol. 1136, Algorithms - ESA '96, pages 62-75. Springer-Verlag, Berlin, Germany, 1996.
280. L. Trevisan. Parallel approximation algorithms by positive linear programming. Algorithmica, 21:72-88, 1998.
281. L. Trevisan. Inapproximabilité des problèmes d'optimisation combinatoire [Inapproximability of combinatorial optimization problems, in French]. In V. T. Paschos, editor, Optimisation combinatoire, vol. 2, Concepts avancés, chapter 3 (Séries Informatique et systèmes d'information). Lavoisier, Paris, France, 2005. English version available at http://www.cs.berkeley.edu/~luca/pubs/inapprox.pdf. Accessed June 2, 2010.
282. L. Trevisan. Approximation algorithms for unique games. Theory of Computing, 4:111-128, 2008. Online journal available at http://theoryofcomputing.org.
283. V. V. Vazirani. Approximation Algorithms. Springer, Berlin, Germany, second edition, 2004.
284. V. G. Vizing. On an estimate of the chromatic class of a p-graph [in Russian]. Diskretnyĭ Analiz, 3:25-30, 1964.
285. H. Whitney. On the abstract properties of linear dependence. American Journal of Mathematics, 57:509-533, 1935.
286. A. Wigderson. Improving the performance guarantee of approximate graph coloring. Journal of the ACM, 30:729-735, 1983.
287. D. P. Williamson. On the Design of Approximation Algorithms for a Class of Graph Problems. PhD thesis, MIT, Cambridge, MA, USA, September 1993. Also appears as Tech Report MIT/LCS/TR-584.
288. D. P. Williamson. The primal-dual method for approximation algorithms. Mathematical Programming, 91:447-478, 2002.
289. D. P. Williamson andA. van Zuylen. Asimpler and better derandomization of an approximation algorithm for single source rent-or-buy. Operations Research Letters, 35:707-712, 2007.
290. H. Wolkowicz, R. Saigal, and L. Vandenberghe, editors. Handbook of Semidefinite Programming: Theory, Algorithms, and Applications. Kluwer Academic Publishers, Boston, MA, USA, 2000.
291. L. A. Wolsey. Heuristic analysis, linear programming and branch and bound. Mathematical Programming Study, 13:121-134, 1980.
292. L. A. Wolsey. Mixed integer programming formulations for production planning and scheduling problems. Invited talk at the 12th International Symposium on Mathematical Programming, MIT, Cambridge, MA, USA, 1985.
293. M. Yannakakis. On the approximation of maximum satisfiability. Journal of Algorithms, 17:475-502, 1994.
294. A. Z. Zelikovsky. An 11/6-approximation algorithm for the network Steiner problem. Algorithmica, 9:463-470, 1993.
295. L. Zhao, H. Nagamochi, and T. Ibaraki. Greedy splitting algorithms for approximating multiway partition problems. Mathematical Programming, 102:167-183, 2005.
296. D. Zuckerman. Linear degree extractors and the inapproximability of max clique and chromatic number. Theory of Computing, 3:103-128, 2007. Online journal available at http://theoryofcomputing.org.