Abstract
Several sequential approximation algorithms are based on the following paradigm: solve a linear or semidefinite programming relaxation, then use randomized rounding to convert fractional solutions of the relaxation into integer solutions for the original combinatorial problem. We demonstrate that such a paradigm can also yield parallel approximation algorithms by showing how to convert certain linear programming relaxations into essentially equivalent positive linear programming [18] relaxations that can be near-optimally solved in NC. Building on this technique, and finding some new linear programming relaxations,we develop improved parallel approximation algorithms for Max Sat, Max Directed Cut, and Max kCSP. We also show a connection between probabilistic proof checking and a restricted version of Max kCSP. This implies that our approximation algorithm for Max kCSP can be used to prove inclusion in P for certain PCP classes.
Research partially supported by the HCM SCOOP project of the European Union. Part of this work was done while the author was visiting the Departament de Llenguatges i Sistemes Informatics of the Universitat Politècnica de Catalunya
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References
N. Alon and J. Spencer. The Probabilistic Method. Wiley Interscience, 1992.
S. Arora, D. Karger, and M. Karpinski. Polynomial time approximation schemes for dense instances of NP-hard problems. In Proceedings of the 27th A CM Symposium on Theory of Computing, pages 284–293, 1995.
C. Bazgan. Personal communication. 1996.
M. Bellare, O. Goldreich, and M. Sudan. Free bits, PCP's and non-approximability — towards tight results (3rd version). Technical Report TR95-24, ECCC, 1995. Extended abstract in Proc. of FOCS'95.
G. Bongiovanni, P. Crescenzi, and S. De Agostino. Descriptive complexity and parallel approximation of optimization problems. Manuscript, 1991.
D.P. Bovet and P. Crescenzi. Introduction to the Theory of Complexity. Prentice Hall, 1993.
B. Chor and M. Sudan. A geometric approach to betweennes. In Proceedings of the 3rd European Symposium on Algorithms, 1995.
U. Feige and M.X. Goemans. Approximating the value of two provers proof systems, with applications to MAX 2SAT and MAX DICUT. In Proceedings of the 3rd Israel Symposium on Theory of Computing and Systems, 1995.
M.X. Goemans and D.P. Williamson. New 3/4-approximation algorithms for the maximum satisfiability problem. SIAM Journal on Discrete Mathematics, 7(4):656–666, 1994. Preliminary version in Proc. of IPCO'93.
M.X. Goemans and D.P. Williamson. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. Journal of the ACM, 42(6):1115–1145, 1995. Preliminary version in Proc. of STOC'94.
D.J. Haglin. Approximating maximum 2-CNF satisfiability. Parallel Processing Letters, 2:181–187, 1992.
D. Hochbaum. Approximation algorithms for set covering and vertex cover problems. SIAM Journal on Computing, 11:555–556, 1982.
H. B. Hunt III, M.V. Marathe, V. Radhakrishnan, S.S. Ravi, D.J. Rosenkrantz, and R.E. Stearns. Every problem in MAX SNP has a parallel approximation algorithm. Manuscript, 1993.
D. Karger, R. Motwani, and M. Sudan. Approximate graph coloring by semidefinite programming. In Proceedings of the 35th IEEE Symposium on Foundations of Computer Science, 1994.
S. Khanna, R. Motwani, M. Sudan, and U. Vazirani. On syntactic versus computational views of approximability. In Proceedings of the 35th IEEE Symposium on Foundations of Computer Science, pages 819–830, 1994.
H.C. Lau and O. Watanabe. Randomized approximation of the constraint satisfaction problem. In Proceedings of the 5th Scandinavian Workshop on Algorithm Theory, 1996.
M. Luby. A simple parallel algorithm for the maximal independent set problem. SIAM Journal on Computing, 15:1036–1053, 1986.
M. Luby and N. Nisan. A parallel approximation algorithm for positive linear programming. In Proceedings of the 25th ACM Symposium on Theory of Computing, pages 448–457, 1993.
M. Luby and A. Wigderson. Pairwise independence and derandomization. Technical Report TR-95-035, International Computer Science Institute, 1995.
C. H. Papadimitriou and M. Yannakakis. Optimization, approximation, and complexity classes. Journal of Computer and System Sciences, 43:425–440, 1991. Preliminary version in Proc. of STOC'88.
C.H. Papadimitriou. Computational Complexity. Addison-Wesley, 1994.
P. Raghavan and C.D. Thompson. Randomized rounding: a technique forprovably good algorithms and algorithmic proofs. Combinatorica, 7:365–374, 1987.
M. Serna. Approximating linear programming is log-space complete for P. Information Processing Letters, 37, 1991.
M. Serna and F. Xhafa. On parallel versus sequential approximation. In Proceedings of the 3rd European Symposium on Algorithms, pages 409–419, 1995.
D. Shmoys. Computing near-optimal solutions to combinatorial optimization problems. In Combinatorial Optimization. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 20, 1995.
L. Trevisan, G.B. Sorkin, M. Sudan, and D.P. Williamson. Gadgets, approximation and linear programming. Manuscript, 1996.
M. Yannakakis. On the approximation of maximum satisfiability. Journal of Algorithms, 17, 1994. Preliminary version in Proc. of SODA '92.
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Trevisan, L. (1996). Positive linear programming, parallel approximation and PCP's. In: Diaz, J., Serna, M. (eds) Algorithms — ESA '96. ESA 1996. Lecture Notes in Computer Science, vol 1136. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-61680-2_47
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DOI: https://doi.org/10.1007/3-540-61680-2_47
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