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15] R. Impagliazzo and D. Zuckerman. How to recycle random bits. In Proc. 30th Symposium on Foundations of Computer Science, pages 248{ 253. IEEE Computer Society Press, Los Alamitos, CA, 1989. 16] D. S. Johnson. Approximation algorithms for combinatorial problems. J. Computer and System Sciences, 9:256{278, 1974. 17] D. Kozen. Lower bounds for natural proof systems. In Proc. 18rd Symposium on Foundations of Computer Science, pages 254{266. IEEE Computer Society Press, Los Alamitos, CA, 1977. 18] C. Lund, L. Fortnow, H. Karlo , and N. Nisan. Algebraic methods for interactive proof systems. J. ACM, 39:859{868, 1992. 19] M. Marathe, H. Hunt III, R. Stearns, and V. Radhakrishnan. Hierarchical speci cations and polynomial-time approximation schemes for PSPACEcomplete problems. In Proc. 26th ACM Symposium on the Theory of Computing, pages 468{477. ACM, New York, 1994. 20] C. Papadimitriou. Games against nature. J. Computer and System Sciences, 31:288{301, 1985. 21] C. Papadimitriou and M. Yannakakis. Optimization, approximation, and complexity classes. J. Computer and System Sciences, 43:425{440, 1991. 22] T. J. Schaefer. On the complexity of some two-person perfect-information games. J. Computer and System Sciences, 16:185{225, 1978. 23] A. Shamir. IP = PSPACE. J. ACM, 39:869{877, 1992.
and q(n) are both arbitrary polynomials, then the universal debater in a PCDS can be replaced by a random debater without loss of generality. In 10], we show that, even if r(n) = log n and q(n) = 1, one can replace the universal debater by a random debater and still retain the power to recognize any language in PSPACE. This fact has implications for the hardness of approximating stochastic PSPACE-hard functions, of the type studied by Papadimitriou 20]. We have described PSPACE-hard functions that do not have PTAS's, unless some unexpected collapse occurs. It is not hard to de ne a PSPACE-hard function that does have a PTAS, but the straightforward examples are arti cial. We were thus led to ask in 8, 9] whether there is a natural PSPACE-hard function that has a PTAS. A positive answer to this question is provided in 19]. Bodlaender 5] has extended our results by showing that MAX-Q-3SAT can be approximated within some 0 < < 1 and by providing a simpler proof of the fact that MAX GGEOG is PSPACE-hard to approximate; his proof that approximating MAX GGEOG is hard does not involve PCDS's. Hunt et al. 14] showed, also using direct reduction arguments, that it is PSPACEhard to approximate several other constrained optimization problems within certain factors. These problems include MAX-Q-FORMULA, a generalization of (log n)-MAX-Q-FORMULA, where the \clauses" are general formulas (with no restrictions on the number of variables per \clause"). It is not known whether characterizations of EXP and NEXP can be found that are similar to the PCP and PCD characterizations of NP and PSPACE, respectively, and that lead to interesting nonappoxability results for problems that are complete for EXP or NEXP.
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3] R. Aumann and S. Hart, editors. Handbook of Game Theory, volume 1. North Holland, Amsterdam, 1992. 4] L. Babai and S. Moran. Arthur-Merlin games: A randomized proof system and a hierarchy of complexity classes. J. Computer and System Sciences, 36:254{276, 1988. 5] H. Bodlaender. Private communication. 6] A. Cohen and A. Wigderson. Dispersers, deterministic ampli cation, and weak random sources. In Proc. 30th Symposium on Foundations of Computer Science, pages 14{19. IEEE Computer Society Press, Los Alamitos, CA, 1989. 7] A. Condon. The complexity of the max word problem and the power of one-way interactive proof systems. Computational Complexity, 3:292{305, 1993. 8] A. Condon, J. Feigenbaum, C. Lund, and P. Shor. Probabilistically checkable debate systems and approximation algorithms for PSPACE-hard functions. Technical memorandum, AT&T Bell Laboratories, January 1993. 9] A. Condon, J. Feigenbaum, C. Lund, and P. Shor. Probabilistically checkable debate systems and approximation algorithms for PSPACE-hard functions (extended abstract). In Proc. 25th ACM Symposium on the Theory of Computing, pages 305{314. ACM, New York, 1993. 10] A. Condon, J. Feigenbaum, C. Lund, and P. Shor. Random debaters and the hardness of approximating stochastic functions. In Proc. 9th Conference on Structure in Complexity Theory, pages 280{293. IEEE Computer Society Press, Los Alamitos, CA, 1994. Final version to appear in SIAM Journal on Computing. 11] U. Feige, S. Goldwasser, L. Lovasz, M. Safra, and M. Szegedy. Approximating clique is almost NP-complete. In Proc. 32nd Symposium on Foundations of Computer Science, pages 2{12. IEEE Computer Society Press, Los Alamitos, CA, 1991. 12] M. R. Garey and D. S. Johnson. Computers and Intractibility: A guide to the theory of NP-Completeness. W.H. Freeman and Company, San Fransisco, 1979. 13] P. Gemmell and M. Sudan. Highly resilient correctors for polynomials. Inf. Proc. Letters, 43:169{174, 1992. 14] H. Hபைடு நூலகம்nt III, M. Marathe, and R. Stearns. Generalized CNF satis ability problems and non-e cient approximability. In Proc. 9th Conference on Structure in Complexity Theory, pages 356{366. IEEE Computer Society Press, Los Alamitos, CA, 1994. 28
and q(n) are both arbitrary polynomials, then the universal debater in a PCDS can be replaced by a random debater without loss of generality. In 10], we show that, even if r(n) = log n and q(n) = 1, one can replace the universal debater by a random debater and still retain the power to recognize any language in PSPACE. This fact has implications for the hardness of approximating stochastic PSPACE-hard functions, of the type studied by Papadimitriou 20]. We have described PSPACE-hard functions that do not have PTAS's, unless some unexpected collapse occurs. It is not hard to de ne a PSPACE-hard function that does have a PTAS, but the straightforward examples are arti cial. We were thus led to ask in 8, 9] whether there is a natural PSPACE-hard function that has a PTAS. A positive answer to this question is provided in 19]. Bodlaender 5] has extended our results by showing that MAX-Q-3SAT can be approximated within some 0 < < 1 and by providing a simpler proof of the fact that MAX GGEOG is PSPACE-hard to approximate; his proof that approximating MAX GGEOG is hard does not involve PCDS's. Hunt et al. 14] showed, also using direct reduction arguments, that it is PSPACEhard to approximate several other constrained optimization problems within certain factors. These problems include MAX-Q-FORMULA, a generalization of (log n)-MAX-Q-FORMULA, where the \clauses" are general formulas (with no restrictions on the number of variables per \clause"). It is not known whether characterizations of EXP and NEXP can be found that are similar to the PCP and PCD characterizations of NP and PSPACE, respectively, and that lead to interesting nonappoxability results for problems that are complete for EXP or NEXP.
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3] R. Aumann and S. Hart, editors. Handbook of Game Theory, volume 1. North Holland, Amsterdam, 1992. 4] L. Babai and S. Moran. Arthur-Merlin games: A randomized proof system and a hierarchy of complexity classes. J. Computer and System Sciences, 36:254{276, 1988. 5] H. Bodlaender. Private communication. 6] A. Cohen and A. Wigderson. Dispersers, deterministic ampli cation, and weak random sources. In Proc. 30th Symposium on Foundations of Computer Science, pages 14{19. IEEE Computer Society Press, Los Alamitos, CA, 1989. 7] A. Condon. The complexity of the max word problem and the power of one-way interactive proof systems. Computational Complexity, 3:292{305, 1993. 8] A. Condon, J. Feigenbaum, C. Lund, and P. Shor. Probabilistically checkable debate systems and approximation algorithms for PSPACE-hard functions. Technical memorandum, AT&T Bell Laboratories, January 1993. 9] A. Condon, J. Feigenbaum, C. Lund, and P. Shor. Probabilistically checkable debate systems and approximation algorithms for PSPACE-hard functions (extended abstract). In Proc. 25th ACM Symposium on the Theory of Computing, pages 305{314. ACM, New York, 1993. 10] A. Condon, J. Feigenbaum, C. Lund, and P. Shor. Random debaters and the hardness of approximating stochastic functions. In Proc. 9th Conference on Structure in Complexity Theory, pages 280{293. IEEE Computer Society Press, Los Alamitos, CA, 1994. Final version to appear in SIAM Journal on Computing. 11] U. Feige, S. Goldwasser, L. Lovasz, M. Safra, and M. Szegedy. Approximating clique is almost NP-complete. In Proc. 32nd Symposium on Foundations of Computer Science, pages 2{12. IEEE Computer Society Press, Los Alamitos, CA, 1991. 12] M. R. Garey and D. S. Johnson. Computers and Intractibility: A guide to the theory of NP-Completeness. W.H. Freeman and Company, San Fransisco, 1979. 13] P. Gemmell and M. Sudan. Highly resilient correctors for polynomials. Inf. Proc. Letters, 43:169{174, 1992. 14] H. Hபைடு நூலகம்nt III, M. Marathe, and R. Stearns. Generalized CNF satis ability problems and non-e cient approximability. In Proc. 9th Conference on Structure in Complexity Theory, pages 356{366. IEEE Computer Society Press, Los Alamitos, CA, 1994. 28