A. Condon, J. Feigenbaum, C. Lund, P. Shor. "Random debaters and the hardness of approximating stochastic functions". Proc. of Ninth Annual Structure in Complexity Theory Conference, 280--293, 1994.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of optimization versions of Pspace hard problems such as Quantified Satisfiability, Generalized Geography and Finite Automata Intersection. With similar techniques, they have also showed 2 nonapproximability of certain stochastic problems including Dynamic Graph Reliability and Mah Jong [9]. Approximation results for propositional planning have not appeared very frequently in the literature. Selman [22] has provided a weak lower bound on the approximability of PL the problem cannot be approximated within any constant in polynomial time. PL for a certain class of planning ....

....a propositional planning problem. ffl The direct reduction techniques used in this article have provided tight lower bounds. However, nonapproximability results based on PCDS s often fail to do so. One immediate question is whether the results in, for instance, Condon et al. 8] or Condon et al. [9] can be strengthened by using direct reduction arguments. Another question is if their proofs can be simplified by employing such techniques. This last question has an affirmative answer in at least one case. Condon et al. 8] have provided nonapproximability results for MAX GGEOG (by using ....

A. Condon, J. Feigenbaum, C. Lund, and P. W. Shor, Random debaters and the hardness of approximating stochastic functions, SIAM J. Comput. 26(2) (1997) 369--400.

....and reading a constant number of bits such that, for any x # L, there exists y with y # q( x ) and with V (x, y) accepting with probability 1 and, for x ## L, there is no y with y # q( x ) and with V (x, y) accepting with probability # 1 4. This idea can also be extended to pspace [10, 9], using a game that alternates between two players instead of a proof presented by a single player. Since the construction of reliable probabilistic verifiers is not very intuitive, it might help to consider some analogies. One is a parallel between probabilistic verifiers and a judge on a ....

....is np hard to approximate up to any constant factor. See [4, 27] for an overview. Let us also mention recent work [18] that obtains several np lower bounds for approximation problems by traditional complexity theoretic techniques. The probabilistic verifier characterization of pspace is given in [10, 9]. A traditional characterization of pspace is that a language L is given by a polynomial p and a polynomial time predicate P on strings, with a string x in L i# #y 1 #y 2 #y 3 . #y p( x ) P(xy 1 y 2 y 3 . y p( x ) where each y i may be taken to be a single bit 0, 1 or, ....

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A. Condon, J. Feigenbaum, C. Lund, and P. Shor, Random debaters and the hardness of approximating stochastic functions, Proceedings of the 9-th Annual IEEE Conference on Structure in Complexity Theory, IEEE Computer Society Press, Los Alamitos, California, 1994, pp. 280--293, full version to appear in SIAM Journal on Computing.

....a compendium that lists the current approximation status of important optimization problems. 3. 2 Other applications of PCP Techniques The PCP Theorem and related techniques have found many other theoretical applications in complexity theory and cryptography, including new definitions of PSPACE [40] and PH [92] probabilistically checkable codes, zero knowledge proofs, checkable VLSI computations, etc. See [3] for a survey. One interesting application first observed by Babai et al. 20] and given prominence in the New York Times [94] article on the PCP Theorem) is that the PCP Theorem ....

A. Condon, J. Feigenbaum, C. Lund and P. Shor. Random debaters and the hardness of approximating stochastic functions. SIAM Journal on Computing, 26(2):369-400, April 1997.

....of coins and reading a constant number of bits such that, for any x 2 L , there exists y with jyj q(jxj) and with V (x; y) accepting with probability 1 and, for x 62 L , there is no y with jyj q(jxj) and with V (x; y) accepting with probability 1=4 . This idea can also be extended to pspace [10, 9], using a game that alternates between two players instead of a proof presented by a single player. Since the construction of reliable probabilistic verifiers is not very intuitive, it might help to consider some analogies. One is a parallel between probabilistic verifiers and a judge on a ....

....is np hard to approximate up to any constant factor. See [4, 27] for an overview. Let us also mention recent work [18] that obtains several np lower bounds for approximation problems by traditional complexity theoretic techniques. The probabilistic verifier characterization of pspace is given in [10, 9]. A traditional characterization of pspace is that a language L is given by a polynomial p and a polynomial time predicate P on strings, with a string x in L iff 9y 1 8y 2 9y 3 : 8y p(jxj) P (xy 1 y 2 y 3 : y p(jxj) where each y i may be taken to be a single bit f0; 1g or, ....

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A. Condon, J. Feigenbaum, C. Lund, and P. Shor. Random debaters and the hardness of approximating stochastic functions. In Proc. 9-th Annual IEEE Conference on Structure in Complexity Theory, pages 280--293. IEEE Computer Society Press, Los Alamitos, California, 1994. Full version to appear in SIAM Journal of Computing.

....fixed ffl 0. A sequence of improvements culminating in Hastad [H96] shows that even approximating within a factor n 1 Gammaffl is hard if NP 6 BPP . Lastly, we mention that PCP style characterizations have been provided for other complexity classes as well, such as PSPACE (Condon et al. CFLS93] and PH (Kiwi et al. KLR 94] Acknowledgments We thank Mike Luby for many discussions in the early stages of this work; specifically, his lower bound on the complexity of the protocol as presented in [FGL 91] is what started this work. We also thank Ron Fagin, Oded Goldreich, Noam ....

A. Condon, J. Feigenbaum, C. Lund, and P. Shor. Random debaters and the hardness of approximating stochastic functions. In Proc. of the 9th Structure in Complexity Theory Conference, pages 280--293, 1993. Also available as DIMACS Techreport TR 93-79.

....formulas. It is shown in [21, 22] that ; ae are just as hard to approximate. Some of these properties may be obtained directly from the np hardness of mll and the pspace hardness of mall, while others involve recent complexitytheoretic techniques for proving lower bounds on optimization problems [4, 3, 30, 8, 7, 14]. 4 But how hard is it to make one good move In this paper we investigate lower bounds on such local proof search heuristics. For instance, we show that on mll formulas, it is np hard to compute a local proof search heuristic up to any constant factor. That is, unless p=np, there is no ....

....logic proof game on mall formulas such that the expected score resulting from proponent s use of the heuristic throughout the play is within factor 1 ffl from the optimal score. Theorem 3. 3 relies on recent complexity theoretic techniques for proving pspace lower bounds on optimization problems [5, 12, 23, 29, 8, 7]. In contrast, the following theorem may be obtained from the pspace hardness of mall. Theorem 3.4 Let ffl 0. Every ffl heuristic for the advanced linear logic proof game on provable mall formulas is pspace hard. 4 Further Work We do not know any significant positive results on proof search ....

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A. Condon, J. Feigenbaum, C. Lund, and P. Shor. Random debaters and the hardness of approximating stochastic functions. In Proc. 9-th Annual IEEE Conference on Structure in Complexity Theory, pages 280--293. IEEE Computer Society Press, Los Alamitos, California, 1994. Full version to appear in SIAM Journal of Computing.

....mll . mall is pspace complete [20] It follows from the np completeness of the pure multiplicative fragment, mll [18, 22] that mll is np complete. These are global hardness properties in that they provide lower bounds on proponent s optimal strategy. Games from complexity theoretic literature [5, 14, 28, 34, 11, 8, 7, 16, 30] may be represented in the linear logic proof game, with the new complexity results obtained as corollaries of the complexity properties of games from the literature just mentioned. A representative case is studied here in detail in Section 7. The reader is referred to [25] for an outline of other ....

....of f . Let D f0; 1g , and let R be the set of real numbers. For a function f : D R , the neighborhood of f consists of all functions g: D R such that, for every string x 2 D , the difference between f(x) and g(x) is relatively small. One measure of error that appears in the literature [10, 4, 3, 8, 7] is Definition 3.1 Let D f0; 1g , and let f be a real valued function on D . Let 0 ffl 1, where ffl may depend on jxj, the length of a string x 2 D . The ffl neighborhood of f is the set ffl nbhd(f) ae g: D R j 8x 2 D; ffl(jxj) g(x) f(x) 1 ffl(jxj) oe For any g 2 ffl ....

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A. Condon, J. Feigenbaum, C. Lund, and P. Shor. Random debaters and the hardness of approximating stochastic functions. In Proc. 9-th Annual IEEE Conference on Structure in Complexity Theory, pages 280--293. IEEE Computer Society Press, Los Alamitos, California, 1994. Full version to appear in SIAM Journal of Computing.

....theorem has been pushed quite far (much further than we envisioned at the time of its discovery ) in proving non approximability results. We feel that it ought to have many other uses in complexity theory (or related areas like cryptography) One result in this direction is due to Condon et al. [34, 35], who use our main theorem (actually, a stronger form of it that we did not state) to prove a PCP style characterization of PSPACE. We hope that there will be many other applications. Acknowledgments This paper was motivated strongly by the work of Arora and Safra [6] and we thank Muli Safra for ....

A. Condon, J. Feigenbaum, C. Lund and P. Shor. Random debaters and the hardness of approximating stochastic functions. SIAM Journal on Computing, 26(2):369-400, April 1997.

....theorem has been pushed quite far (much further than we envisioned at the time of its discovery ) in proving non approximability results. We feel that it ought to have many other uses in complexity theory (or related areas like cryptography) One result in this direction is due to Condon et al. [34, 35], who use our main theorem (actually, a stronger form of it that we did not state) to prove a PCP style characterization of PSPACE. We hope that there will be many other applications. Acknowledgments This paper was motivated strongly by the work of Arora and Safra [6] and we thank Muli Safra for ....

A. Condon, J. Feigenbaum, C. Lund and P. Shor. Random debaters and the hardness of approximating stochastic functions. SIAM Journal on Computing, 26(2):369-400, April 1997.

....# Received by the editors December 1, 1993; accepted for publication (in revised form) May 12, 1995. These results first appeared in Random debaters and the hardness of approximating stochastic functions (extended abstract) Technical Memorandum, AT T Bell Laboratories, Murray Hill, NJ, 1993 [11]. They were presented in preliminary form at the 9th Annual IEEE Conference on Structure in Complexity Theory, Amsterdam, The Netherlands, June 1994. http: www.siam.org journals sicomp 26 2 26073.html Computer Sciences Department, University of Wisconsin, 1210 West Dayton Street, Madison, WI ....

A. Condon, J. Feigenbaum, C. Lund, and P. Shor, Random debaters and the hardness of approximating stochastic functions (extended abstract), Technical Memorandum, AT&T Bell Laboratories, Murray Hill, NJ, 1993.

....with PCD(r(n) q(n) The characterizations of PSPACE presented in Section 2 are those in which r(n) 0 and q(n) is an arbitrary polynomial. Specifically, Alternating Polynomial Time is, by definition, PCD(0, poly(n) and poly(n) round Arthur Merlin Games are RPCD(0, poly(n) Condon et al. [6, 7] study the potential tradeoff between random bits and query bits. If the referee V is allowed to flip coins, might it still be able to determine the winner of the game without reading the entire transcript The results in [6, 7] show that, as in the PCP characterization of NP, the best possible ....

....poly(n) round Arthur Merlin Games are RPCD(0, poly(n) Condon et al. 6, 7] study the potential tradeoff between random bits and query bits. If the referee V is allowed to flip coins, might it still be able to determine the winner of the game without reading the entire transcript The results in [6, 7] show that, as in the PCP characterization of NP, the best possible tradeoff between Documenta Mathematica Delta Extra Volume ICM 1998 Delta 1 1000 4 Feigenbaum r(n) and q(n) is obtainable. Furthermore, this tradeoff is obtainable both when the opponents are two strategic players (a PCDS) ....

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A. Condon, J. Feigenbaum, C. Lund, and P. Shor, "Random Debaters and the Hardness of Approximating Stochastic Functions," SIAM Journal on Computing, 26 (1997), pp. 369-400.

....ffl 1, it is PSPACE hard to approximate the function H MAX 3SAT to within ratio ffl. Thus, if there is a polynomial time algorithm that computes a function g such that g(F ) is in the range [fflH MAX SAT(F ) 1=ffl)H MAX SAT(F ) then PSPACE = P. To prove this, we use a result of Condon et al. [5], which characterizes PSPACE in terms of resource bounded debate systems. We reduce the problem of determining if such a debate system accepts a language L to the problem of approximating the H MAX 3SAT function. As an immediate application, we obtain nonapproximability results for functions on ....

....that it is PSPACE hard to approximate H MAX CUT and H MAX INDEPENDENT SET to within some constant factor. We note that hardness of approximation results for several other PSPACE hard problems (but not for hierarchically defined problems) based on reductions from debate systems can be found in [4, 5]. A hardness of approximability result for a PSPACE hard hierarchically defined linear programming problem can be found in [17] Our second result is that there is an efficient approximation algorithm for the H MAX SAT problem with performance guarantee 2 3. Specifically, given any H CNF formula F ....

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A. Condon, J. Feigenbaum, C. Lund, and P. Shor, Random Debaters and the Hardness of Approximating Stochastic Functions, SIAM Journal on Computing, Vol. 26, No. 2, March 1997, pages 369-400.

.... arise in many VLSI and scheduling applications (see, for example, 18] for references to these and other applications) Because such representations can implicitly describe a structure of exponential size, using just polynomial space, These results first appeared in our Technical Memorandum [11]. They were presented in preliminary form at the 9th Annual IEEE Conference on Structure in Complexity Theory, Amsterdam, The Netherlands, June 1994. y University of Wisconsin, Computer Sciences Department, 1210 West Dayton Street, Madison, WI 57306 USA, condon cs.wisc.edu. Supported in part by ....

A. Condon, J. Feigenbaum, C. Lund, and P. Shor, Random Debaters and the Hardness of Approximating Stochastic Functions (Extended Abstract), AT&T Bell Laboratories Technical Memorandum, Murray Hill NJ, May 1993.

....versions of Dynamic Graph Reliability, Stochastic Satisfiability, Mah Jongg, Stochastic Generalized Geography, and other games against nature of the type introduced in [Papadimitriou, J. Comput. System Sci. 31 (1985) pp. 288 301] These results first appeared in our Technical Memorandum [11]. They were presented in preliminary form at the 9th Annual IEEE Conference on Structure in Complexity Theory, Amsterdam, The Netherlands, June 1994. y University of Wisconsin, Computer Sciences Department, 1210 West Dayton Street, Madison, WI 57306 USA, condon cs.wisc.edu. Supported in part by ....

A. Condon, J. Feigenbaum, C. Lund, and P. Shor, Random Debaters and the Hardness of Approximating Stochastic Functions (Extended Abstract), AT&T Bell Laboratories Technical Memorandum, Murray Hill NJ, May 1993.

....languages in the polynomialtime hierarchy. Other uses of competing players to study complexity classes include [Reif84, PR79] Feige, Shamir and Tennenholtz [FST88] proposed an interactive proof system in which the notion of competition is present. Recently, Condon, Feigenbaum, Lund and Shor [CFLS93a, CFLS93b] characterized PSPACE by systems in which a verifier with O(log n) random bits can read only a constant number of bits of a polynomial round debate between two players. We show that Sigma P k is the class of languages that can be recognized by a verifier with similar access to a k round debate. ....

A. Condon, J. Feigenbaum, C. Lund, and P. Shor. "Random debaters and the hardness of approximating stochastic functions". DIMACS TR 93-79, Rutgers University, Piscataway NJ, 1993.

....the class of languages accepted by such Arthur Merlin games. In this context, the fact that AM(poly(n) PSPACE (cf. 18, 23] means that, if r(n) 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 ....

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.

No context found.

A. Condon, J. Feigenbaum, C. Lund, P. Shor. "Random debaters and the hardness of approximating stochastic functions". Proc. of Ninth Annual Structure in Complexity Theory Conference, 280--293, 1994.

No context found.

A. Condon, J. Feigenbaum, C. Lund and P. Shor, "Random Debaters and the Hardness of Approximating Stochastic functions for PSPACE-Hard Functions," SIAM Journal on Computing, 26(2), April 1997, pp. 369-400.

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