H. B. Hunt III, M. V. Marathe, and R. E. Stearns, Generalized CNF satisfiability problems and non-e#cient approximability, in Proceedings of the Ninth Annual Structure in Complexity Theory Conference, Amsterdam, The Netherlands, IEEE Computer Society Press, Los Alamitos, CA, 1994, pp. 356--366.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....complexity, approximation, function generation, Pspace 1 Introduction Approximation of NP complete optimization problems has been a very active area of research for some years. However, approximability results for Pspace hard problems have only recently begun to appear in the literature [1,3,5]. In this paper, we study an optimization version of the Finite Function Generation (FFG) problem. FFG is defined as follows: Instance: A triple hA; F; hi where A is a finite set, F is a collection of total functions on A and h is a total function on A. Question: Does there exist a sequence hf i ....

H. Hunt III, M. Marathe, and R. Stearns. Generalized CNF satisfiability problems and non-efficient approximability, in: Proc. 9th Structure in Complexity Theory Conf. (1994) 356--366.

....have then been studied quite extensively since such problems arise in scheduling and VLSI applications. Marathe et al. 19] contains several results of this type. Nonapproximability results for some other problems, including Max QDones and Finite Function Generation, can be found in Hunt et al. [16] and Jonsson [17] The techniques of probabilistically checkable proofs which have been so important in understanding the approximability of NP hard problems have also influenced the study of Pspace hard problems. Condon et al. 8] introduced the probabilistically checkable debate systems (PCDS) ....

H. Hunt III, M. Marathe, and R. Stearns, Generalized CNF satisfiability problems and non-efficient approximability, in: Proc. 9th Conference on Structure in Complexity Theory (IEEE Computer Society Press, 1994) 356--366.

....are hard to approximate in the sense that computing an approximation would also solve every known recognition problem in np. For instance, the size of the largest clique in a graph 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 ....

....In [9] it is also observed that Theorem 5.4 ( 9] There exists a positive constant c such that prob stoc on stochastic formulas is pspace hard to approximate within multiplicative ratio 2 n c , where n is the length of a stochastic formula. Using the latter theorem and the techniques of [18], it can be shown that 1 pspace hardness of approximation was defined in [9] and [10] in terms of Turingreducibility, but their results as well as ours satisfy the more restrictive definition in terms of many one reducibility used here. It is possible that some approximation problems exist that ....

H. B. Hunt III, M. V. Marathe, and R. E. Stearns, Generalized CNF satisfiability problems and non-e#cient approximability, Proceedings of the 9-th Annual IEEE Conference on Structure in Complexity Theory, IEEE Computer Society Press, Los Alamitos, California, 1994, pp. 356--366.

....of variables as a basic primitive. This allows us to talk about problems such as Max EkSat. In Section 5.3 we extend Schaefer s results to establish the hardness of satisfiable Max CSP problems. Similar results, again with replication of variables being allowed, were first shown by Hunt et al. [26]. 5.1 Containment results for Max CSP We start with the polynomial time solvable cases. Proposition 5.1 Weighted Max CSP(F) Weighted Min CSP(F) is in PO if F is 0 valid (1 valid) Proof: Set each variable to zero (resp. one) this satisfies all the constraints. 2 Before proving the ....

H. B. Hunt III, M. V. Marathe, and R. E. Stearns. Generalized CNF satisfiability problems and non-efficient approximability (preliminary version). Proceedings of the Ninth Annual Structure in Complexity Theory Conference, pages 356--366, Amsterdam, The Netherlands, 28 June-1 July 1994. IEEE Computer Society Press.

....of variables as a basic primitive. This allows us to talk about problems such as Max EkSat. In Section 5.3 we extend Schaefer s results to establish the hardness of satisfiable Max CSP problems. Similar results, again with replication of variables being allowed, were first shown by Hunt et al. [26]. 5.1 Containment results for Max CSP We start with the polynomial time solvable cases. Proposition 5.1 Weighted Max CSP(F ) Weighted Min CSP(F) is in PO if F is 0 valid (1 valid) Proof: Set each variable to zero (resp. one) this satisfies all the constraints. 2 Before proving the ....

H. B. Hunt III, M. V. Marathe, and R. E. Stearns. Generalized CNF satisfiability problems and non-efficient approximability (preliminary version). Proceedings of the Ninth Annual Structure in Complexity Theory Conference, pages 356--366, Amsterdam, The Netherlands, 28 June-1 July 1994. IEEE Computer Society Press.

....are hard to approximate in the sense that computing an approximation would also solve every known recognition problem in np. For instance, the size of the largest clique in a graph 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 ....

....In [9] it is also observed that Theorem 5.4 ( 9] There exists a positive constant c such that prob stoc on stochastic formulas is pspace hard to approximate within multiplicative ratio 2 Gamman c , where n is the length of a stochastic formula. Using the latter theorem and the techniques of [18], it can be shown that Theorem 5.5 There exists a positive real number c such that the function max stoc on generalized stochastic formulas is pspace hard to approximate within multiplicative ratio n Gammac , where n is the length of the generalized stochastic formula. 6 Linear logic proof ....

H.B. Hunt III, M.V. Marathe, and R.E. Stearns. Generalized CNF satisfiability problems and non-efficient approximability. In Proc. 9-th Annual IEEE Conference on Structure in Complexity Theory, pages 356--366. IEEE Computer Society Press, Los Alamitos, California, 1994.

....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 ....

H.B. Hunt III, M.V. Marathe, and R.E. Stearns. Generalized CNF satisfiability problems and non-efficient approximability. In Proc. 9-th Annual IEEE Conference on Structure in Complexity Theory, pages 356--366. IEEE Computer Society Press, Los Alamitos, California, 1994.

....Quantified Satisfiability, Generalized Geography, and Finite Automata Intersection. Building on the techniques of [2, 3, 13] we showed that it is in fact PSPACE hard to approximate these problems closely (where closely depends on the problem) Using direct reduction arguments, Hunt et al. [14] showed that some generalized quantified satisfiability problems (for example, satisfiability of quantified formulas in which clauses are not restricted to be disjunctions of literals) are PSPACE hard to approximate. An important class of PSPACE hard problems not previously addressed in this ....

....Our new results on RPCDSs are used to obtain such proofs in section 3. There has been other very recent work, both on approximation algorithms and on nonapproximability results for PSPACE hard problems. Using direct reductions from variations of the Quantified Satisfiability problem, Hunt et al. [14] and Marathe et al. 18] showed that several PSPACE hard problems are hard to approximate, unless PSPACE = P. These include algebraic problems and graph problems on hierarchically defined graphs. Marathe et al. 19] proved that several graph problems such as vertex cover and independent set, when ....

H. Hunt III, M. Marathe, and R. Stearns, Generalized CNF satisfiability problems and non-e#cient approximability, in Proc. 9th Conference on Structure in Complexity Theory, IEEE Computer Society Press, Los Alamitos, CA, 1994, pp. 356--366.

....Quantified Satisfiability, Generalized Geography, and Finite Automata Intersection. Building on the techniques of [2, 3, 13] we showed that it is in fact PSPACE hard to approximate these problems closely (where closely depends on the problem) Using direct reduction arguments, Hunt et al. [14] showed that some generalized quantified satisfiability problems (for example, satisfiability of quantified formulas in which clauses are not restricted to be disjunctions of literals) are PSPACE hard to approximate. An important class of PSPACE hard problems not previously addressed in this ....

....Our new results on RPCDS s are used to obtain such proofs in Section 3. There has been other very recent work, both on approximation algorithms and on nonapproximability results for PSPACE hard problems. Using direct reductions from variations of the Quantified Satisfiability problem, Hunt et al. [14] and Marathe et al. 18] showed that several PSPACE hard problems are hard to approximate, unless PSPACE = P. These include algebraic problems and graph problems on hierarchically defined graphs. Marathe et al. 19] proved that several graph problems such as vertex cover and independent set, when ....

H. Hunt III, M. Marathe and R. Stearns, Generalized CNF Satisfiability Problems and Non-Efficient Approximability, Proc. 9th Conference on Structure in Complexity Theory, IEEE Computer Society Press, Los Alamitos, 1994, pp. 356-366.

....Quantified Satisfiability, Generalized Geography, and Finite Automata Intersection. Building on the techniques of [2, 3, 13] we showed that it is in fact PSPACE hard to approximate these problems closely (where closely depends on the problem) Using direct reduction arguments, Hunt et al. [14] showed that some generalized quantified satisfiability problems (for example, satisfiability of quantified formulas in which clauses are not restricted to be disjunctions of literals) are PSPACE hard to approximate. An important class of PSPACE hard problems not previously addressed in this ....

....Our new results on RPCDS s are used to obtain such proofs in Section 3. There has been other very recent work, both on approximation algorithms and on nonapproximability results for PSPACE hard problems. Using direct reductions from variations of the Quantified Satisfiability problem, Hunt et al. [14] and Marathe et al. 18] showed that several PSPACE hard problems are hard to approximate, unless PSPACE = P. These include algebraic problems and graph problems on hierarchically defined graphs. Marathe et al. 19] proved that several graph problems such as vertex cover and independent set, when ....

H. Hunt III, M. Marathe and R. Stearns, Generalized CNF Satisfiability Problems and Non-Efficient Approximability, Proc. 9th Conference on Structure in Complexity Theory, IEEE Computer Society Press, Los Alamitos, 1994, pp. 356-366.

....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 ....

....real number c such that the function max stoc on generalized stochastic formulas is pspace hard to approximate within multiplicative ratio n Gammac , where n is the length of the generalized stochastic formula. This appears to be a new observation, but it follows an analogous result in [16] regarding a counting game on generalized classical formulas. Note the trade off with Theorem 5.2: Theorem 6.1 obtains a better bound but on a much larger class. The proof of Theorem 5.1 in [7] uses the methods of [11] to obtain a class of stochastic formulas OE for which prob stoc(OE) is either ....

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H.B. Hunt III, M.V. Marathe, and R.E. Stearns. Generalized CNF satisfiability problems and non-efficient approximability. In Proc. 9-th Annual IEEE Conference on Structure in Complexity Theory, pages 356--366. IEEE Computer Society Press, Los Alamitos, California, 1994.

....in [19] Bodlaender [5] has extended our results by showing that MAX Q 3SAT can be approximated within some 0 ffl 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 ....

H. Hunt III, M. Marathe, and R. Stearns. Generalized CNF satisfiability problems and non-efficient approximability. In Proc. 9th Conference on Structure in Complexity Theory, pages 356--366. IEEE Computer Society Press, Los Alamitos, CA, 1994.

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H. B. Hunt III, M. V. Marathe and R. E. Stearns, "Generalized CNF Satisfiability Problems and Non-Efficient Approximability," Proc. 9th ACM Conf. on Structure in Complexity Theory, June-July 1994, pp. 356-366.

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H. B. Hunt III, M. V. Marathe, and R. E. Stearns, Generalized CNF satisfiability problems and non-e#cient approximability, in Proceedings of the Ninth Annual Structure in Complexity Theory Conference, Amsterdam, The Netherlands, IEEE Computer Society Press, Los Alamitos, CA, 1994, pp. 356--366.

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