M. Luby and B. Velickovi'c. On deterministic approximation of dnf. In Proceedings of the 23rd Annual Symposium on the Theory of Computing, pages 430--438, 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....problems for constant depth circuits. Luby, Velickovic and Wigderson [LVW93] optimize the constructions of Nisan and Wigderson [Nis91, NW94] to the case of depth 2 circuits, thus solving the counting and satisfiability problem in time n O( log n) for general CNF and DNF. Luby and Velickovic [LV91] show how to reduce arbitrary CNF and DNF to formula in a simplified format, and show that the counting and satisfiability problems can be solved in polynomial time for k CNF even if k = O( log n) 1 8 ) is more than a constant. Hirsch [Hir98] in seemingly independent work, shows how to solve ....

Michael Luby and Boban Velickovic. On deterministic approximation of DNF. In Proceedings of STOC'91, pages 430--438, 1991.

....) with an additive error of exp( n) This error is optimal, up to the logarithmic factor implicit in the notation. The problem of enumerating satisfying assignments of a boolean formula in DNF form F = 1 C i is an instance of the general problem that had been extensively studied [7]. Here A i is the set of assignments that satisfy C i , and Pr( i2S A i ) a S =2 where i2S C i is satis ed by a S assignments. Judging from the general results, it is hard to expect a decent approximation of F s number of satisfying assignments, without knowledge of the numbers a S ....

....involved in the approximation) The problem of enumerating the satisfying assignments of a DNF formula is an instance of the general problem. Already this special case is known to be complete for the class #P . Much attention has been given to ecient algorithms for approximating this number, see [7] and the references therein. To put the DNF problem in the general context of our problem, let the probability space be f0; 1g under uniform distribution. Associated with every clause in the DNF formula is the event that this clause is satis ed. Each such event is, in fact, a subcube of the ....

M. Luby and B. Velickovic, On deterministic approximation of DNF, STOC 23 (1991), 430-438.

....n k ) where rk is the largest root of x k Gamma x k Gamma1 Gamma : Gamma x Gamma 1 (such as r2 = 1:618, r3 = 1:839, r4 = 1:928; limk 1 rk = 2. Very hard in general, the problem of counting models turns out to be quite tractable in many practical cases. Luby and Velickovic [22] have presented an algorithm for approximating the number of models of a DNF propositional formula with m clauses on n variables. The running time of the algorithm is estimated by a polynomial in m and n multiplied by n O( 1 ffi (log 2 m log 2 1 ffl ) log log m log 1 ffl log 1 ....

Luby, M., and Velickovic, B. On deterministic approximation of DNF. Proc. 23rd ACM Symp. on Theory of Computing, New Orleans, LA, May 1991, ACM Press, 430-438.

....ff k is the largest root of x k Gamma x k Gamma1 Gamma : Gamma x Gamma 1 (such as ff 2 = 1:618, ff 3 = 1:839, ff 4 = 1:928; lim k 1 ff k = 2) Very hard in general, the problem of counting models turns out to be quite tractable in many practical cases. Luby and Velickovic [53] have presented an algorithm for approximating the number of models of a DNF propositional formula with m clauses on n variables. The running time of the algorithm is estimated by a polynomial in m and n multiplied by n O( 1 ffi (log 2 m log 2 1 ffl ) log log m log 1 ffl log 1 ....

Luby, M., and Velickovic, B., 1991, On deterministic approximation of DNF. In Proceedings, 23rd ACM Symp. on Theory of Computing, New Orleans, LA, ACM Press, 430--438.

....not an isolated discovery. In several fields results of a similar flavour have been obtained, for instance: i) the VC dimension [VC 71] and its applications in computational learning theory [LLW 91] AB 92] and computational geometry [HW 87] ii) Monte Carlo approximations [KL 83] KLM 89] LV 91] iii) uniformity and irregularities (discrepancies) of set systems and matrices [BF 81] LSV 86] iv) stochastic information theory [Ahl 78] Ahl 79] On the other hand there are some statistics papers by Wald and others [DWW 51] WW 51] with similarly sounding titles but different ....

Luby, M. and Velickovic, B. 1991. On deterministic approximation of DNF, in Proc. 23rd ACM Symposium on Theory of Computing, New Orleans: 430--438.

....AND P. D. MACKENZIE Proof. For p # W # j , let D(p) be the set of processors which p one a#ects. Then D(p) # k t . A sunflower is defined as a collection of sets such that if an element is in two of the sets, then it is contained in all of the sets. The Erdos Rado theorem ( 4] see also [17]) says the following: let t and m be positive integers and let F be a family of sets such that every element of F has size at most t and F t (m 1) t . Then F contains a sunflower of size m. If we let F be the family of sets D(p) for p # W # j , then F contains a sunflower of size ....

M. Luby and B. Veli ckovi c, On deterministic approximation of DNF, in Proceedings of the ACM Symposium on Theory of Computing 23, 1991, pp. 430--438.

....assignments of truth values to the variables of a formula F in time O(m2 n ) To satisfy practical applications, one has either to resort to a good approximation of (F ) or use an algorithm for computing (F ) exactly with a reasonable average time complexity. Regarding the first alternative, Luby and Velickovi c (1991) presented a deterministic algorithm for approximating the proportion of truth assignments that satisfy a DNF formula F. Let P r[F ] denote this proportion (P r[F ] F ) 2 n where n is the number of variables of F) Given small numbers ffl; ffi 0, the algorithm computes an estimate Y of P ....

Luby, M., & Velickovi'c, B. (1991). On deterministic approximation of DNF. In Proceedings of STOC'91.

....(for a constant ) by some O(d d ) sparse function, even if the DNF has Omega Gamma n d ) terms. This can be contrasted with the results based on the Sunflower Theorem that show that a DNF with terms of size at most d can be approximated by a DNF with O(2 d 2 ) terms of size at most d (see [LV91]) Our results are based on the lower bound techniques that were developed for proving lower bound for polynomial size constant depth circuit [Ajt83, FSS84, Yao85, Has86] Those techniques work almost identically for DNF and CNF; for this reason all our results apply also to CNF. The paper is ....

M. Luby and B. Velickovic. On deterministic approximation of DNF. In Proceedings of the 23 rd Annual ACM Symposium on Theory of Computing, pages 430--438, May 1991.

....and (k=ffl) k . In contrast to previous results, when k = O(log(n) and ffl = poly(1=n) the size of the sample space in our construction is polynomial in n. This case is important to some applications (e.g. this construction improves the running time of some of the algorithms presented in [16]) Two natural examples where we obtain a significant improvement follow. Example 1 Suppose we wish to approximate n independently distributed 0 1 random variables, each assigned 1 with probability 1 2 2ffl and 0 otherwise. Using previously known techniques, one obtains a sample space of size ....

.... (1) z 1 ( z n (1) z n ( is mapped to a point, y 1 ; y n , in the new sample space (where y i = 0 iff z i (1) Delta Delta Delta z i ( p i (1) Delta Delta Delta p i ( Our analysis of the above construction is somewhat analogous to the proof of Theorem 3 in [16]. We fix k variables Y i 1 ; Y i k out of Y 1 ; Y n , and consider the quality of the approximation which they provide. Without loss of generality, we consider the random variables Y 1 ; Y k . Rather than bounding the Max Norm performance of the approximation (as required in the ....

Luby, M., Velickovi'c, B., "On Deterministic Approximation of DNF", Algorithmica (special issue devoted to randomized algorithms), Vol. 16, No. 4/5, October/November 1996, pp. 415-- 433.

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Luby, M., Velickovi'c, B., "On Deterministic Approximation of DNF", Proc. 23rd STOC, 1991, pp. 430--438.

....and (k=ffl) k . In contrast to previous results, when k = O(log(n) and ffl = poly(1=n) the size of the sample space in our construction is polynomial in n. This case is important to some applications (e.g. this construction improves the running time of some of the algorithms presented in [18]) Two natural examples where we obtain a significant improvement follow. Example 1 Suppose we wish to approximate n independently distributed 0 1 random variables, each assigned 1 with probability 1 2 2ffl and 0 otherwise. Using previously known techniques, one obtains a sample space of size ....

.... (1) z 1 ( z n (1) z n ( is mapped to a point, y 1 ; y n , in the new sample space (where y i = 0 iff z i (1) Delta Delta Delta z i ( p i (1) Delta Delta Delta p i ( Our analysis of the above construction is somewhat analogous to the proof of Theorem 3 in [18]. We fix k variables Y i 1 ; Y i k out of Y 1 ; Y n , and consider the quality of the approximation which they provide. Without loss of generality, we consider the random variables Y 1 ; Y k . Rather than bounding the Max Norm performance of the approximation (as required in the ....

Luby, M., Velickovi'c, B., "On Deterministic Approximation of DNF", Algorithmica (special issue devoted to randomized algorithms), Vol. 16, No. 4/5, October/November 1996, pp. 415-- 433.

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Luby, M., Velickovi'c, B., "On Deterministic Approximation of DNF", Proc. 23rd STOC, 1991, pp. 430--438.

....approximated in time exp(exp( p log(n=ffl) Nisan [12] considerably strengthened this result, obtaining exp(log(n) O(1) running time for approximation algorithms for the same class of circuits. The case of DNF formulas, which is a subclass of AC 0 circuits, was treated in [11] and more efficient approximation algorithms are described for this case. No nontrivial simulation of probabilistic constant depth circuits with modular gates was known. This may seem surprising at first, as exponential lower bounds on size existed for both classes of circuits, and these lower ....

....exp(O(log(n=ffl) 4 ) where n is the size of the formula. The algorithm is a slight modification of the algorithm presented in [12] but uses our refined notion of (n; l; d; r) design for a given (m; n; k) system C. The running time is better than the running time of the algorithm presented in [11] for formulas which have unrestricted clause length. Since the arguments in this case are very similar to the ones in [12] or in x4 we shall be sketchy in our description. As in [12] we use the following lower bound result of Boppana and Hastad [5, Chapter 8] Theorem 3 Let F be any DNF formula ....

Luby, M., Velickovi'c, B., "On Deterministic Approximation of DNF", STOC 1991, pp. 430--438.

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Luby, M., Velickovi'c, B., "On Deterministic Approximation of DNF", to appear in the proceedings of STOC 1991.

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Luby, M., Velickovi'c, B., "On Deterministic Approximation of DNF", Proceedings of 23rd ACM Symposium on Theory of Computing, 1991, pp. 430--438.

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M. Luby and B. Velickovi'c. On deterministic approximation of dnf. In Proceedings of the 23rd Annual Symposium on the Theory of Computing, pages 430--438, 1991.

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Luby, M., and Velickovic, B. (1991), On deterministic approximation of DNF, in "Proceedings of 23th ACM Symposium on Theory of

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