D. Achlioptas. Setting 2 variables at a time yields a new lower bound for random 3-SAT. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pages 28--37, Portland, OR, May 2000.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....of a given prioritised algorithm. A more precise definition is in Section 3. Note that many deprioritised algorithms can be derived by using the operations of the original (prioritised) algorithm. The first use of randomised choices between operations, in a related setting, was by Achlioptas [1]. His algorithm mTT contains some deprioritisation, but it also contains a prioritised choice. The net e#ect is quite di#erent from what is achieved with the fully deprioritised algorithms in the present paper. This paper has objectives at several levels. One is to show that the intuitive idea ....

D. Achlioptas, Setting 2 variables at a time yields a new lower bound for random 3-SAT, In 32nd Annual ACM Symposium on Theory of Computing (STOC 32) pp. 28--37 (2000).

....of random k SAT instances (or k CNF formulas) over n propositional variables. All our discussion refers to k xed and then letting n be suciently large. The probability space of random k SAT instances has been widely studied in recent years for several good reasons. The most recent literature is [Ac2000] Fr99] Be et al..98] AcSo2000] One of the reasons for studying random k SAT instances is that they have the following sharp threshold behaviour [Fr99] There exists a constant c = c k such that for any 0 formulas with at most (1 ) c n clauses are satis able whereas formulas with at ....

....k CNF formulas) over n propositional variables. All our discussion refers to k xed and then letting n be suciently large. The probability space of random k SAT instances has been widely studied in recent years for several good reasons. The most recent literature is [Ac2000] Fr99] Be et al..98] [AcSo2000]. One of the reasons for studying random k SAT instances is that they have the following sharp threshold behaviour [Fr99] There exists a constant c = c k such that for any 0 formulas with at most (1 ) c n clauses are satis able whereas formulas with at least (1 ) c n are unsatis ....

[Article contains additional citation context not shown here]

Dimitris Achlioptas. Setting 2 variables at a time yields a new lower bound for random 3-SAT. In Proceedings SToC 2000, ACM, 28-37.

....for IVPs for ODEs verify that a unique solution to a problem exists and produce bounds that are guaranteed to contain this solution. There are situations when guaranteed bounds on the mathematically correct result are desired or needed. For example, such bounds can be used to prove a theorem [1, 29, 31]. Also, some calculations may be critical to the safety or reliability of a system [9] Therefore, it may be necessary to ensure that the true result is within computed bounds. Furthermore, methods that produce guaranteed bounds may be used to check a sample calculation to be performed by a ....

....the right side of (26) should be speci ed in a .h le as shown in Figure 4. In this le, we also include FILE VDP.h #ifndef INCLUDED VDP H #define INCLUDED VDP H #include odenum.h template typename Y TYPE void VDPtemplate( Y TYPE yp, const Y TYPE y ) double MU = 2. 0; yp[0] y[1]; yp[1] MU (1 sqr(y[0] y[1] y[0] #include declode.h DECLARE ODE PROBLEM(VDP,2,VDPtemplate, Van Der Pol s Equation ) #endif Figure 4. Specifying the function for computing the right side of (26) odenum.h contains the declaration of ODE NUMERIC. the le declode.h and then the ....

[Article contains additional citation context not shown here]

D. Achlioptas, Setting 2 variables at a time yields a new lower bound for random 3-SAT, Technical report MSR-TR-99-96, Microsoft Research, Microsoft Corp., One Microsoft Way, Redmond, WA 98052, December 1999.

....able 3 CNF require exponential size proofs. Their lower bound was later improved and simpli ed by [Beame and Pitassi ( 96) The existence and value of a satis ability threshold constant is an important combinatorial question, outside the scope of this paper. We refer the interested reader to [Achlioptas ( 00), Janson et al. 00) Dubois, et al. 00) for bounds on the satis ability threshold. 24 and nally improved up to a ratio = o( n) by [Beame et al. 98) and by [Ben Sasson and Wigderson ( 99) Following is a simple proof of the best known lower bound, based on the expansion of a ....

D. Achlioptas. Setting 2 variables at time yields a new lower bound for random 3-SAT. In Proceedings of 32th STOC. pp. 28-37 (2000).

...., where is called the clause density, and denote by C C a random CNF from this distribution. Many interesting facts are known about C , and in our brief survey we will only discuss , noting that all results can be generalized to larger k. It is known that as goes from 3:145 : [1] to 4:5793 : 15] the fraction of satisfiable formulas in C goes from 1 o(1) to o(1) and that the threshold is sharp [12] Knowing this, one may ask what is the range of for which the satisfiability problem is hard to solve. When 3:145 this amount to finding a satisfying assignment ....

D. Achlioptas. Setting 2 variables at time yields a new lower bound for random 3-SAT. In Proceedings of 32th STOC. pp. 28-37 (2000).

....proofs to be refuted. The importance of their work was in showing that in fact Resolution is a very weak proof system, because in some sense almost all unsatis able 3CNF require exponential size proofs to be refuted. Their lower bound was Currently, the best lower bound on 3 is 3:145 3 of [A00] and the best upper bound is 3 4:5793 [JSY00] and a recent 3 4:506 claimed by [DBM00] later improved and simpli ed by Beame and Pitassi in [BP96] and nally improved up to a ratio = o( n) by Beame, Karp, Pitassi and Saks in [BKPS98] and reformulated in terms of a general technique ....

D. Achlioptas. Setting 2 variables at time yields a new lower bound for random 3-SAT. In Proceedings of 32th STOC. pp. 2837 (2000).

....random k SAT instances (or k CNF formulas) over n propositional variables. All our discussion refers to k xed and then letting n be suciently large. The probability space of random k SAT instances has been widely studied in recent years for several good reasons. The most recent literature is [Ac2000] Fr99] Be et al..98] One of the reasons for studying random k SAT instances is that they have the following sharp threshold behaviour [Fr99] There exists a constant c = c k such that for any 0 formulas with at most (1 ) c n clauses are satis able whereas formulas with at least (1 ) ....

....least 2 k (ln 2) n clauses the expected number of satisfying assignments of a random formula tends to 0 and the formulas are unsatis able with high probability. For 3 SAT instances much e ort is spent to approximate the value of c 3 . The currently best results are that c 3 is at least 3:125 [Ac2000] and at most 4:601 [KiKrKrSt98] In [Du et al..2000] it is claimed that c 3 4:501. For k = 2 we have c 2 = 1 [ChRe92] Go96] Partially supported by a USA Israeli BSF grant The algorithmic interest in this threshold is due to the empirical obeservation that random k SAT instances at the ....

[Article contains additional citation context not shown here]

Dimitris Achlioptas. Setting 2 variables at a time yields a new lower bound for random 3-SAT. In Proceedings SToC 2000, ACM.

....d 3 1. Using the same approach, Broder, Frieze, and Upfal [BFU93] improved the lower bound to d 3 1:63. By introducing two new heuristics and analyzing their probabilities of success on M(n; m; 3) Frieze and Suen [FSS96] proved that d 3 3:003. The best lower bound belongs to Achlioptas [ACH99a], who proved that d 3 3:145 by an improvement to the heuristics of Frieze and Suen [FSS96] 5) The (2 p) Model M(n; m; 2 p) The 2 p model was introduced by Monasson et al. MZK99a, MZK99b] in an e ort to understand the di erence between the phase transition in 2 SAT(polynomial problem) ....

....k; z)is solubleg = 8 : 1; if z z c ; 0; if z z c : The value z c is called the threshold (or critical value) of the corresponding probability model. An important task in the study of the phase transitions is to locate the threshold for various random combinatorial problems. See [ACH99a, FSS96, KKK98] for further references. Throughout this section, we assume k = 2 in the xed ratio model N(n; k; z) 3.3.1 A Linear Algorithm for the Case z 3:0 Consider a random instance f = n P i=1 f i of the xed ratio model N(n; 2; z) with z = 3:0 3:0. Without loss of generality, we may write f ....

D.Achlioptas, \Setting 2 Variables at a Time Yields a New Lower Bound for Random 3-SAT", MicroSoft Tech. Report MSR-TR-9996, 1999.

....size of a treelike refutation, for clause density greater than p n (theorem 5.5) This bound is nearly tight. Previously such lower bounds were known only for special types of treelike proofs, formed by a Ordered DLL algorithm [BKPS98] 1 Currently, the best lower bound on 3 is 3:145 3 of [A00] and the best upper bound is 3 4:5793 [JSY00] and a recent 3 4:506 claimed by [DBM00] 2 1.4 Proof Outline A 3 CNF has an obvious interpretation in terms of bipartite graphs. Under this intepretation, a matching in the graph G(F ) associated with F corresponds to a partial assignment ....

D. Achlioptas. Setting 2 variables at time yields a new lower bound for random 3-SAT. In Proceedings of 32th STOC. pp. 28-37 (2000).

....It can be argued that this has led to better SAT solvers: for example, the winners of recent SAT solver competitions use such heuristics but it is hard to nd a discussion of these heuristics which predates the probabilistic results. Notable contributions in this area have some from Achlioptas [1], Broder and others [11] Purdom, Brown, Haven, Bugrara and others [48, 37, 12, 52] Chao, Franco and Paul [30, 27, 15, 16, 28] Chvatal and Reed [19] Frieze and Suen [33] among others. 3.3 Diculty of resolution for unsatis ability: probabilistic lower bounds It has been known for some time ....

....constant c 1 , and satis able with probability tending to 1 if m=n c 2 2 k =k [19] for some constant c 2 . For 3 SAT, these two bounds have been steadily tightened by numerous ideas and results [25] to something close to m=n 4:54 for unsatis ability with high probability and m=n 3:145 [1] for satis ability with high probability. Recently, it became known [32] that for any k 2 there is a sequence r k (n) such that the probability of satis ability tends to 1 if m=n r k (n) e and of unsatis ability tends to 1 if m=n r k (n) e, where e is an arbitrarily small constant. This is ....

Dimitris Achlioptas. Setting 2 variables at a time yields a new lower bound for random 3-sat. Manuscript, 1999.

No context found.

D. Achlioptas. Setting 2 variables at a time yields a new lower bound for random 3-SAT. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pages 28--37, Portland, OR, May 2000.

No context found.

D. Achlioptas, \Setting 2 variables at a time yields a new lower bound for random 3-SAT." STOC `00.

No context found.

D. Achlioptas. Setting 2 variables at a time yields a new lower bound for random 3-SAT. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pages 28--37, Portland, OR, May 2000.

No context found.

D. Achlioptas. Setting 2 variables at a time yields a new lower bound for random 3-SAT. In Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, pages 28--37, Portland, OR, May 2000.

.... Phase Transition in 1 in k SAT and NAE 3 SAT Dimitris Achlioptas Arthur Chtcherba y Gabriel Istrate z Cristopher Moore x 1 Introduction Determining bounds for the random k SAT threshold has been an active area of research in recent years [1, 3]. Yet, in spite of signi cant e orts, neither a tight analysis nor the structural properties of this threshold have been determined. In this paper we study random instances of two other canonical variations of satis ability, 1 in k SAT and Not All Equal 3 SAT. Like random k SAT, each generative ....

D. Achlioptas, \Setting 2 variables at a time yields a new lower bound for random 3-SAT." STOC `00.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC