M.-T. Chao and J. Franco. Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k-satis ability problem. Inform. Sci., 51(3):289-314, 1990. 20  Home/Search   Document Not in Database   Summary   Related Articles   Check

....pretty much understood. For k 3 the story is very di#erent. It is now known that a threshold for satisfiability exists in some (not completely satisfactory) sense, Friedgut [10] There has been considerable work on trying to find estimates for this threshold in the case k = 3 Chao and Franco [5, 6], Broder, Frieze and Upfal [4] Frieze and Suen [12] Achlioptas [1] Achlioptas and Sorkin [2] the last mentioned paper giving a lower bound of 3.26. Upper bounds have been pursued with the same vigour Kirousis, Kramakis, Krizanc and Stamatiou [15] Janson, Stamatiou and Vamvakari [14] ....

M.T. Chao and J. Franco, Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k satisfiable problem, Information Science 51 (1990) 289--314.

....that a sequence of events En occurs almost certainly if Pr[E n ] tends to 1 as n 1. For k = 2, a sharp threshold was proved in [9, 16] a random instance of 2 SAT is almost certainly satisfiable if r 1 and almost certainly unsatisfiable if r 1. For k = 3, there has been a series of results [6, 7, 5, 13, 19, 15, 21, 11] narrowing the area for which we do not have almost certain (un)satisfiability. The best bounds currently known come from [15] where it was proven that a random instance of 3 SAT is almost certainly satisfiable if r 3:003, and [21] where it was proven that a random instance of 3 SAT is almost ....

....known come from [15] where it was proven that a random instance of 3 SAT is almost certainly satisfiable if r 3:003, and [21] where it was proven that a random instance of 3 SAT is almost certainly unsatisfiable if r 4:602. For general k, the best known lower bound for r is Theta(2 =k) [7, 9, 15] while 2 is an easy upper bound and was improved by a constant factor in [21] by extending the techniques used for 3 SAT. Very recently, Friedgut [14] made great progress towards establishing the existence of a threshold for k SAT for all values of k, yet his approach does not provide any ....

[Article contains additional citation context not shown here]

M.-T. Chao and J. Franco, "Probabilistic analysis of a generalization of the unitclause literal selection heuristic for the k-satisfiability problem," Information Science, Vol. 51 (1990) 289--314.

....gives the currently best known lower bound for r 3 . For general k, an upper bound of r k = O(2 ) can be proved by the first moment method while the technique of [16] gives a constant factor improvement for this bound. On the other hand, a lower bound of r k = Omega Gamma1 =k) is given in [6, 5] and it is improved by a constant factor in [12] Determining the asymptotic order of r k seems an interesting, and challenging, open problem. A recent and very important development pertaining to monotone properties in general (such as satisfiability) is due to Friedgut [11] who provided a ....

.... of two deterministic functions, solve the corresponding differential equations and use the solutions as approximations of C 3 (t) C 2 (t) This use of differential equations to describe the mean path of Markov chains (in this case defined implicitly) has found many applications including [1, 4, 5, 12, 15, 20, 23, 24]. Although the step is quite intuitive, the justification of the fact that the chain a.s. stays close to its mean path, and hence the solutions of the differential equations provide a good approximation, is highly nontrivial. A rigorous, unifying framework regarding the applicability of this ....

M.-T. Chao, and J. Franco, Probabilistic analysis of a generalization of the unit-clause literal selection heuristic for the k-satisfiability problem, Information Science, 51 (1990), 289--314.

....lower bounds of [3] from order k to order k. In particular, they proved that there exists a constant c 0 such that if r c 2 =k then a simple, linear time algorithm w.h.p. nds a 2 coloring of H k (n; rn) Their algorithm was motivated by algorithms for random k SAT due to Chao and Franco [7] and Chv atal and Reed [8] In fact, those algorithms give a similar m =k) lower bound on the random k SAT threshold which, like r k , can also be easily bounded as O(2 ) Very recently, the authors eliminated the gap for the random k SAT threshold, determining its value within a factor of ....

Chao, M.-T., and Franco, J. Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k-satis ability problem. Inform. Sci. 51, 3 (1990), 289-314.

....ln 2. To see this, fix any truth assignment and observe that a random k clause is satisfied by it with probability 1 2 . Therefore, the expected number of satisfying truth assignments of F k (n; m = rn) is [2(1 2 = o(1) for r 2 ln 2. Shortly afterwards, Chao and Franco [6] complemented this result by proving that for all k 3, if r 2 =k then the following lineartime algorithm, called UNIT CLAUSE (UC) finds a satisfying truth assignment with probability at least = r) 0: If there exist unit clauses, pick one randomly and satisfy it; else pick a random ....

....from 1 to 0. In the following, we will take the liberty of writing r k r if F k (n; rn) is satisfiable w.h.p. for all r r (and analogously for r k r ) Chvatal and Reed [7] besides proving r 2 = 1, gave the first lower bound for r k , strengthening the positiveprobability result of [6]. In particular, they considered a generalization of UC, called SC, which in the absence of unit clauses satisfies a random literal in a random 2 clause (and in the absence of 2 clauses satisfies a uniformly random literal) They proved that for all k 3, if r (3=8)2 =k then SC finds a ....

M.-T. Chao and J. Franco. Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k-satisfiability problem. Inform. Sci., 51(3):289--314, 1990.

....80 s observed that r # k ln 2. To see this, fix any truth assignment and observe that a random k clause is satisfied by it with probability 1 2 k . Therefore, the expected number of satisfying truth assignments of F k (n, rn) is [2(1 2 k ) o(1) for r ln 2. In 1990, Chao and Franco [3] complemented this by proving that that for r 2 k a simple algorithm, called Unit Clause(uc) finds a satisfying truth assignment with uniformly positive probability. At around the same time, experimental results by Cheeseman, Kanefsky and Taylor [4] and Mitchell, Selman and Levesque [16] ....

M.-T. Chao and J. Franco. Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k-satisfiability problem. Inform. Sci., 51(3):289--314, 1990.

....pretty much understood. For k 3 the story is very different. It is now known that a threshold for satisfiability exists in some (not completely satisfactory) sense, Friedgut [10] There has been considerable work on trying to find estimates for this threshold in the case k 3 Chao and Franco [5, 6], Broder, Frieze and Upfal [4] Frieze and Suen [12] Achlioptas [1] Achlioptas and Sotkin [2] the last mentioned paper giving a lower bound of 3.26. Upper bounds have been pursued with the same vigour Kirousis, Kramakis, Krizanc and Stamatiou [15] Janson, Stamatiou and Vamvakari [14] ....

M.T. Chao and J. Franco, Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k satisfiable problem, Information Science 51 (1990) 289-314.

....useful model for evaluating the effectiveness of a particular propositional proof system: strong lower bounds on proof size for random k CNF formulas attest to the fact that the proof system in question is ineffective on average. A fundamental conjecture about the random k CNF formula model, see [CS88, BFU93, CF90, CR92, FS96, KKKS98]) says that there is a constant q k , the satisfiability threshold, such that a random k CNF formula of clause density D is almost certainly satisfiable for D q k (as n gets large) and almost certainly unsatisfiable if D q k . There is considerable empirical and analytic evidence for this. ....

M.T. Chao and J. Franco. Probabilistic analysis of a generalization of the unit-clause literal selection heuristics. Information Science, 51:289--314, 1990.

....of variables approaches infinity. We prove this result by analyzing the behaviour of an algorithm for Model GB. In what follows, 1 C denote the set of constraints of arity 1. This algorithm, basically a natural extension of the Unit Clause heuristic for k SAT introduced by Chao and Franco [3] is as follows: Algorithm: UC Input: A random CSP instance Output: a solution exists or can not determine whether a solution exists 1. Set 0 j . 2. Repeat. 3. If f 1 C , then choose, at random, a constraint l from 1 C and assign a value to the variable u in l to make l satisfied . 4. ....

....then UC algorithm verifies that a solution exists for random CSP instances generated by Model GB with probability greater than e for some 0 e as the number of variables tends to infinity. For 2 = k , this condition amounts to . 1 r The basic idea behind the proof of this theorem, as given in [3], is as follows: after the algorithm has successfully assigned values to the first j variables, there are ) j C i constraints of arity i that are uniformly distributed among all possible constraints of arity i on the unset variables. Moreover, if the average number of constraints of arity 1 into ....

M.T. Chao, J. Franco. Probabilistic analysis of a generalization of the unit-clause literal selection heuristic for the k-satisfiability problem. Information Science. 51 (1990) 289-314.

....now pretty much understood. For k 3 the story is very di erent. It is now known that a threshold for satis ability exists in some (not completely satisfactory) sense, Friedgut [10] There has been considerable work on trying to nd estimates for this threshold in the case k = 3 Chao and Franco [5, 6], Broder, Frieze and Upfal [4] Frieze and Suen [12] Achlioptas [1] Achlioptas and Sorkin [2] the last mentioned paper giving a lower bound of 3.26. Upper bounds have been pursued with the same vigour Kirousis, Kramakis, Krizanc and Stamatiou [15] Janson, Stamatiou and Vamvakari [14] ....

M.T. Chao and J. Franco, Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k satis able problem, Information Science 51 (1990) 289-314.

....is as m is increased, when does F (k; n; m) stop being w.h.p. satis able Again, by considering the expected number of solutions (here, satisfying assignments) it is easy to show if m = c2 k n, where c ln 2, then w.h.p. F (k; n; m) is unsatis able. In the opposite direction, Chao and Franco [6] proved that, for k 4, a random k SAT formula with m = c(2 k =k)n clauses is w.h.p. satis able, if c 1=4. Chv atal and Reed [7] extended the result 2 of [6] to all k 2 (and simpli ed it) while Frieze and Suen [12] inter alia, improved the constant to c 1. The similarity between the ....

....it is easy to show if m = c2 k n, where c ln 2, then w.h.p. F (k; n; m) is unsatis able. In the opposite direction, Chao and Franco [6] proved that, for k 4, a random k SAT formula with m = c(2 k =k)n clauses is w.h.p. satis able, if c 1=4. Chv atal and Reed [7] extended the result 2 of [6] to all k 2 (and simpli ed it) while Frieze and Suen [12] inter alia, improved the constant to c 1. The similarity between the two problems is quite apparent, though probably cannot be translated into a formal statement. This similarity stimulated Alon and Spencer [1] to try and derive a ....

[Article contains additional citation context not shown here]

Chao, M.-T., and Franco, J. Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k-satisability problem. Inform. Sci. 51, 3 (1990), 289-314.

....execution with a set of functions s 1 (x) s k (x) de ned by a system of di erential equations. It follows from the main theorem of [34] that for 0 x 1, after roughly xn vertices have been colored, a.s. jS i j = s i (x)n o(n) This is a common technique for other applications see [8, 9, 17, 21, 28, 30, 34] and we omit a detailed justi cation in this extended abstract. We show that if s 0 1 (x) 0 for all 0 x 1, then the algorithm will succeed with constant positive probability. Intuitively, this means that if vertices enter S 1 slowly enough (at a rate less than one per step so that s 0 ....

M.-T. Chao, J. Franco, Probabilistic analysis of a generalization of the unit-clause literal selection heuristic for the k-satisability problem, Information Science, 51 (1990), 289-314.

.... [13] and the rate of decline in complexity with increasing ratio has been analyzed as well [6] Our new results show that the lower end of this characterization is also somewhat misleading; in fact, our results show that the exponentially hard region for the Unit Clause (UC) algorithm studied in [11] (and the basis for the back tracking algorithm [33] begins at least at ratio 3:81, well before ratio 4:2. This concurs with recent experimental evidence that even the best of current Davis Putnam implementations seem to have bad behavior below the threshold [14] An extension of random 3 SAT ....

M.T. Chao and J. Franco. Probabilistic analysis of a generalization of the unit-clause literal selection heuristics. Information Science, 51:289-314, 1990.

....algorithms will succeed almost certainly if they exhaust the 3 clauses without c 2 ever reaching 1. In fact, if they reach ( 1 )n; 2n=3) without c 2 ever reaching 1 they will succeed with constant probability [4] One can extend the card type algorithms, such as UC (unit clauses first) and GUC [14], into full backtracking DPLL algorithms in a variety of different ways so that the execution of the original algorithm is the first path explored in the tree of recursive calls. If the original card type algorithm reaches ( 1 )n; c 3 n) where c 3 is at least 2.28, the resulting formula is almost ....

....with probability at least p A . 2. For each A 2 fUC,ORDERED DLL,GUCg, an execution of algorithm A FS on a random 3 CNF formula with r An clauses reaches a bad t stage with t n=2 such that the residual formula is uniformly distributed w.h.p. 19 Proof. The lemma follows readily from results in [14, 1, 21]. Below we outline these results and show how they can be combined. The original analyses in these papers were largely geared towards understanding the ratios between clauses and variables at which random k CNF formulas remain satisfiable almost surely, particularly in the case that k = 3. In ....

[Article contains additional citation context not shown here]

M.T. Chao and J. Franco. Probabilistic analysis of a generalization of the unit-clause literal selection heuristics. Information Science, 51:289--314, 1990.

....an instance of SAT. When k = 2 a sharp threshold has been proved in [9, 15] A random instance of 2 SAT is almost surely satisfiable if r 1 and almost surely unsatisfiable if r 1. For k = 3, such a sharp threshold behaviour is widely conjectured to exist and there has been a series of papers [6, 7, 5, 12, 18, 14, 19] narrowing the area for which we do not have almost surely (un)satisfiability. The best bounds currently known come from [14] where it was proven that a random instance of 3 SAT is almost surely satisfiable if r 3:003, and [19] where it was proven that a random instance of 3 SAT is almost surely ....

....known come from [14] where it was proven that a random instance of 3 SAT is almost surely satisfiable if r 3:003, and [19] where it was proven that a random instance of 3 SAT is almost surely unsatisfiable if r 4:598. For general k, the best known lower bound for r is Theta(2 k =k) [7, 14] while 2 k is an easy upper bound and was improved by a constant factor in [19] by extending the techniques used for 3 SAT. Very recently, Friedgut [13] made great progress towards establishing the existence of a threshold for k SAT for all values of k, although he does not determine it s ....

M.-T. Chao and J. Franco, "Probabilistic analysis of a generalization of the unit-clause literal selection heuristic for the k-satisfiability problem," Information Science, Vol. 51, (1990), 289--314.

....in some sense easy , but they do not establish that complete SAT solvers have polynomial median running time in this region. The analytical results of Franco and his collaborators suggest that in this region we might expect a polynomial median running time for certain heuristic algorithms, cf. [10], but they do not prove it de nitively. In [14] the authors reported linear median running time of Tableau, their SAT solver, for densities 1, 2, and 3, and an exponential median running time for densities 4.26 and 10. In [38] the authors reported linear median running time of their DLL SAT ....

M. Chao and J. V. Franco. Probabilistic analysis of a generalization of the unitclause literal selection heuristics for the k-satisability problem. Information Sciences, 51:289-314, 1990.

....Franco and Paull [12] denote this distribution as f(m; n; k) We omit k as it is clear. This model has been used in many studies of k SAT including Brown and Purdom [3] Purdom and Brown [19] Franco and Paull[12] Franco [10] Franco, Plotkin, and Rosenthal [13] M T Chao and Franco [5] and [6], Chv atal and Reed [7] and Goerdt [15] It has also been used in 2 the study of k Exact SAT in Rosenthal, Speckenmeyer, and Kemp [22] and Rosenthal [21] The reader should be cautioned that among these papers there is confusing variation in the symbols used for the number of variables, the ....

....of clauses, and the number of literals per clause. The pure literal heuristic is based on the pure literal rule which is part of the Davis Putnam Procedure (DPP ) 8] an algorithm for satisfiability (SAT ) and k SAT . Franco and Paull [12] provide a recent description. 12] 10] 13] 5] [6], etc. study aspects of DPP using the constant degree model. Goldberg [16] Goldberg, Purdom, and Brown [16] Purdom and Brown [20] Bugrara, Pan, and Purdom [4] Franco [11] etc. study aspects of DPP for SAT using another distribution, the constant density model. A lot of the work in the ....

M-T Chao and J. Franco. Probabilistic Analysis of a Generalization of the Unit Clause Literal Selection Heuristic for the k-Satisfiability Problem. Information Sciences, 51, 289-314, 1990.

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M.-T. Chao and J. Franco. Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k-satis ability problem. Inform. Sci., 51(3):289-314, 1990. 20

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M.-T. Chao and J. Franco. Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k-satisfiability problem. Inform. Sci., 51(3):289--314, 1990. MR1072035 (91g:68076)

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M.-T. Chao and J. Franco. Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the k-satisfiability problem. Inform. Sci., 51(3):289--314, 1990.

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M.T. Chao and J. Franco, Probabilistic analysis of a generalization of the unitclause literal selection heuristic for the k satisfiability problem, Information Science 51 (1990) 289-314.

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M.T. Chao and J. Franco. Probabilistic analysis of a generalization of the unit-clause literal selection heuristics. Information Science, 51:289-314, 1990.

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M.T. Chao and J. Franco. Probabilistic analysis of a generalization of the unit-clause literal selection heuristics. Information Science, 51:289--314, 1990.

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M. T. Chao and J. Franco. Probabilistic analysis of a generalization of the unit-clause literal selection heuristics. Information Science, 51:289--314, 1990.

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Chao, Ming-Te, and John Franco, "Probabilistic analysis of a generalization of the unit-clause literal selection heuristics for the K satisfiable problem", Information science 51 (1990), 289--314.

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