L. M. Kirousis, E. Kranakis, D. Krizanc, and Y. Stamatiou, Approximating the unsatisfiability threshold of random formulas, Random Structures & Algorithms, 12 (1998), 253--269.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....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] Dubois, Boufkhad and Mandler [8] the last mentioned paper giving an upper bound of 4.506. For larger values of k, even less is known. It was shown in [7] that if m n and k is constant then a random instance of k SAT is satisfiable with probability ....

L.M. Kirousis, E. Kranakis, D. Krizanc and Y.C. Stamatiou, Approximating the unsatisfiability threshold of random formulas, Random Structures and Algorithms 12 (1998) 253--269.

....last ten years, the satisfiability threshold conjecture has received attention in theoretical computer science, mathematics and, more recently, statistical physics. A large fraction of this attention has been devoted to the first computationally non trivial case, k = 3 and a long series of results [4, 5, 16, 1, 3, 21, 10, 22, 19, 23, 20, 11, 13] has narrowed the potential range of r 3 . Currently this is pinned between 3:42 by Kaporis, Kirousis and Lalas [21] and 4:506 by Dubois and Boufkhad [10] All upper bounds for r 3 come from probabilistic counting arguments, refining the idea of counting the expected number of satisfying truth ....

.... bound for the random NAE k SAT threshold for each value of k as the solution to a transcendental equation (yet one without an attractive closed form, hence Theorem 2) It is, perhaps, worth comparing our lower bound for the NAE k SAT threshold with the upper bound derived using the technique of [23] for small values of k. Even for k = 3, our lower bound is competitive with the best known lower bound of 1:514, obtained by analyzing a generalization of UC that minimizes the number of unit clauses [2] For larger k, the gap between the upper and the lower bound rapidly converges to 1=4. k 3 ....

L. M. Kirousis, E. Kranakis, D. Krizanc, and Y. Stamatiou. Approximating the unsatisfiability threshold of random formulas. Random Structures Algorithms, 12(3):253--269, 1998.

....moment method. Our proof actually yields an explicit lower bound for r k for each k 3. Already for k 4 our result improves upon all previously known lower bounds for r k . Below, we compare our lower bound with the best known algorithmic lower bound [10, 12] and the best known upper bound [7, 15] for some small values of k. k 3 4 5 7 10 20 21 Upper bound 4.51 10.23 21.33 87.88 708.94 726, 817 1, 453, 635 Our lower bound 2.68 7.91 18.79 84.82 704.94 726, 809 1, 453, 626 Algorithmic lower bound 3.42 5.54 9.63 33.23 172.65 95, 263 181, 453 Theorem 2 establishes that the random k SAT ....

....assignments that have at least one satisfied literal in every clause yet, in total, satisfy only as many literal occurrences as a random truth assignment. Our proof leaves a O(k) gap between the upper and lower bounds. With respect to the gap it is worth pointing out that the best known techniques [15] for improving the first moment upper bound only give (1 ln 2) 2. However, already for r = 2 k(ln 2) 2 w.h.p. there are no satisfying truth assignments that satisfy only km 2 o(n) literal occurrences. That is, any improvement over our lower bound would mean that tendencies toward ....

L. M. Kirousis, E. Kranakis, D. Krizanc, and Y. Stamatiou. Approximating the unsatisfiability threshold of random formulas. Random Structures Algorithms, 12(3):253--269, 1998. 11

....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] Dubois, Boufkhad and Mandler [8] the last mentioned paper giving an upper bound of 4.506. 2 k For larger values of k, even less is known. It was shown in [7] that if m ( n and k is constant then a random instance of k SAT is satisfiable with probability ....

L.M. Kirousis, E. Kranakis, D. Krizanc and Y.C. Stamatiou, Approximating the unsatisfiability threshold of random formulas, Random Structures and Algorithms 12 (1998) 253-269.

....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. ....

....q k (n) with the above property, but he does not rule out the possibility that q k (n) varies with n. It is known that q 2 = 1 is independent of n [CR92, Goe96] and that for each k q k (n) is bounded between two constants b k and d k that are independent of n, e.g. and 3.003 q 3 (n) 4. 601 [FS96, KKKS98]. The threshold indicates three distinct ranges of clause density for investigating complexity. For D at the threshold, an effective algorithm must be able to distinguish between unsatisfiable and satisfiable instances. Below the threshold, a random formula is almost certainly satisfiable and the ....

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L. M. Kirousis, E. Kranakis, D. Krizanc, and Y. C. Stamatiou. Approximating the unsatisfiability threshold of random formulas. Random Structures and Algorithms, 12(3):253-- 269, 1998.

....were also found for other values of k in random k SAT. Up to now, only the phase transition point for 2 SAT has been proved to be 1 by Chvtal and Reed [5] and Goerdt [9] For random 3 SAT, the best known lower bound and upper bound for the phase transition point are 3.145 and 4. 602 [11] respectively. The interest in the phase transition behaviour has been furthered enhanced by the observation that the instances in the Research supported by National 973 Project of China Grant No.G1999032701. For the first author, there is a permanent email address that is ke.xu 263.net. ....

Kirousis, L. M., Kranakis, E., Krizanc, D., & Stamatiou, Y. C. Approximating the unsatisfiability threshold of random formulas, Random Structures and Algorithms 12 (1998) 253-269.

....1; f( c ffl)n) 0. This was shown to be true for k = 2 with the constant c = 1, see [10] 17] but for k 3 was not known. For k = 3 a series of upper and lower bounds have seemed to be slowly converging to the value suggested by simulations (c = 4:2: see [19] 11] 9] 8] 16] 22] [21]. We now show that the existence of a threshold for any given k can be demonstrated by the proof of theorem 1.1. I would like to thank Svante Janson for pointing out the following subtlety to me: What I actually show is not the existence of a constant c but of a function c(n) such that the phase ....

M. Kirouris, E. Kranakis and D. Krizanc, Approximating the unsatisfiability threshold of random formulas, in: Algorithms - ESA' 96, Lecture Notes in Computer Science 1136

....set in G then the graph induced by V n U very rarely contains a perfect matching. For simplicity we restrict our analysis to the case c 1. The following Lemma gives an upper bound on (G) by using a strengthened version of the Markov inequality. Such technique has been applied elsewhere [2, 9, 11]. Lemma 1. For each c 1 there exists a constant f c 1 such that (G) f c n for a.e. G 2 G(n; c=n) Proof. An independent set I in a graph G = V; E) is maximal if for every v 2 V n I there exists u 2 I such that fu; vg 2 E. If G has no maximal independent set of size at least k then it ....

L. M. Kirousis, E. Kranakis, D. Krizanc, and Y. C. Stamatiou. Approximating the unsatisfiability threshold of random formulas. Random Structures and Algorithms, 12(3):253--269, 1998.

....a threshold has been recently confirmed [Fri97] its exact location has yet to be found. Initial upper bounds on (n) for k = 3 were obtained in [CS88] where it was proved that almost all m clause, n variable instances with m 5:19 n are unsatisfiable. The best results to date are Theorem 34 [KKKS98] Almost all formulae with more than 4:601 n clauses are not satisfiable. 64 Theorem 35 [FS96] Almost all formulae with less than 3:003 n clauses are satisfiable. Phase transition phenomena are not restricted to decision problems. In fact the last two sections in Chapter 4 will provide examples ....

....E(X ] 7 8 m X Pr[ 2 A ] j 2 A ] The final bit of work involves obtaining a good upper bound on the probability that a specific is a maximal satisfying assignment conditioned on the fact that is satisfied by . Let K c = 1 e 3c=7 for every c 2 IR . Theorem 39 [KKKS98] If is a random formula on n variables and m = cn clauses, and sets s variables to zero then Pr[ 2 A ] j 2 A ] K c o(1) s . Theorem 40 [KKKS98] If is a random formula on n variables and m = cn clauses, the expected value of X ] is at most (7=8) cn (1 K c o(1) n . It ....

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L. M. Kirousis, E. Kranakis, D. Krizanc, and Y. C. Stamatiou. Approximating the Unsatisfiability Threshold of Random Formulas. Random Structures and Algorithms, 12(3):253--269, 1998.

....how a random 3 SAT formula can be satisfied by binary assignments (possibly of a particular kind) to its variables. Thus, two lower bounds of were calculated : 1:63 and 3:003 [2] And starting from an easy upper bound, 5:19, three others were successively obtained : 4:76 [3] 4:643 [1] and 4:601 [4]. However this semantic approach may be subject to diminishing returns in view of the complexity of the calculations for the latest bounds. In this paper, in order to estimate better, a new syntactic approach is proposed: This is suggested, e.g. simply by practical experience with a ....

L. M. Kirousis, E. Kranakis, D. Krizanc, and Y. C. Stamatiou. Approximating the unsatisfiability threshold of random formulas. Random Structures and Algorithms, 12(3):253--269, 1998. 2

....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. ....

....(n) with the above property, but he does not rule out the possibility that q k (n) varies with n. It is known that q 2 = 1 is independent 5 of n [CR92, Goe96] and that for each k q k (n) is bounded between two constants b k and d k that are independent of n, e.g. and 3.003 # q 3 (n) # 4. 601 [FS96, KKKS98]. The threshold indicates three distinct ranges of clause density for investigating complexity. For D at the threshold, an effective algorithm must be able to distinguish between unsatisfiable and satisfiable instances. Below the threshold, a random formula is almost certainly satisfiable and the ....

[Article contains additional citation context not shown here]

L. M. Kirousis, E. Kranakis, D. Krizanc, and Y. C. Stamatiou. Approximating the unsatisfiability threshold of random formulas. Random Structures and Algorithms, 12(3):253-- 269, 1998.

....to variables is approximately 4.3 [45] Unfortunately, theory has often proved more difficult. A recent result proves that the width of the phase transition in random 3 Sat narrows as problems increase in size [15] However, we only have rather loose but hard won bounds on its actual location [16, 37]. For random constraint satisfaction problems (CSP s) Achlioptas et al. recently provided a more negative theoretical result [1] They show that, as the number of variables in problems increases, the conventional random models almost surely contain flawed variables and are therefore trivially ....

L.M. Kirousis, E. Kranakis, and D. Krizanc. Approximating the unsatisfiability threshold of random formulas. In Proceedings of the 4th Annual European Symposium on Algorithms (ESA'96), pages 27--38, 1996.

....3 occurs at around 4:2, and a recent theoretical result by Friedgut (1997) shows that for each n, there is a sharp 0 1 threshold ff k (n) that possibly varies with n. It is known that ff 2 = 1 (Chv atal Reed, 1992; Goerdt, 1996) and that for k = 3, 3:003 ff k (n) 4:601 (Frieze Suen, 1996; Kirousis et al. 1998). An interesting feature of the threshold is that the dual problems of bounding the threshold value from above and below appear to be asymmetric. At slightly below the 0 1 threshold, there are simple lineartime algorithms for finding a satisfying assignment, but at slightly above the 0 1 ....

....instance that is not positive. Empirical estimates of the 0 1 threshold in Section 3.1 confirm the form of Equation 1. While the above calculation gives an upper bound on m as a function of t and n, this upper bound is likely larger than the actual number. In several papers (Kamath et al. 1995; Kirousis et al. 1998), it has been demonstrated that the expected number of satisfying assignments is larger than the number of satisfying assignments almost certainly, because for those formulae that are satisfiable, they actually have many satisfying assignments (thus skewing the average) In particular, Kamath et ....

[Article contains additional citation context not shown here]

Kirousis, L. M.; Kranakis, E.; Krizanc, D.; and Stamatiou, Y. C. 1998. Approximating the unsatisfiability threshold of random formulas. Random Structures and Algorithms 12(3):253--269.

....in [GPFW97] Lastly, the random model is important for proving lower bounds for propositional proof systems. Lower bounds 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, KKK96]) says that there is a constant c k , the satisfiability threshold, such that a random k CNF formula of clause density D is almost certainly satisfiable for D c k (as n gets large) and almost certainly unsatisfiable if if D c k . There is considerable empirical and analytic evidence for ....

....c k (n) with the above property, but he does not rule out the possibility that c k (n) varies with n. It is known that c 2 = 1 is independent of n [CR92, Goe96] and that for each k c k (n) is bounded between two constants b k and d k that are independent of n, e.g. and 3:003 c 3 (n) 4:598 [FS96, KKK96]. The threshold indicates three distinct ranges of clause density for investigating complexity. For D at the threshold, an effective algorithm must be able to distinguish between unsatisfiable and satisfiable instances. Below the threshold, a random formula is almost certainly satisfiable and the ....

[Article contains additional citation context not shown here]

L. M. Kirousis, E. Kranakis, and D. Krizanc. Approximating the unsatisfiability threshold of random formulas. In Proceedings of the Fourth Annual European Symposium on Algorithms, pages 27--38, Barcelona, Spain, September 1996.

....3 occurs at around 4:2, and a recent theoretical result by Friedgut (1997) shows that for each n, there is a sharp 0 1 threshold ff k (n) that possibly varies with n. It is known that ff 2 = 1 (Chv atal Reed, 1992; Goerdt, 1996) and that for k = 3, 3:003 ff k (n) 4:601 (Frieze Suen, 1996; Kirousis et al. 1998). An interesting feature of the threshold is that the dual problems of bounding the threshold value from above and below appear to be asymmetric. At slightly below the 0 1 threshold, there are simple lineartime algorithms for finding a satisfying assignment, but at slightly above the 0 1 ....

....instance that is not positive. Empirical estimates of the 0 1 threshold in Section 3.1 confirm the form of Equation 1. While the above calculation gives an upper bound on m as a function of t and n, this upper bound is likely larger than the actual number. In several papers (Kamath et al. 1995; Kirousis et al. 1998), it has been demonstrated that most formulae will have a much smaller number of satisfying assignments than the expected number of satisfying assignments, because for those formulae that are satisfiable, they actually have many satisfying assignments (thus skewing the average) In particular, ....

[Article contains additional citation context not shown here]

Kirousis, L. M.; Kranakis, E.; Krizanc, D.; and Stamatiou, Y. C. 1998. Approximating the unsatisfiability threshold of random formulas. Random Structures and Algorithms 12(3):253--269.

.... transition For random 2 Sat, the phase transition has been proven to occur at L=N = 1 [9, 42] For random 3 Sat, the phase transition has been shown to occur experimentally around L=N = 4:3 [75, 12] Theoretical bounds put the phase transition for random 3 Sat in the interval 3:003 L=N 4:598 [23, 66]. For random 4 Sat, the phase transition has been shown to occur experimentally around L=N = 9:8 [35] For large k, the phase transition for random k Sat occurs at a value of L=N close to Gamma1= log 2 (1 Gamma 1 2 k ) 65] A recent result of Friedgut proves that the width of the phase ....

L.M. Kirousis, E. Kranakis, and D. Krizanc. Approximating the unsatisfiability threshold of random formulas. In Proceedings of the 4th Annual European Symposium on Algorithms (ESA'96), pages 27--38, 1996.

....proven to occur at l=n = 1 [8, 25] which corresponds to 0:42. For random 3 Sat, the phase transition has been shown to occur experimentally around l=n = 4:3 [43, 10] which corresponds to 0:82. Theoretical bounds put the phase transition for random 3 Sat in the interval 3:003 l=n 4:598 [15, 34]. This corresponds to the interval 0:58 0:89. For random 4 Sat, the phase transition has been shown to occur experimentally around l=n = 9:8 [21] which corresponds to 0:91. For large k, the phase transition for random k Sat occurs at a value of l=n close to Gamma1= log 2 (1 Gamma 1 2 ....

L.M. Kirousis, E. Kranakis, and D. Krizanc. Approximating the unsatisfiability threshold of random formulas. In Proceedings of the 4th Annual European Symposium on Algorithms (ESA'96), pages 27--38, 1996.

....= r k (n) such that (1) holds. While it is widely believed that lim n 1 r k (n) exists, determining this limit (r k ) even for k = 3, seems very challenging. In this paper we first go over the (first moment) argument that enables the known upper bounds for r k and then show how the main idea of [6], developed for random k SAT, can be seen as a refinement of this argument, yielding an improved upper bound for r k . y Authors address: Department of Computer Science, University of Toronto, Toronto, Ontario, Canada M5S 3G4. Email: foptas,molloyg cs.toronto.edu 2 The first moment method ....

....graph is k colorable could be small while the expected number of k colorings is not. If one views the set of all k partitions as the n dimensional k ary cube f1; kg n then the above argument suggests that when k colorings exist, they tend to appear in large clusters . The main idea of [6], when translated to k coloring, amounts to examining the expectation of only those k colorings that satisfy some local maximality condition. That is, instead of counting all the k colorings in a cluster (as the first moment does) we need only count a representative one (locally maximum) 3 A ....

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L. M. Kirousis, E. Kranakis, D. Krizanc, and Y. Stamatiou, Approximating the unsatisfiability threshold of random formulas, Random Structures & Algorithms, to appear. A preliminary version appeared in Proc. of the 4th European Symposium on Algorithms, Springer LNCS, 1136: 27--38, 1996.

....to variables is approximately 4.3 [43] Unfortunately, theory has often proved more difficult. A recent result proves that the width of the phase transition in random 3 Sat narrows as problems increase in size [14] However, we only have rather loose but hard won bounds on its actual location [15, 35]. For random constraint satisfaction problems, Achlioptas et al. recently provided a more negative theoretical result [1] They show that, as problem size increases, the conventional random models almost surely contain flawed variables and are therefore trivially insoluble. This paper studies the ....

L.M. Kirousis, E. Kranakis, and D. Krizanc. Approximating the unsatisfiability threshold of random formulas. In Proceedings of the 4th Annual European Symposium on Algorithms (ESA'96), pages 27--38, 1996.

....Molloy Department of Computer Science University of Toronto Abstract We show that if a random graph on n vertices has m = r o(1) n edges, where r=2.522, then almost surely it is not 3 colorable. The previous best such value for r was 2. 571[5] Our result follows from applying ideas of [4] to the k coloring problem. The development of the proof is for k colorings, where k is an arbitrary constant, and we present similar improvements for small values of k. 1 Introduction The term almost all in the title has the meaning introduced by Erdos and R enyi [3] If N(n; m; A) stands for ....

....with very small probability, contribute substantially to the expectation. In other words, by substituting the number of k colorings for the variable indicating the existence of a k coloring (which is the essence of the first moment method) we might give away a lot . The technique introduced in [4], as we apply it to coloring, amounts to taking the expectation of only those k colorings that satisfy a certain local maximality condition. The underlying intuition is that for a random k coloring of G, if we choose a node at random and assign it a different color, the probability that we will ....

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L. M. Kirousis, E. Kranakis, D. Krizanc, Approximating the unsatisfiability threshold of random formulas, Proceedings of the 4th European Symposium on Algorithms, (1992) 27--38.

....one of the most important open problems in the field of random graphs and it is closely related to the problem of determining the ratio of clauses to variables at which a random instance of k SAT turns from a.s. satisfiable to a.s. unsatisfiable. For recent progress in this latter problem see [2, 14, 17, 22]. Luczak [24] proved that asymptotically c k 2k ln k by showing the existence of suitably large, disjoint independent sets in G n;p=ck=n . Algorithmically, substantially less progress has been made. One of the first heuristics to be analyzed [18] 4] GIC, forms color classes by repeatedly ....

L. M. Kirousis, E. Kranakis, D. Krizanc, Approximating the unsatisfiability threshold of random formulas, Proceedings of the 4th European Symposium on Algorithms, (1992) 27--38.

....these inequalities are somewhat better than Suen s, although the difference is negligible in many cases. See Section 8 below. Those inequalities have been used by several different authors for a variety of problems; there are, however, many situations where they are not applicable (see [8, 5] for two examples) and then Suen s inequality is a very attractive choice. The purpose of the present note is to present some improvements and modifications of Suen s original inequality which (we hope) will be easy to apply in different situations. The estimates considered here are exponential ....

....as in Theorem 1, if furthermore the variables fI i g are positively correlated, P(S = 0) e Gamma 2 = max(48 Delta;4 ) We do not know whether the terms involving ffi really are needed in the estimates above, cf. Section 8. In most applications they are harmless, but in at least one [5] they affect the final result significantly. For that reason, we give two slightly stronger versions of Theorems 1 and 2, where this term is reduced as much as we have been able to achieve. Define 1 (x) 2 Z 1 0 te tx dt = 2 xe x Gamma e x 1 x 2 = e x e Gammax Gamma 1 x ....

L.M. Kirousis, E. Kranakis, D. Krizanc & Y.C. Stamatiou, Approximating the unsatisfiability threshold of random formulas. To appear.

....holds) Many authors have given numerical bounds for fl or fl using various techniques. The best bounds that have appeared so far to our knowledge are 3:003 fl fl 4:601 ; where the lower bound is due to Frieze and Suen [6] and the upper to Kirousis, Kranakis, Krizanc and Stamatiou [5]. The purpose of the present paper is to present a small improvement in the upper bound. Theorem 1. fl 4:596. The proof is based on the proof in [5] improving one estimate only. The improvement concerns the sum S n = X 1 ; n2f0;1g exp i a n X i=1 i b n XX 1i jn i ....

....are 3:003 fl fl 4:601 ; where the lower bound is due to Frieze and Suen [6] and the upper to Kirousis, Kranakis, Krizanc and Stamatiou [5] The purpose of the present paper is to present a small improvement in the upper bound. Theorem 1. fl 4:596. The proof is based on the proof in [5], improving one estimate only. The improvement concerns the sum S n = X 1 ; n2f0;1g exp i a n X i=1 i b n XX 1i jn i (1 Gamma j ) j ; 1) summing over the 2 n sequences 1 ; n of 0 s and 1 s. We will prove the following asymptotic formula for S n . All ....

[Article contains additional citation context not shown here]

L.M. Kirousis, E. Kranakis, D. Krizanc & Y.C. Stamatiou, Approximating the unsatisfiability threshold of random formulas. Random Struct. Alg. 12 (1998), 253--269.

.... with extra terms involving ffi and somewhat worse constants in (10) In applications, ffi is usually small and the bounds (9) and (10) are often as useful as (5) and (6) Nevertheless, in at least one application (a recent bound for the 3 SAT problem by Kirousis, Kranakis, Krizanc and Stamatiou [14], where ffi 0:089) the extra factor e 2ffi in (9) affects the final result significantly, and it is desirable to reduce it as much as possible. It is shown in [12] using a proof by Spencer (personal communication) that if 0 ffi , this factor e 2ffi can be replaced by the smallest root ....

....affects the final result significantly, and it is desirable to reduce it as much as possible. It is shown in [12] using a proof by Spencer (personal communication) that if 0 ffi , this factor e 2ffi can be replaced by the smallest root of = e (ffi ) this is the version used in [14]. It is an open problem whether the factor e 2ffi in (9) can be eliminated completely, i.e. whether (5) or (6) 7) holds in the general case too. ON CONCENTRATION OF PROBABILITY 5 3. Azuma s inequality The final inequalitites that we consider apply to random variables of the form X = f(Z ....

L.M. Kirousis, E. Kranakis, D. Krizanc & Y.C. Stamatiou, Approximating the unsatisfiability threshold of random formulas. Random Struct. Alg. 12 (1998), 253--269.

.... variants of DPLL succeed in finding solutions whenever GUCB does, and in the same time, and therefore will almost always find solutions to these instances in polynomial time (although polynomial average time does not necessarily follow) The upper bound is due to Kirousis, Kranakas and Krizanc [KKK96] which improves previous bounds of 4.64 by Dubois and Boufkhad [DB96] 4.78 by Kamath et al. [KMPS94] and 5.19 reported by several authors. The easy 5.19 bound comes from observing that a fixed assignment t satisfies a random 3 literal clause with probability 7 8 and hence t satisfies a random ....

.... 1 for c log 8 7 2 = 5:191: The argument also shows that for c 5:19 the expected number of assignments satisfying a random instance grows exponentially with n. This may help explain the success of local search methods on satisfiable instances. The improved upper bounds given in [DB96, KKK96] are based on the observation that if an instance OE is satisfiable, then some assignment t maximally satisfies OE, meaning t satisfies OE, but flipping the value of t on any single variable from false to true falsifies OE. Dubois and Boufkhad obtained the bound c 4:642 by estimating the ....

L.M. Kirousis, E. Kranakis, and D. Krizanc. Approximating the unsatisfiability threshold of random formulas. Preprint, 1996.

....to variables is approximately 4.3 [14] Unfortunately, theory has often proved more difficult. A recent result proves that the width of the phase transition in random 3 Sat narrows as problems increases in size [3] However, we only have rather loose but hard won bounds on its actual location [4, 13]. For random constraint satisfaction problems, Achlioptas et al. recently provided a more negative theoretical result [1] They show that the conventional random models are almost surely trivially insoluble for large enough problems. This paper studies the impact of this theoretical result on ....

L.M. Kirousis, E. Kranakis, and D. Krizanc. Approximating the unsatisfiability threshold of random formulas. In Proceedings of the 4th Annual European Symposium on Algorithms (ESA'96), pages 27--38, 1996.

....NP complete problems, most important of which is the satisfiability problem. However, with a small number of exceptions (such as the Hamiltonian cycle [KS83] finding the exact location of the phase transition, or even proving its existence, is very difficult. In most problems (e.g. 3 SAT [FS96, KKKS96] and graph coloring [Chv91, AM97] there still exists a large gap between the best proven lower and upper bounds. An approach to this problem would be to study phase transitions in tractable subcases. The satisfiability has been well studied in this regard. Out of the six maximally tractable ....

L. Kirousis, P. Kranakis, D. Krizanc, and Y. Stamatiou. Approximating the unsatisfiability threshold of random formulas. Technical Report TR96-027, School of Computer Science, Carleton University, November 1996.

....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 ....

....= 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 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 ....

[Article contains additional citation context not shown here]

L.M. Kirousis, E. Kranakis, D. Krizanc, and Y.C. Stamatiou, "Approximating the unsatisfiability threshold of random formulas," Random Structures and Algorithms, Vol. 12 (1998) 253--269.

....et al. 14] lowered 5.08 to 4.758 by proving that if a random 3 SAT formula is satisfiable then, with extremely high probability, it has an exponential number of satisfying truth assignments. Hence, providing some account for the wastefulness of the first moment method. Finally, Kirousis et al. [16] introduced a general refinement of the first moment method whereby one counts not all the satisfying truth assignments ( bad objects) but only those that specify a local maximality condition. We will use their technique in section 2, and elaborate on it therein. With this method they showed [16] ....

....[16] introduced a general refinement of the first moment method whereby one counts not all the satisfying truth assignments ( bad objects) but only those that specify a local maximality condition. We will use their technique in section 2, and elaborate on it therein. With this method they showed [16] that r 3 4:598 : which is the best bound currently known. Independently, Dubois et al. 7] using a similar method obtained r 3 4:64. We will say that a sequence of events En occurs almost surely if limn 1 Pr[En ] 1. This idea of using the expectation of a non negative variable, ....

[Article contains additional citation context not shown here]

L. M. Kirousis, and E. Kranakis, and D. Krizanc, Approximating the unsatisfiability threshold of random formulae, Proceedings of the 4th European Symposium on Algorithms, (1992) 27-- 38.

....We then present a new model that is suitable for asymptotic analysis and, in the spirit of random SAT, we derive lower and upper bounds for its parameters so that the instances generated are almost surely over and underconstrained, respectively. Finally, we apply the technique introduced in [19], to one of the popular models in Artificial Intelligence and derive sharper estimates for the probability of being overconstrained as a function of the number of variables. 1 Introduction A constraint network comprises n variables, with their respective domains, and a set of constraint ....

....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 ....

[Article contains additional citation context not shown here]

L. M. Kirousis, E. Kranakis, and D. Krizanc, "Approximating the Unsatisfiability Threshold of Random Formulas," in Proceedings of the Fourth Annual European Symposium on Algorithms, ESA'96, Barcelona, Spain, (1992) 27--38.

....the currently best upper bound has been announced by Dubois et al. 7] and it is 4.506. In this paper, we address the upper bound question for the unsatisfiability threshold from a new perspective that combines the idea of local maximum satisfying truth assignments proposed by Kirousis et al. [14], with the use of sharp estimates on some of the probabilities involved based on results about the so called coupon collector experiment (see for instance [17] and references thereafter) We obtain an upper bound of 4.571 thus improving over the best, with an available complete proof, previous ....

....(a generalization of the binomial coefficients) Despite the extensive literature on q binomial coefficients (see, e.g. 9, 11, 15] no such bound was, to the best of our knowledge, known. 2 The method of local maxima In this section, we will state briefly the methodology followed in [14] and obtain the starting upper bound on the probability that a random formula is satisfiable. Let S be the class of all truth assignments to n variables and An the (random) class of truth assignments that satisfy a random formula . For a given A 2 S, a single flip sf is the change in A of exactly ....

[Article contains additional citation context not shown here]

L.M. Kirousis, E. Kranakis, D. Krizanc, Y.C. Stamatiou, "Approximating the unsatisfiability threshold of random formulas," Random Structures and Algorithms 12, pp 253--269, 1998.

....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 [10, 17] 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 [7, 8, 6, 14, 20, 16, 22, 12] narrowing the area for which we do not have almost certain (un)satisfiability. The best bounds currently known come from [16] where it was proven that a random instance of 3 SAT is almost certainly satisfiable if r 3:003, and [22] where it was proven that a random instance of 3 SAT is almost ....

....= 3, there has been a series of results [7, 8, 6, 14, 20, 16, 22, 12] narrowing the area for which we do not have almost certain (un)satisfiability. The best bounds currently known come from [16] where it was proven that a random instance of 3 SAT is almost certainly satisfiable if r 3:003, and [22] 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 =k) 8, 10, 16] while 2 k is an easy upper bound and was improved by a constant factor in [22] by extending the techniques used for ....

[Article contains additional citation context not shown here]

L.M. Kirousis, E. Kranakis, D. Krizanc, and Y.C. Stamatiou, "Approximating the unsatisfiability threshold of random formulas," Random Structures and Algorithms, Vol. 12 (1998) 253--269.

....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 [11, 19] 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 [8, 9, 7, 15, 22, 17, 25, 13] narrowing the area for which we do not have almost certain (un)satisfiability. The best bounds currently known come from [17] where it was proven that a random instance of 3 SAT is almost certainly satisfiable if r 3:003, and [25] where it was proven that a random instance of 3 SAT is almost ....

....= 3, there has been a series of results [8, 9, 7, 15, 22, 17, 25, 13] narrowing the area for which we do not have almost certain (un)satisfiability. The best bounds currently known come from [17] where it was proven that a random instance of 3 SAT is almost certainly satisfiable if r 3:003, and [25] 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 =k) 9, 11, 17] while 2 k is 2 an easy upper bound and was improved by a constant factor in [25] by extending the techniques used ....

[Article contains additional citation context not shown here]

L.M. Kirousis, E. Kranakis, D. Krizanc, and Y.C. Stamatiou, "Approximating the unsatisfiability threshold of random formulas," Random Structures and Algorithms, Vol. 12 (1998) 253--269. 16

....and have applications in mathematics and physics unimaginable by the uninitiated. We refer to the classical book by Gasper and Rahman [1] for a comprehensive account. We encountered a basic hypergeometric series in our research on probabilistic satisfiability of random Boolean formulas (see [2] for details) Specifically we needed an upper bound for the series P n k=0 Gamma n k Delta q x k expressed, preferrably, as a product. Of course, when q 1, 1 x) n is such an upper bound, but this was not sufficiently small for our purposes. On the other hand, again assuming q 1, ....

L.M. Kirousis, E. Kranakis, and D. Krizanc, Approximating the Unsatisfiability Threshold of Random Formulas, Technical Report TR-95-26, School of Computer Science, Carleton University, Ottawa, Canada, 1995.

....factor that is a function only of the p i s and the sum P i j q ij , where i j denotes that the intersection A i T A j 6= and that i 6= j. Notice that p i = 1 Gamma (1 Gamma (jA i j=j Sigmaj) m and q ij = 1 Gamma (jA i [ A j j=j Sigmaj) m : Our motivation comes from the work in [4] concerning the satisfiability problem of random Boolean formulas, where the question of bounding the probability that a random formula is not satisfied by a given collection of truth assignments was encountered. For each i = 1; r, let E i be the event of at least one letter from A i ....

L.M. Kirousis, E. Kranakis, and D. Krizanc, Approximating the Unsatisfiability Threshold of Random Formulas, Technical Report TR-95-26, School of Computer Science, Carleton University, Ottawa, Canada, 1995.

....sequence of events E n occurs almost surely if Pr(E n ) tends to one as n tends to infinity. For k = 2 a sharp threshold was proved in [9, 16] A random instance of 2 SAT is almost surely satisfiable if r 1 and almost surely 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 surely (un)satisfiability. The best bounds currently known come from [15] where it was proven that a random instance of 3 SAT is almost surely satisfiable if r 3:003, and [21] where it was proven that a random instance of 3 SAT is almost surely ....

....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 surely (un)satisfiability. The best bounds currently known come from [15] where it was proven that a random instance of 3 SAT is almost surely satisfiable if r 3:003, and [21] 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, 9, 15] while 2 k is an easy upper bound and was improved by a constant factor in [21] by extending the techniques used for ....

[Article contains additional citation context not shown here]

L. M. Kirousis, E. Kranakis, and D. Krizanc, "Approximating the unsatisfiability threshold of random formulas," Proceedings of the Fourth Annual European Symposium on Algorithms, ESA '96, (1996) 27--38.

....of random words, when constructed under the two different letter selection schemes defined by the above processes. The motivation behind the study of this problem was the study of the problem of approximating the unsatisfiability threshold of random formulas consisting of clauses of k literals ([1]) where the usual process for constructing a random formula OE of m clauses is Process 1 defined above, where Omega is the set of all possible 2 k Gamma n k Delta possible clauses of k literals. In this research note we study properties that are reducible, in the sense that a word ....

L.M. Kirousis, E. Kranakis, and D. Krizanc, "Approximating the Unsatisfiability Threshold of Random Formulas," Proc. 4th European Symposium on Algorithms, pp 27--38, 1996. This article was processed using the L A T E X macro package with LLNCS style

....et al. 14] lowered 5.08 to 4.758 by proving that if a random 3 SAT formula is satisfiable then, with extremely high probability, it has an exponential number of satisfying truth assignments, hence providing some account for the wastefulness of the first moment method. Finally, Kirousis et al. [16] introduced a general refinement of the first moment method whereby one counts not all the satisfying truth assignments ( bad objects) but only those that satisfy a local maximality condition. In section 2 we will apply their technique to (2 p) SAT and elaborate on it. Applying this method, ....

....a general refinement of the first moment method whereby one counts not all the satisfying truth assignments ( bad objects) but only those that satisfy a local maximality condition. In section 2 we will apply their technique to (2 p) SAT and elaborate on it. Applying this method, they showed [16] that r 3 4:598, which is the best bound currently known. Independently, Dubois and Boufkhad [7] using a similar method obtained r 3 4:64. Unlike upper bounds, which are probabilistic counting arguments, all known lower bounds for r 3 are algorithmic. Chao and Franco [4] analyzed the Unit ....

[Article contains additional citation context not shown here]

L. M. Kirousis, E. Kranakis, and D. Krizanc, Approximating the unsatisfiability threshold of random formulas, Proceedings of the 4th European Symposium on Algorithms, (1992) 27--38.

....in an instance of k SAT. For k = 2 a sharp threshold was 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 behavior 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 ....

....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 unsatisfiable if r 4:598. For general k, the best known lower bound for r is Theta(2 k =k) 7, 9, 14] while 2 k is an easy upper bound and was improved by a constant factor in [19] by extending the techniques used for ....

[Article contains additional citation context not shown here]

L. M. Kirousis, E. Kranakis, and D. Krizanc, "Approximating the unsatisfiability threshold of random formulas," Proceedings of the Fourth Annual European Symposium on Algorithms, ESA '96, (1992) 27--38.

....a non computational proof is equal to 4.87. Notice that we can further improve our bound by selecting even smaller subsets of Sn , by, e.g. increasing the degree of locality in selecting the maxima that represent Sn . This is the object of current investigations to be presented elsewhere (see [6] for a preliminary report) Finally, our method readily generalizes to k SAT, for k 3. 2. The Results Recall, An is the class of all truth assignments, and Sn is the random class of truth assignments that satisfy a random formula OE. We now define a class even smaller than Sn . Definition ....

....of Sn which consists of the lexicographically local maxima at a fixed Hamming distance 2, or even beyond. In this case, certain dependency complications arise in the computation of the expected value of the subclass of Sn . We have done preliminary work for the case of Hamming distance 2 (see [6]) It is conceivable that one might obtain interesting results by letting the degree of locality in selecting the local maxima increase unboundedly. Finally, observe that the estimate can be probably improved further if instead of the Markov type inequality in Lemma 2.2, we use the harmonic mean ....

[Article contains additional citation context not shown here]

L. Kirousis, E. Kranakis, and D. Krizanc, "Approximating the Unsatisfiability Threshold of Random Formulas," Technical Report, Carleton University, TR-95-26, Ottawa, Canada, 1995.

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L. M. Kirousis, E. Kranakis, D. Krizanc, and Y. Stamatiou, Approximating the unsatisfiability threshold of random formulas, Random Structures & Algorithms, 12 (1998), 253--269.

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L.M. Kirousis, E. Kranakis, D. Krizanc, and Y. Stamatiou, Approximating the unsatisfiability threshold of random formulas, Random Structures Algorithms 12 (1998), no. 3, 253--269.

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L. M. Kirousis, E. Kranakis, D. Krizanc, and Y. Stamatiou. Approximating the unsatisfiability threshold of random formulas. Random Structures Algorithms, 12(3):253--269, 1998. MR1635256 (2000c:68069)

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L.M. Kirousis, E. Kranakis, D. Krizanc, and Y.C. Stamatiou, Approximating the unsatisfiability threshold of random formulas, Random Structures and Algorithms 17 (2000) 103-116. 7

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Lefteris M. Kirousis, Evangelos Kranakis, Danny Krizanc, and Yiannis Stamatiou, Approximating the unsatisfiability threshold of random formulas, Random Structures Algorithms 12 (1998), no. 3, 253--269.

No context found.

L. M. Kirousis, E. Kranakis, D. Krizanc, and Y. C. Stamatiou. Approximating the unsatisfiability threshold of random formulas. Random Structures and Algorithms, 12(3):253--269, 1998.

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Kirousis L M, Kranakis E, Krizanc D et al. Approximating the unsatisfiability threshold of random formulas. Random Structures and algorithms. Vol.12, 1998:253-269

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