Results 1  10
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22
Optimal myopic algorithms for random 3SAT
 In Proceedings of the 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... Let F 3 (n; m) be a random 3SAT formula formed by selecting uniformly, independently, and with replacement, m clauses among all 8 \Gamma n 3 \Delta possible 3clauses over n variables. It has been conjectured that there exists a constant r 3 such that for any ffl ? 0, F 3 (n; (r 3 \Gamma ffl)n ..."
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Cited by 67 (8 self)
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Let F 3 (n; m) be a random 3SAT formula formed by selecting uniformly, independently, and with replacement, m clauses among all 8 \Gamma n 3 \Delta possible 3clauses over n variables. It has been conjectured that there exists a constant r 3 such that for any ffl ? 0, F 3 (n; (r 3 \Gamma ffl)n) is almost surely satisfiable, but F 3 (n; (r 3 + ffl)n) is almost surely unsatisfiable. The best lower bounds for the potential value of r 3 have come from analyzing rather simple extensions of unitclause propagation. Recently, it was shown [2] that all these extensions can be cast in a common framework and analyzed in a uniform manner by employing differential equations. Here, we determine optimal algorithms expressible in that framework, establishing r 3 ? 3:26. We extend the analysis via differential equations, and make extensive use of a new optimization problem we call "maxdensity multiplechoice knapsack". The structure of optimal knapsack solutions elegantly characterizes the choi...
Lower bounds for random 3SAT via differential equations
 THEORETICAL COMPUTER SCIENCE
, 2001
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Random maps, coalescing saddles, singularity analysis, and Airy phenomena
 Random Structures & Algorithms
, 2001
"... A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponentialcubic type, corresponding to distributions that involve the ..."
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Cited by 46 (6 self)
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A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type, that is, Gaussian. We exhibit a class of "universal" phenomena that are of the exponentialcubic type, corresponding to distributions that involve the Airy function. In this paper, such Airy phenomena are related to the coalescence of saddle points and the confluence of singularities of generating functions. For about a dozen types of random planar maps, a common Airy distribution (equivalently, a stable law of exponent 3/2) describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and fine optimization of random generation algorithms for multiply connected planar graphs. Based on an extension of the singularity analysis framework suggested by the Airy case, the paper also presents a general classification of compositional schemas in analytic combinatorics.
Setting 2 variables at a time yields a new lower bound for random 3SAT (Extended Abstract)
 STOC
, 2000
"... Let X be a set of n Boolean variables and denote by C(X) the set of all 3clauses over X, i.e. the set of all 8(3) possible disjunctions of three distinct, noncomplementary literais from variables in X. Let F(n, m) be a random 3SAT formula formed by selecting, with replacement, m clauses uniformly ..."
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Cited by 35 (4 self)
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Let X be a set of n Boolean variables and denote by C(X) the set of all 3clauses over X, i.e. the set of all 8(3) possible disjunctions of three distinct, noncomplementary literais from variables in X. Let F(n, m) be a random 3SAT formula formed by selecting, with replacement, m clauses uniformly at random from C(X) and taking their conjunction. The satisfiability threshold conjecture asserts that there exists a constant ra such that as n+ c¢, F(n, rn) is satisfiable with probability that tends to 1 if r < ra, but unsatisfiable with probability that tends to 1 if r:> r3. Experimental evidence suggests rz ~ 4.2. We prove rz> 3.145 improving over the previous best lower bound r3> 3.003 due to Frieze and Suen. For this, we introduce a satisfiability heuristic that works iteratively, permanently setting the value of a pair of variables in each round. The framework we develop for the analysis of our heuristic allows us to also derive most previous lower bounds for random 3SAT in a uniform manner and with little effort.
Frozen Development in Graph Coloring
 THEORETICAL COMPUTER SCIENCE
, 2000
"... We define the `frozen development' of coloring random graphs. We identify two nodes in a graph as frozen if they are the same color in all legal colorings. This is analogous to studies of the development of a backbone or spine in SAT (the Satisability problem). We first describe in detail the algori ..."
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Cited by 34 (5 self)
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We define the `frozen development' of coloring random graphs. We identify two nodes in a graph as frozen if they are the same color in all legal colorings. This is analogous to studies of the development of a backbone or spine in SAT (the Satisability problem). We first describe in detail the algorithmic techniques used to study frozen development. We present strong empirical evidence that freezing in 3coloring is sudden. A single edge typically causes the size of the graph to collapse in size by 28%. We also use the frozen development to calculate unbiased estimates of probability of colorability in random graphs, even where this probability is as low as 10^300. We investigate the links between frozen development and the solution cost of graph coloring. In SAT, a discontinuity in the order parameter has been correlated with the hardness of SAT instances, and our data for coloring is suggestive of an asymptotic discontinuity. The uncolorability threshold is known to give rise to har...
Results Related to Threshold Phenomena Research in Satisfiability: Lower Bounds
, 2000
"... We present a history of results related to the threshold phenomena which arise in the study of random Conjunctive Normal Form (CNF) formulas. In a companion paper [1] in this volume the major ideas used to achieve many of the lower bounds results on the location of the threshold are described in an ..."
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Cited by 20 (1 self)
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We present a history of results related to the threshold phenomena which arise in the study of random Conjunctive Normal Form (CNF) formulas. In a companion paper [1] in this volume the major ideas used to achieve many of the lower bounds results on the location of the threshold are described in an informal, intuitive manner.
Rigorous results for random (2 + p)SAT
, 2001
"... In recent years there has been significant interest in the study of random kSAT formulae. For a given set of n Boolean variables, let Bk denote the set of all possible disjunctions of k distinct, noncomplementary literals from its variables (kclauses). A random kSAT formula Fk(n; m) is formed by ..."
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Cited by 17 (3 self)
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In recent years there has been significant interest in the study of random kSAT formulae. For a given set of n Boolean variables, let Bk denote the set of all possible disjunctions of k distinct, noncomplementary literals from its variables (kclauses). A random kSAT formula Fk(n; m) is formed by selecting uniformly and independently m clauses from Bk and taking their conjunction. Motivated by insights from statistical mechanics that suggest a possible relationship between the “order” of phase transitions and computational complexity, Monasson and Zecchina (Phys. Rev. E 56(2) (1997) 1357) proposed the random (2+p)SAT model: for a given p ∈ [0; 1], a random (2 + p)SAT formula, F2+p(n; m), has m randomly chosen clauses over n variables, where pm clauses are chosen from B3 and (1 − p)m from B2. Using the heuristic “replica method” of statistical mechanics, Monasson and Zecchina gave a number of nonrigorous predictions on the behavior of random (2 + p)SAT formulae. In this paper we give the first rigorous results for random (2 + p)SAT, including the following surprising fact: for p 6 2=5, with probability 1 − o(1), a random (2 + p)SAT formula is satis able i its 2SAT subformula is satisfiable. That is, for p 6 2=5, random (2 + p)SAT behaves like random 2SAT.
Phase transition and finitesize scaling for the integer partitioning problem
, 2001
"... Dedicated to D. E. Knuth on the occasion of his 64th birthday. Abstract. We consider the problem of partitioning n randomly chosen integers between 1 and 2 m into two subsets such that the discrepancy, the absolute value of the difference of their sums, is minimized. A partition is called perfect if ..."
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Cited by 17 (2 self)
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Dedicated to D. E. Knuth on the occasion of his 64th birthday. Abstract. We consider the problem of partitioning n randomly chosen integers between 1 and 2 m into two subsets such that the discrepancy, the absolute value of the difference of their sums, is minimized. A partition is called perfect if the optimum discrepancy is 0 when the sum of all n integers in the original set is even, or 1 when the sum is odd. Parameterizing the random problem in terms of κ = m/n, we prove that the problem has a phase transition at κ = 1, in the sense that for κ < 1, there are many perfect partitions with probability tending to 1 as n → ∞, while for κ> 1, there are no perfect partitions with probability tending to 1. Moreover, we show that this transition is firstorder in the sense the derivative of the socalled entropy is discontinuous at κ = 1. We also determine the finitesize scaling window about the transition point: κn = 1 − (2n) −1 log 2 n + λn/n, by showing that the probability of a perfect partition tends to 1, 0, or some explicitly computable p(λ) ∈ (0, 1), depending on whether λn tends to −∞, ∞, or λ ∈ (−∞, ∞), respectively. For λn → − ∞ fast enough, we show that the number of perfect partitions is Gaussian in the limit. For λn → ∞, we prove that with high probability the optimum partition is unique, and that the optimum discrepancy is Θ(2 λn). Within the window, i.e., if λn  is bounded, we prove that the optimum discrepancy is bounded. Both for λn → ∞ and within the window, we find the limiting distribution of the (scaled) discrepancy. Finally, both for the integer partitioning problem and for the continuous partitioning problem, we find the joint distribution of the k smallest discrepancies above the scaling window.
Ten challenges redux: Recent progress in propositional reasoning and search
 In Proceedings of CP ’03
, 2003
"... Abstract. In 1997 we presented ten challenges for research on satisfiability testing [1]. In this paper we review recent progress towards each of these challenges, including our own work on the power of clause learning and randomized restart policies. 1 ..."
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Cited by 16 (1 self)
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Abstract. In 1997 we presented ten challenges for research on satisfiability testing [1]. In this paper we review recent progress towards each of these challenges, including our own work on the power of clause learning and randomized restart policies. 1
Planar Maps and Airy Phenomena
, 2000
"... A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type (e x 2 ), that is, Gaussian. We exhibit here a new class of \universal" phenomena that are of the exponentialcubic type (e ix 3 ), corresponding to ..."
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Cited by 13 (4 self)
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A considerable number of asymptotic distributions arising in random combinatorics and analysis of algorithms are of the exponentialquadratic type (e x 2 ), that is, Gaussian. We exhibit here a new class of \universal" phenomena that are of the exponentialcubic type (e ix 3 ), corresponding to nonstandard distributions that involve the Airy function. Such Airy phenomena are expected to be found in a number of applications, when conuences of critical points and singularities occur. About a dozen classes of planar maps are treated in this way, leading to the occurrence of a common Airy distribution that describes the sizes of cores and of largest (multi)connected components. Consequences include the analysis and ne optimization of random generation algorithms for multiply connected planar graphs.