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Random kSAT: two moments suffice to cross a sharp threshold
 CoRR
, 2006
"... Abstract. Many NPcomplete constraint satisfaction problems appear to undergo a “phase transition” from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by showing that abo ..."
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Cited by 41 (4 self)
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prove that the threshold for both random hypergraph 2colorability (Property B) and random NotAllEqual kSAT is 2 k−1 ln 2 − O(1). As a corollary, we establish that the threshold for random kSAT is of order Θ(2 k), resolving a longstanding open problem.
Random kSAT: Two Moments Suffice to Cross a Sharp Threshold
"... Abstract Many NPcomplete constraint satisfaction problems appear to undergo a "phase transition"from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by ..."
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by showingthat above a certain density the first moment (expectation) of the number of solutions tends to zero. We show that in the case of certain symmetric constraints, considering the secondmoment of the number of solutions yields nearly matching lower bounds for the location of the threshold. Specifically
Sharp Thresholds of Graph properties, and the ksat Problem
 J. Amer. Math. Soc
, 1998
"... Given a monotone graph property P , consider p (P ), the probability that a random graph with edge probability p will have P . The function d p (P )=dp is the key to understanding the threshold behavior of the property P . We show that if d p (P )=dp is small (corresponding to a nonsharp thres ..."
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Cited by 214 (7 self)
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Given a monotone graph property P , consider p (P ), the probability that a random graph with edge probability p will have P . The function d p (P )=dp is the key to understanding the threshold behavior of the property P . We show that if d p (P )=dp is small (corresponding to a nonsharp
Thresholds for Random Geometric kSAT
, 2013
"... We study two geometric models of random ksatisfiability which combine random kSAT with the Random Geometric Graph: boolean literals are placed uniformly at random or according to a Poisson process in a cube, and for each set of k literals contained in a ball of a given radius, a clause is formed. ..."
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We study two geometric models of random ksatisfiability which combine random kSAT with the Random Geometric Graph: boolean literals are placed uniformly at random or according to a Poisson process in a cube, and for each set of k literals contained in a ball of a given radius, a clause is formed
A Threshold of ln n for Approximating Set Cover
 JOURNAL OF THE ACM
, 1998
"... Given a collection F of subsets of S = f1; : : : ; ng, set cover is the problem of selecting as few as possible subsets from F such that their union covers S, and max kcover is the problem of selecting k subsets from F such that their union has maximum cardinality. Both these problems are NPhar ..."
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Cited by 778 (5 self)
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o(1)) ln n), and previous results of Lund and Yannakakis, that showed hardness of approximation within a ratio of (log 2 n)=2 ' 0:72 lnn. For max kcover we show an approximation threshold of (1 \Gamma 1=e) (up to low order terms), under the assumption that P != NP .
An iterative thresholding algorithm for linear inverse problems with a sparsity constraint
, 2008
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Thresholds in Random Graphs and ksat
, 2006
"... In their seminal work [6],[7], Erdos and Renyi invented the notion of random graphs and made the fundamental observation that when the parameter controlling the edge probability varies, the random system undergoes a dramatic and swift qualitative change. “Much like water that freezes abruptly as it ..."
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In their seminal work [6],[7], Erdos and Renyi invented the notion of random graphs and made the fundamental observation that when the parameter controlling the edge probability varies, the random system undergoes a dramatic and swift qualitative change. “Much like water that freezes abruptly
Where the REALLY Hard Problems Are
 IN J. MYLOPOULOS AND R. REITER (EDS.), PROCEEDINGS OF 12TH INTERNATIONAL JOINT CONFERENCE ON AI (IJCAI91),VOLUME 1
, 1991
"... It is well known that for many NPcomplete problems, such as KSat, etc., typical cases are easy to solve; so that computationally hard cases must be rare (assuming P != NP). This paper shows that NPcomplete problems can be summarized by at least one "order parameter", and that the hard p ..."
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Cited by 681 (1 self)
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problems occur at a critical value of such a parameter. This critical value separates two regions of characteristically different properties. For example, for Kcolorability, the critical value separates overconstrained from underconstrained random graphs, and it marks the value at which the probability
Results 1  10
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305,013