Pruning processes and a new characterization of convex geometries

Ardila, Federico; Maneva, Elitza

doi:10.1016/j.disc.2008.08.020

Cited by 7 publications

(10 citation statements)

References 26 publications

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“…The corresponding generalization of Theorem 4 is proved in [2]. It is quite possible that the value of E[Z ] in this case is significantly smaller.…”

Section: Discussionmentioning

confidence: 92%

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On the satisfiability threshold and clustering of solutions of random 3-SAT formulas

Maneva

Sinclair

2008

Theoretical Computer Science

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a b s t r a c tWe study the structure of satisfying assignments of a random 3-Sat formula. In particular, we show that a random formula of density α ≥ 4.453 almost surely has no non-trivial ''core'' assignments. Core assignments are certain partial assignments that can be extended to satisfying assignments, and have been studied recently in connection with the Survey Propagation heuristic for random Sat. Their existence implies the presence of clusters of solutions, and they have been shown to exist with high probability below the satisfiability threshold for k-Sat with k ≥ 9 [D. Achlioptas, F. Ricci-Tersenghi, On the solution-space geometry of random constraint satisfaction problems, in: Proc. 38th ACM Symp. Theory of Computing, STOC, 2006, pp. 130-139]. Our result implies that either this does not hold for 3-Sat, or the threshold density for satisfiability in 3-Sat lies below 4.453. The main technical tool that we use is a novel simple application of the first moment method.

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“…The corresponding generalization of Theorem 4 is proved in [2]. It is quite possible that the value of E[Z ] in this case is significantly smaller.…”

Section: Discussionmentioning

confidence: 92%

“…(A more general statement and a connection to a combinatorial object known as ''convex geometry'' was developed in [2].) Specifically, the total weight of partial assignments consistent with a given satisfying assignment is exactly 1.…”

Section: Definitionmentioning

confidence: 99%

On the satisfiability threshold and clustering of solutions of random 3-SAT formulas

Maneva

Sinclair

2008

Theoretical Computer Science

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“…Then we give sufficient conditions on a weighting system on solutions only, such that a transfer between this weighting system and the previous one may be possible. Doing this we establish a general framework for putting weights onto solutions, and use it to derive two particular weighting systems: the first one addresses general CSPs and the second one is built to improve on the weighting system introduced by Maneva et al [15], Maneva and Sinclair [1], Ardila and Maneva [17]. The purpose of such a transfer is to estimate the global weight in the weighting system on solutions by means of the global weight of the weighting system on all valuations (which is easier to compute).…”

Section: Weighting Of Solutionsmentioning

confidence: 99%

Estimating satisfiability

Boufkhad

Hugel

2012

Discrete Applied Mathematics

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International audienceThe problem of estimating the proportion of satisfiable instances of a given CSP (constraint satisfaction problem) can be tackled through weighting. It consists in putting onto each solution a non-negative real value based on its neighborhood in a way that the total weight is at least 1 for each satisfiable instance. We define in this paper a general weighting scheme for the estimation of satisfiability of general CSPs. First we give some sufficient conditions for a weighting system to be correct. Then we show that this scheme allows for an improvement on the upper bound on the existence of non-trivial cores in 3-SAT obtained by Maneva and Sinclair (2008) [17] to 4.419. Another more common way of estimating satisfiability is ordering. This consists in putting a total order on the domain, which induces an orientation between neighboring solutions in a way that prevents circuits from appearing, and then counting only minimal elements. We compare ordering and weighting under various conditions

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“…It is well known [16] that GR(ℋ is de�ned uniquely, that is, if a deletion rule is applicable to a vertex or edge, applying a rule elsewhere does not make the �rst inapplicable; so, the reduction procedure proceeds to do everything that is ever possible, independent of the order of reductions chosen. In other words, the Graham reduction process enjoys the socalled Church-Rosser property [16], which makes it a pruning process according to the terminology used in [17]. It is also well known [16] that if ℋ is an acyclic, simple and connected hypergraph, then (i) GR(ℋ is an acyclic, simple, and connected hypergraph;…”

Section: Preliminariesmentioning

confidence: 99%

“…Let and be two convex sets with . A descending path [17] from to in the lattice of is an ordering ( 1 … of the elements of such that, for each , 1 ≤ ≤ , the set 1 2 … is convex.…”

Section: Lemma 14 (Eg See [17]) a Convexity Space Is A Convex Geomentioning

confidence: 99%

Decomposable Convexities in Graphs and Hypergraphs

Malvestuto

2013

ISRN Combinatorics

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Given a connected hypergraph ℋwith vertex set V, a convexity space on ℋ is a subset of the powerset of V that contains ∅, V, and the singletons; furthermore, is closed under intersection and every set in is connected in ℋ. e members of are called convex sets. e convex hull of a subset X of V is the smallest convex set containing X. By a cluster of ℋ we mean any nonempty subset of V in which every two vertices are separated by no convex set. We say that a convexity space on ℋ is decomposable if it satis�es the following three axioms: (i) the maximal clusters of ℋ form an acyclic hypergraph, (ii) every maximal cluster of ℋ is a convex set, and (iii) for every nonempty vertex set X, a vertex does not belong to the convex hull of X if and only if it is separated from X by a convex cluster. We prove that a decomposable convexity space on ℋ is fully speci�ed by the maximal clusters of ℋ in that (1) there is a closed formula which expresses the convex hull of a set in terms of certain convex clusters of ℋ and (2) is a convex geometry if and only if the subspaces of induced by maximal clusters of ℋ are all convex geometries. Finally, we prove the decomposability of some known convexities in graphs and hypergraphs taken from the literature (such as "monophonic" and "canonical" convexities in hypergraphs and "all-paths" convexity in graphs).

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Pruning processes and a new characterization of convex geometries

Cited by 7 publications

References 26 publications

On the satisfiability threshold and clustering of solutions of random 3-SAT formulas

On the satisfiability threshold and clustering of solutions of random 3-SAT formulas

Estimating satisfiability

Decomposable Convexities in Graphs and Hypergraphs

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