Selecting Complementary Pairs of Literals

Kaporis, Alexis C.; Kirousis, Lefteris M.; Lalas, Efthimios G.

doi:10.1016/s1571-0653(04)00462-7

Cited by 54 publications

(26 citation statements)

References 18 publications

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“…In Table 1, we illustrate this gap for some small values of k. For k = 3, the upper bound on r * k comes from [14], while for k > 3 from [13,20]. The best algorithmic lower bound for k = 3 is from [16,19], while for k > 3 it is from [15]. Similar results (and gaps) exist for a number of other constraint satisfaction problems, such as random NAE k-SAT and hypergraph 2-coloring, regular random graph coloring, random Max k-SAT, and others (see [2,4,5,6]).…”

Section: Introductionmentioning

confidence: 99%

Random Formulas Have Frozen Variables

Achlioptas¹,

Ricci-Tersenghi²

2009

SIAM J. Comput.

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For a large number of random constraint satisfaction problems, such as random k-SAT and random graph and hypergraph coloring, we have very good estimates of the largest constraint density for which solutions exist. Yet, all known polynomial-time algorithms for these problems fail to find solutions even at much lower densities. To understand the origin of this gap one can study how the structure of the space of solutions evolves in such problems as constraints are added. In particular, it is known that much before solutions disappear, they organize into an exponential number of clusters, each of which is relatively small and far apart from all other clusters. Here we further prove that inside every cluster a majority of variables are frozen, i.e., take only one value. The existence of such frozen variables gives a satisfying intuitive explanation for the failure of the polynomial-time algorithms analyzed so far. At the same time, our results lend support to one of the two main hypotheses underlying Survey Propagation, a heuristic introduced by physicists in recent years that appears to perform extraordinarily well on random constraint satisfaction problems.

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Section: Introductionmentioning

confidence: 99%

Random Formulas Have Frozen Variables

Achlioptas¹,

Ricci-Tersenghi²

2009

SIAM J. Comput.

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“…To our knowledge the best result found by myopic algorithms is due to Achlioptas [1], but the greedy algorithm of Kaporis et al [5] pushes this bound a bit further. We emphasize that this contribution does not claim to come up with an even sharper proven lower bound.…”

Section: Introductionmentioning

confidence: 88%

Observed Lower Bounds for Random 3-SAT Phase Transition Density Using Linear Programming

Heule

Maaren

2005

Theory and Applications of Satisfiability Testing

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Abstract. We introduce two incomplete polynomial time algorithms to solve satisfiability problems which both use Linear Programming (LP) techniques. First, the FlipFlop LP attempts to simulate a Quadratic Program which would solve the CNF at hand. Second, the WeightedLinearAutarky LP is an extended variant of the LinearAutarky LP as defined by Kullmann [6] and iteratively updates its weights to find autarkies in a given formula. Besides solving satisfiability problems, this LP could also be used to study the existence of autark assignments in formulas. Results within the experimental domain (up to 1000 variables) show a considerably sharper lower bound for the uniform random 3-Sat phase transition density than the proved lower bound of the myopic algorithm (> 3.26) by Achlioptas [1] and even than that of the greedy algorithm (> 3.52) proposed by Kaporis [5].

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“…The existence of this phase transition at a critical density α c was proven by Friedgut in 1999 [Fri99]; 2 however only in the k = 2 case its value is known exactly (α c = 1 [CR92, Goe92, BBC + 01]). A long line of works for k = 3 have narrowed it down to 3.52 ≤ α c ≤ 4.49 [KKL03,HS03,DKMP08] (with evidence that α c ≈ 4.267 [MPZ02]), and in the large k limit it has been shown that 2 k ln 2…”

Section: Random Qsatmentioning

confidence: 99%

A quantum Lovász local lemma

Ambainis

Kempe

Sattath

2012

J. ACM

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The Lovász Local Lemma (LLL) is a powerful tool in probability theory to show the existence of combinatorial objects meeting a prescribed collection of "weakly dependent" criteria. We show that the LLL extends to a much more general geometric setting, where events are replaced with subspaces and probability is replaced with relative dimension, which allows to lower bound the dimension of the intersection of vector spaces under certain independence conditions.Our result immediately applies to the k-QSAT problem: For instance we show that any collection of rank 1 projectors with the property that each qubit appears in at most 2 k /(e · k) of them, has a joint satisfiable state.We then apply our results to the recently studied model of random k-QSAT. Recent works have shown that the satisfiable region extends up to a density of 1 in the large k limit, where the density is the ratio of projectors to qubits. Using a hybrid approach building on work by Laumann et al. [LLM + 09] we greatly extend the known satisfiable region for random k-QSAT to a density of Ω(2 k /k 2 ). Since our tool allows us to show the existence of joint satisfying states without the need to construct them, we are able to penetrate into regions where the satisfying states are conjectured to be entangled, avoiding the need to construct them, which has limited previous approaches to product states.

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Selecting Complementary Pairs of Literals

Cited by 54 publications

References 18 publications

Random Formulas Have Frozen Variables

Random Formulas Have Frozen Variables

Observed Lower Bounds for Random 3-SAT Phase Transition Density Using Linear Programming

A quantum Lovász local lemma

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