On the possibility of faster SAT algorithms

Pătraşcu, Mihai; Williams, Ryan

doi:10.1137/1.9781611973075.86

Cited by 142 publications

(146 citation statements)

References 20 publications

(27 reference statements)

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“…Pǎtraşcu and Williams [29] give several tight lower bounds (matching the known upper bounds) for problems such as k-dominating set (for any constant k ≥ 3), 2SAT with two extra unrestricted length clauses, and HornSAT with k extra unrestricted length clauses.…”

Section: Hardness Under Sethmentioning

confidence: 95%

“…The best algorithm for k-dominating set for k ≥ 7 runs in n k+o(1) time, and obtaining O(n k−ε ) time would break SETH [29]. The k-dominating set problem is well-studied in the area of fixed-parameter complexity.…”

Section: Hardness Under Sethmentioning

confidence: 99%

“…To prove the statement for CNF-SAT, one can apply the reduction from [29], and one would obtain that a O(n 2−ε ) time algorithm for diameter approximation would imply an O * (2 n(1−ε 2 /4) ) time algorithm for CNF-SAT. Here we show a direct reduction from CNF-SAT to diameter that gives the runtime given in the theorem.…”

Section: Hardness Under Sethmentioning

confidence: 99%

See 2 more Smart Citations

Fast approximation algorithms for the diameter and radius of sparse graphs

Roditty

Williams²

2013

Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing

193

290

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The diameter and the radius of a graph are fundamental topological parameters that have many important practical applications in real world networks. The fastest combinatorial algorithm for both parameters works by solving the all-pairs shortest paths problem (APSP) and has a running time of O(mn) in m-edge, n-node graphs. In a seminal paper, Aingworth, Chekuri, Indyk and Motwani [SODA'96 and SICOMP'99] presented an algorithm that computes in O(m √ n + n 2 ) time an estimateD for the diameter D, such that ⌊2/3D⌋ ≤D ≤ D. Their paper spawned a long line of research on approximate APSP. For the specific problem of diameter approximation, however, no improvement has been achieved in over 15 years. Our paper presents the first improvement over the diameter approximation algorithm of Aingworth et al. , producing an algorithm with the same estimate but with an expected running time of O(m √ n). We thus show that for all sparse enough graphs, the diameter can be 3/2-approximated in o(n 2 ) time. Our algorithm is obtained using a surprisingly simple method of neighborhood depth estimation that is strong enough to also approximate, in the same running time, the radius and more generally, all of the eccentricities, i.e. for every node the distance to its furthest node.We also provide strong evidence that our diameter approximation result may be hard to improve. We show that if for some constant ε > 0 there is an O(m 2−ε ) time (3/2 − ε)-approximation algorithm for the diameter of undirected unweighted graphs, then there is an O * ((2 − δ) n ) time algorithm for CNF-SAT on n variables for constant δ > 0, and the strong exponential time hypothesis of [Impagliazzo, Paturi, Zane JCSS'01] is false.* Work supported by the Israel Science Foundation (grant no. 822/10). † Partially supported by NSF Grants CCF-0830797 and CCF-1118083 at UC Berkeley, and by NSF Grants IIS-0963478 and IIS-0904325, and an AFOSR MURI Grant, at Stanford University.Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Motivated by this negative result, we give several improved diameter approximation algorithms for special cases. We show for instance that for unweighted graphs of constant diameter D not divisible by 3, there is an O(m 2−ε ) time algorithm that gives a (3/2 − ε) approximation for constant ε > 0. This is interesting since the diameter approximation problem is hardest to solve for small D.

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Section: Hardness Under Sethmentioning

confidence: 95%

Section: Hardness Under Sethmentioning

confidence: 99%

See 1 more Smart Citation

Fast approximation algorithms for the diameter and radius of sparse graphs

Roditty

Williams²

2013

Proceedings of the Forty-Fifth Annual ACM Symposium on Theory of Computing

193

290

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“…We make use of a recent result of Patrascu and Williams [30] (and engineer low-dimensional gadgets inspired by the gadgets of Vavasis [36]) to show that under the Exponential Time Hypothesis [20], there is no exact algorithm for NMF that runs in time (nm) o(r) . This intractability result holds also for the SF problem.…”

Section: Theorem 13 There Is An Algorithm For the Exact Nmf Problemmentioning

confidence: 99%

Computing a Nonnegative Matrix Factorization---Provably

Arora¹,

Ge²,

Kannan³

et al. 2016

SIAM J. Comput.

157

284

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In the Nonnegative Matrix Factorization (NMF) problem we are given an n × m nonnegative matrix M and an integer r > 0. Our goal is to express M as AW where A and W are nonnegative matrices of size n × r and r × m respectively. In some applications, it makes sense to ask instead for the product AW to approximate M -i.e. (approximately) minimize M − AW F where F denotes the Frobenius norm; we refer to this as Approximate NMF.This problem has a rich history spanning quantum mechanics, probability theory, data analysis, polyhedral combinatorics, communication complexity, demography, chemometrics, etc. In the past decade NMF has become enormously popular in machine learning, where A and W are computed using a variety of local search heuristics. Vavasis recently proved that this problem is NP-complete. (Without the restriction that A and W be nonnegative, both the exact and approximate problems can be solved optimally via the singular value decomposition.)We initiate a study of when this problem is solvable in polynomial time. Our results are the following:1. We give a polynomial-time algorithm for exact and approximate NMF for every constant r. Indeed NMF is most interesting in applications precisely when r is small.2. We complement this with a hardness result, that if exact N M F can be solved in time (nm) o(r) , 3-SAT has a sub-exponential time algorithm. This rules out substantial improvements to the above algorithm.3. We give an algorithm that runs in time polynomial in n, m and r under the separablity condition identified by Donoho and Stodden in 2003. The algorithm may be practical since it is simple and noise tolerant (under benign assumptions). Separability is believed to hold in many practical settings.To the best of our knowledge, this last result is the first example of a polynomial-time algorithm that provably works under a non-trivial condition on the input and we believe that this will be an interesting and important direction for future work.

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“…While ETH is now a widely believed assumption, and has been used as a starting point to prove running time lower bounds for numerous problems [5,4,11,18,17], SETH remains largely untouched (with one exception [21]). The reason for this is two-fold.…”

Section: Complexity Assumptionmentioning

confidence: 99%