Simulating branching programs with edit distance and friends: or: a polylog shaved is a lower bound made

Abboud, Amir; Hansen, Thomas Dueholm; Williams, Virginia Vassilevska; Williams, Ryan

doi:10.1145/2897518.2897653

Cited by 84 publications

(131 citation statements)

References 52 publications

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“…Thus far, fine-grained complexity has remained focused on specific problems, rather than organizing problems into classes as in traditional complexity. As the field has grown, many fundamental relationships between problems have been discovered, making the graph of known results a somewhat tangled web of reductions ( [36,5,9,11,12,1,13,2,27]). …”

Section: Introductionmentioning

confidence: 99%

Completeness for First-Order Properties on Sparse Structures with Algorithmic Applications

Gao¹,

Impagliazzo²,

Kolokolova³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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Properties definable in first-order logic are algorithmically interesting for both theoretical and pragmatic reasons. Many of the most studied algorithmic problems, such as Hitting Set and Orthogonal Vectors, are first-order, and the first-order properties naturally arise as relational database queries. A relatively straightforward algorithm for evaluating a property with k + 1 quantifiers takes time O(m k ) and, assuming the Strong Exponential Time Hypothesis (SETH), some such properties require O(m k− ) time for any > 0. (Here, m represents the size of the input structure, i.e. the number of tuples in all relations.)We give algorithms for every first-order property that improves this upper bound to m k /2 Θ( √ log n) , i.e., an improvement by a factor more than any poly-log, but less than the polynomial required to refute SETH. Moreover, we show that further improvement is equivalent to improving algorithms for sparse instances of the well-studied Orthogonal Vectors problem. Surprisingly, both results are obtained by showing completeness of the Sparse Orthogonal Vectors problem for the class of first-order properties under fine-grained reductions. To obtain improved algorithms, we apply the fast Orthogonal Vectors algorithm of [3,16].While fine-grained reductions (reductions that closely preserve the conjectured complexities of problems) have been used to relate the hardness of disparate specific problems both within P and beyond, this is the first such completeness result for a standard complexity class.

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Section: Introductionmentioning

confidence: 99%

Completeness for First-Order Properties on Sparse Structures with Algorithmic Applications

Gao¹,

Impagliazzo²,

Kolokolova³

et al. 2017

Proceedings of the Twenty-Eighth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…This notation is borrowed from[AHWW16], which studied the Satisfying Pair problem for Branching Programs.…”

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confidence: 99%

An Equivalence Class for Orthogonal Vectors

Chen

Williams

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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The Orthogonal Vectors problem (OV) asks: given n vectors in {0, 1} O(log n) , are two of them orthogonal? OV is easily solved in O(n 2 log n) time, and it is a central problem in fine-grained complexity: dozens of conditional lower bounds are based on the popular hypothesis that OV cannot be solved in (say) n 1.99 time. However, unlike the APSP problem, few other problems are known to be non-trivially equivalent to OV.We show OV is truly-subquadratic equivalent to several fundamental problems, all of which (a priori) look harder than OV. A partial list is given below:

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“…For example, in [AVY15], the authors prove hardness for several problems, basing on at least one of the SETH, the APSP conjecture, or the 3-SUM Conjecture being true. In [AHVW16], Abboud et al introduce a hierarchy of C-SETH assumptions: the C-SETH asserts that there is no 2 (1−ε)n time satisfiability algorithm for circuits from C.10 They show that the quadratic time hardness of Edit-Distance, LCS and other related sequence alignment problems can be based on the much weaker and much more plausible assumption NC-SETH. However, this has not been shown for approximation version of fine-grained problems.…”

Section: Weaker Complexity Assumptions For Approximation Hardnessmentioning

confidence: 99%

Fine-grained Complexity Meets IP = PSPACE

Chen¹,

Goldwasser²,

Lyu³

et al. 2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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In this paper we study the fine-grained complexity of finding exact and approximate solutions to problems in P. Our main contribution is showing reductions from an exact to an approximate solution for a host of such problems.As one (notable) example, we show that the Closest-LCS-Pair problem (Given two sets of strings A and B, compute exactly the maximum LCS(a, b) with (a, b) ∈ A × B) is equivalent to its approximation version (under near-linear time reductions, and with a constant approximation factor). More generally, we identify a class of problems, which we call BP-Pair-Class, comprising both exact and approximate solutions, and show that they are all equivalent under near-linear time reductions.Exploring this class and its properties, we also show:• Under the NC-SETH assumption (a significantly more relaxed assumption than SETH), solving any of the problems in this class requires essentially quadratic time.• Modest improvements on the running time of known algorithms (shaving log factors) would imply that NEXP is not in non-uniform NC 1 .• Finally, we leverage our techniques to show new barriers for deterministic approximation algorithms for LCS.A very important consequence of our results is that they continue to hold in the data structure setting. In particular, it shows that a data structure for approximate Nearest Neighbor Search for LCS (NNS LCS ) implies a data structure for exact NNS LCS and a data structure for answering regular expression queries with essentially the same complexity.At the heart of these new results is a deep connection between interactive proof systems for boundedspace computations and the fine-grained complexity of exact and approximate solutions to problems in P. In particular, our results build on the proof techniques from the classical IP = PSPACE result. * MIT, lijieche@mit.edu.The study of the fine-grained hardness of problems in P is one of the most interesting developments of the last few years in complexity theory. The study was initially aimed at the complexity of exact versions of important problem in P, such as Longest Common Subsequence (LCS), Edit Distance, All Pair Shortest Path (APSP), and 3-SUM. This was the natural starting point. There are several main thrusts of the study: establishing equivalence classes of problems that are "equivalent" to each other in the sense that a substantial improvement in one would imply a similar improvement in the other; showing fine-grained hardness under complexity assumptions, most notably the SETH1; and showing implications of even slight algorithmic improvements, such as "shaving-logs" off algorithms for P time problems, to circuit lower bounds.However, for many of these problems, approximate solutions are of interest as well, as they originate in natural problems which arise in pattern matching and bioinformatics [AVW14, BI15, BI16, BGL17, BK18], dynamic data structures [Pat10, AV14, AV14, HKNS15, KPP16, AD16, HLNV17, GKLP17], graph algorithms [RW13, GIKW17, AVY15, KT17], computational geometry [Bri14, Wil18a, DKL18, Che18] and ...

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Simulating branching programs with edit distance and friends: or: a polylog shaved is a lower bound made

Cited by 84 publications

References 52 publications

Completeness for First-Order Properties on Sparse Structures with Algorithmic Applications

Completeness for First-Order Properties on Sparse Structures with Algorithmic Applications

An Equivalence Class for Orthogonal Vectors

Fine-grained Complexity Meets IP = PSPACE

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