Nondeterministic Extensions of the Strong Exponential Time Hypothesis and Consequences for Non-reducibility

Carmosino, Marco; Gao, Jiawei; Impagliazzo, Russell; Mihajlin, Ivan; Paturi, Ramamohan; Schneider, Stefan

doi:10.1145/2840728.2840746

Cited by 92 publications

(85 citation statements)

References 32 publications

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“…The following theorem states that Sparse k-OV is complete for M C(∃ k ∀) (and its negation form M C(∀ k ∃)), and hard for M C(∀∃ k−1 ∀) (and its negation form M C(∃∀ k−1 ∃)) under fine-grained reductions. 14]). We will also show that the 2-Set Cover problem and the Sperner Family problem, both in M C(∃∃∀), are equivalent to Sparse OV under fine-grained reductions, and thus complete for first-order properties underfine-grained reductions.…”

Section: Resultsmentioning

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: Resultsmentioning

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

Self Cite

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“…Subsequent work of Carmoniso et al [22] gave evidence that basing the HS conjecture on SETH is unlikely. Using their framework one can show that SETH-hardness for Radius is similarly unlikely.…”

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confidence: 99%

Approximation and Fixed Parameter Subquadratic Algorithms for Radius and Diameter in Sparse Graphs

Abboud¹,

Williams²,

Wang³

2015

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

136

409

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The radius and diameter are fundamental graph parameters, with several natural definitions for directed graphs. Each definition is well-motivated in a variety of applications. All versions of diameter and radius can be solved via solving all-pairs shortest paths (APSP), followed by a fast postprocessing step. However, solving APSP on n-node graphs requires Ω(n 2 ) time even in sparse graphs. We study the question: when can diameter and radius in sparse graphs be solved in truly subquadratic time, and when is such an algorithm unlikely? Motivated by our conditional lower bounds on computing these measures exactly in truly subquadratic time, we search for approximation and fixed parameter subquadratic algorithms, and alternatively, for reasons why they do not exist.We find that:• Most versions of Diameter and Radius can be solved in truly subquadratic time with optimal approximation guarantees, under plausible assumptions. For example, there is a 2-approximation algorithm for directed Radius with one-way distances that runs inÕ(m √ n) time, while a (2 − δ)-approximation algorithm in O(n 2−ε ) time is considered unlikely.• On graphs with treewidth k, we can solve all versions in 2 O(k log k) n 1+o(1) time. We show that these algorithms are near optimal since even a (3/2 − δ)-approximation algorithm that runs in time 2 o(k) n 2−ε would refute plausible assumptions.Two conceptual contributions of this work that we hope will incite future work are: the introduction of a Fixed Parameter Tractability in P framework, and the statement of a differently-quantified variant of the Orthogonal Vectors Conjecture, which we call the Hitting Set Conjecture. * The full version of the paper can be found at: http://arxiv.org/abs/1506. 01799. A.A and V

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“…Furthermore, assuming the existence of computational PIR schemes [33], every set in BPP has a two-message doubly-efficient argument system [30,31,11]. 12 On the other hand, any set having a doublyefficient argument system is in BPP (since we can decide membership in such a set by emulating the interaction between the prescribed prover and verifier strategies). 13 The fact that argument systems are always asserted by relying on an intractability assumption is no coincidence, since these asserted systems do not satisfy the information theoretic soundness requirement.…”

Section: On Doubly-efficient Argument Systemsmentioning

confidence: 99%

On Doubly-Efficient Interactive Proof Systems

Goldreich

2018

FNT in Theoretical Computer Science

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Nondeterministic Extensions of the Strong Exponential Time Hypothesis and Consequences for Non-reducibility

Cited by 92 publications

References 32 publications

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

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

Approximation and Fixed Parameter Subquadratic Algorithms for Radius and Diameter in Sparse Graphs

On Doubly-Efficient Interactive Proof Systems

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