Subquadratic Algorithms for Succinct Stable Matching

Künnemann, Marvin; Moeller, Daniel; Paturi, Ramamohan; Schneider, Stefan

doi:10.1007/978-3-319-34171-2_21

Cited by 11 publications

(13 citation statements)

References 42 publications

(22 reference statements)

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“…Fine-Grained Complexity classifies the time complexity of fundamental problems under popular conjectures, the most productive of which has been the Strong Exponential Time Hypothesis 1 (SETH). The list of "SETH-Hard" problems is long, including central problems in pattern matching and bioinformatics [AWW14, BI15,BI16], graph algorithms [RV13,GIKW17], dynamic data structures [AV14], parameterized complexity and exact algorithms [PW10, LMS11, CDL + 16], computational geometry [Bri14], time-series analysis [ABV15,BK15], and even economics [MPS16] (a longer list can be found in [Wil15]).…”

Section: Introductionmentioning

confidence: 99%

Distributed PCP Theorems for Hardness of Approximation in P

Abboud

Rubinstein

Williams

2017

2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)

123

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We present a new distributed model of probabilistically checkable proofs (PCP). A satisfying assignment x ∈ {0, 1} n to a CNF formula ϕ is shared between two parties, where Alice knows x 1 , . . . , x n/2 , Bob knows x n/2+1 , . . . , x n , and both parties know ϕ. The goal is to have Alice and Bob jointly write a PCP that x satisfies ϕ, while exchanging little or no information. Unfortunately, this model as-is does not allow for nontrivial query complexity. Instead, we focus on a non-deterministic variant, where the players are helped by Merlin, a third party who knows all of x.Using our framework, we obtain, for the first time, PCP-like reductions from the Strong Exponential Time Hypothesis (SETH) to approximation problems in P. In particular, under SETH we show that there are no truly-subquadratic approximation algorithms for Bichromatic Maximum Inner Product over {0, 1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first two problems we obtain nearly-polynomial factors of 2 (log n) 1−o(1) ; only (1 + o(1))-factor lower bounds (under SETH) were known before.1 SETH is a pessimistic version of P = NP, stating that for every ε > 0 there is a k such that k-SAT cannot be solved in O((2 − ε) n ) time.2 See the end of this section for a discussion of "bichromatic" vs "monochromatic" closest pair problems. 3 In SODA'17, two entire sessions were dedicated to algorithms for similarity search.

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Section: Introductionmentioning

confidence: 99%

Distributed PCP Theorems for Hardness of Approximation in P

Abboud

Rubinstein

Williams

2017

2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)

123

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“…If a subproblem of an algorithmic problem can be modeled by a simple circuit, and that circuit can be transformed into a "nice" polynomial (or "nice" distribution of polynomials), then fast algebraic algorithms can be applied to evaluate or manipulate the polynomial quickly. This approach has led to advances on problems such as All-Pairs Shortest Paths [Wil14a], Orthogonal Vectors and Constraint Satisfaction [WY14, AWY15,Wil14d], All-Nearest Neighbor problems [AW15], and Stable Matching [MPS16].…”

Section: Introductionmentioning

confidence: 99%

Polynomial Representations of Threshold Functions and Algorithmic Applications

Alman

Chan

Williams

2016

2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)

102

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We design new polynomials for representing threshold functions in three different regimes: probabilistic polynomials of low degree, which need far less randomness than previous constructions, polynomial threshold functions (PTFs) with "nice" threshold behavior and degree almost as low as the probabilistic polynomials, and a new notion of probabilistic PTFs where we combine the above techniques to achieve even lower degree with similar "nice" threshold behavior. Utilizing these polynomial constructions, we design faster algorithms for a variety of problems: • Offline Hamming Nearest (and Furthest) Neighbors: Given n red and n blue points in ddimensional Hamming space for d = c log n, we can find an (exact) nearest (or furthest) blue neighbor for every red point in randomized time n 2−1/O( √ c log 2/3 c) or deterministic time n 2−1/O(c log 2 c) .These improve on a randomized n 2−1/O(c log 2 c) bound by Alman and Williams (FOCS'15), and also lead to faster MAX-SAT algorithms for sparse CNFs.• Offline Approximate Nearest (and Furthest) Neighbors: Given n red and n blue points in ddimensional ℓ 1 or Euclidean space, we can find a (1 + ε)-approximate nearest (or furthest) blue neighbor for each red point in randomized time near dn + n 2−Ω(ε 1/3 / log(1/ε)) . This improves on an algorithm by Valiant (FOCS'12) with randomized time near dn + n 2−Ω(

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“…Request permissions from Permissions@acm.org. ITCS'16, January [14][15][16]2016 3-sum conjecture [13] from computational geometry or the Strong Exponential Time Hypothesis for the complexity of SAT [16,15], it follows that the known algorithms for many basic problems within P, including Fréchet distance [9], edit distance [5], string matching [1], k-dominating set [23], orthogonal vectors [26], stable marriage for low dimensional ordering functions [21], and many others [8], are essentially optimal.…”

Section: Introductionmentioning

confidence: 99%

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

Carmosino

Gao

Impagliazzo

et al. 2016

Proceedings of the 2016 ACM Conference on Innovations in Theoretical Computer Science

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We introduce the Nondeterministic Strong Exponential Time Hypothesis (NSETH) as a natural extension of the Strong Exponential Time Hypothesis (SETH). We show that both refuting and proving NSETH would have interesting consequences. In particular we show that disproving NSETH would give new nontrivial circuit lower bounds. On the other hand, NSETH implies non-reducibility results, i.e. the absence of (deterministic) fine-grained reductions from SAT to a number of problems. As a consequence we conclude that unless this hypothesis fails, problems such as 3-sum, APSP and model checking of a large class of first-order graph properties cannot be shown to be SETH-hard using deterministic or zero-error probabilistic reductions.

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Subquadratic Algorithms for Succinct Stable Matching

Cited by 11 publications

References 42 publications

Distributed PCP Theorems for Hardness of Approximation in P

Distributed PCP Theorems for Hardness of Approximation in P

Polynomial Representations of Threshold Functions and Algorithmic Applications

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

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