An Equivalence Class for Orthogonal Vectors

Chen, Lijie; Williams, Ryan

doi:10.1137/1.9781611975482.2

Cited by 18 publications

(15 citation statements)

References 51 publications

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“…Studying the fine-grained approximability of polynomial-time optimization problems (hardness of approximation in P), is a recent and influential trend: After a breakthrough result by Abboud, Rubinstein, and Williams [3] establishing the Distributed PCP in P framework, a number of works gave strong conditional lower bounds, including results for nearest neighbor search [28] or a tight characterization of the approximability of maximum inner product [13,15]. Further results include work on approximating graph problems [25,7,10,22], the Fréchet distance [8], LCS [1, 2], monochromatic inner product [23], earth mover distance [26], as well as equivalences for fine-grained approximation in P [15,14,10]. Related work studies the inapproximability of parameterized problems, ruling out certain approximation guarantees within running time f (k)n g (k) under parameter k (such as FPT time f (k) poly(n), or n o(k) ), see [17] for a recent survey.…”

Section: Hardness Of Approximation In Pmentioning

confidence: 99%

“…Since Maximum Inner Product has received significant interest for improved algorithms (see particularly [13,15]), we turn to the question whether our completeness result also yields lower-order algorithmic improvements for all problems in the class. Indeed, by combining the best known Maximum/Minimum Inner Product algorithms with our reductions, we obtain the following general results for MaxSP and MinSP.…”

Section: Algorithms: Lower-order Improvementsmentioning

confidence: 99%

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Fine-Grained Completeness for Optimization in P

Bringmann,

Cassis,

Fischer

et al. 2021

Preprint

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We initiate the study of fine-grained completeness theorems for exact and approximate optimization in the polynomial-time regime.Inspired by the first completeness results for decision problems in P (Gao, Impagliazzo, Kolokolova, Williams, TALG 2019) as well as the classic class MaxSNP and MaxSNP-completeness for NP optimization problems (Papadimitriou, Yannakakis, JCSS 1991), we define polynomial-time analogues MaxSP and MinSP, which contain a number of natural optimization problems in P, including Maximum Inner Product, general forms of nearest neighbor search and optimization variants of the k-XOR problem. Specifically, we define MaxSP as the class of problems definable as maxx 1 ,...,x k #{(y1, . . . , y ℓ ) : ϕ(x1, . . . , x k , y1, . . . , y ℓ )}, where ϕ is a quantifier-free first-order property over a given relational structure (with MinSP defined analogously). On m-sized structures, we can solve each such problem in time O(m k+ℓ−1 ). Our results are:We determine (a sparse variant of) the Maximum/Minimum Inner Product problem as complete under deterministic fine-grained reductions: A strongly subquadratic algorithm for Maximum/Minimum Inner Product would beat the baseline running time of O(m k+ℓ−1 ) for all problems in MaxSP/MinSP by a polynomial factor. This completeness transfers to approximation: Maximum/Minimum Inner Product is also complete in the sense that a strongly subquadratic c-approximation would give a (c + ε)approximation for all MaxSP/MinSP problems in time O(m k+ℓ−1−δ ), where ε > 0 can be chosen arbitrarily small. Combining our completeness with (Chen, Williams, SODA 2019), we obtain the perhaps surprising consequence that refuting the OV Hypothesis is equivalent to giving a O(1)-approximation for all MinSP problems in faster-than-O(m k+ℓ−1 ) time.By fine-tuning our reductions, we obtain mild algorithmic improvements for solving and approximating all problems in MaxSP and MinSP, using the fastest known algorithms for Maximum/Minimum Inner Product.

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Section: Hardness Of Approximation In Pmentioning

confidence: 99%

Section: Algorithms: Lower-order Improvementsmentioning

confidence: 99%

Fine-Grained Completeness for Optimization in P

Bringmann,

Cassis,

Fischer

et al. 2021

Preprint

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“…In [32], Σ 2 communication protocols are utilized to show the subquadratic-time equivalence between OV n,O(log n) , Max-IP n,O(log n) , Approximate Bichromatic Closest Pair and several other problems.…”

Section: Related Workmentioning

confidence: 99%

Untitled

Chen¹

2020

Theory of Comput.

Self Cite

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In this paper we study the (Bichromatic) Maximum Inner Product Problem (Max-IP), in which we are given sets A and B of vectors, and the goal is to find a ∈ A and b ∈ B maximizing inner product a • b. Max-IP is a basic question and serves as the base problem in the recent breakthrough of [Abboud et al., FOCS 2017] on hardness of approximation for polynomial-time problems. It is also used (implicitly) in the argument for hardness of exact 2-Furthest Pair (and other important problems in computational geometry) in poly-loglog dimensions in [Williams, SODA 2018]. We have three main results regarding this problem. * Supported by an Akamai Fellowship.

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“…This equivalence is arguably the first fine-grained equivalence between natural problems with different running time complexities: MonoConvolution is a problem in O n 3/2 time, whereas 3SUM is in O n 2 time, and a polynomial improvement on one of these running times would result in a polynomial improvement over the other. All previous fine-grained equivalences were between problems with the same running time exponent: the problems equivalent to APSP [45,1] are all solvable in O N 1.5 time where N is the size of their input, the problems equivalent to Orthogonal Vectors [13] or to (min, +)-convolution [15] are all in quadratic time, the problems equivalent to CNF-SAT [14] are all in O(2 n ) time, etc. While tight fine-grained reductions between problems with different running times are well-known, there was no such equivalence until our result, largely since it often seems difficult to reduce a problem with a smaller asymptotic running time to one with a larger running time, something our Theorem 8 overcomes.…”

Section: Our Contributionsmentioning

confidence: 99%

Monochromatic Triangles, Intermediate Matrix Products, and Convolutions

Lincoln,

Polak,

Williams

2020

Preprint

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The most studied linear algebraic operation, matrix multiplication, has surprisingly fast O(n ω ) time algorithms for ω < 2.373. On the other hand, the (min, +) matrix product which is at the heart of many fundamental graph problems such as All-Pairs Shortest Paths, has received only minor n o(1) improvements over its brute-force cubic running time and is widely conjectured to require n 3−o(1) time. There is a plethora of matrix products and graph problems whose complexity seems to lie in the middle of these two problems. For instance, the Min-Max matrix product, the Minimum Witness matrix product, All-Pairs Shortest Paths in directed unweighted graphs and determining whether an edge-colored graph contains a monochromatic triangle, can all be solved in O(n (3+ω)/2 ) time. While slight improvements are sometimes possible using rectangular matrix multiplication, if ω = 2, the best runtimes for these "intermediate" problems are all O(n 2.5 ).A similar phenomenon occurs for convolution problems. Here, using the FFT, the usual (+, ×)-convolution of two n-length sequences can be solved in O(n log n) time, while the (min, +)convolution is conjectured to require n 2−o(1) time, the brute force running time for convolution problems. There are analogous intermediate problems that can be solved in O(n 1.5 ) time, but seemingly not much faster: Min-Max convolution, Minimum Witness convolution, etc.Can one improve upon the running times for these intermediate problems, in either the matrix product or the convolution world? Or, alternatively, can one relate these problems to each other and to other key problems in a meaningful way?This paper makes progress on these questions by providing a network of fine-grained reductions. We show for instance that APSP in directed unweighted graphs and Minimum Witness product can be reduced to both the Min-Max product and a variant of the monochromatic triangle problem, so that a significant improvement over n (3+ω)/2 time for any of the latter problems would result in a similar improvement for both of the former problems. We also show that a natural convolution variant of monochromatic triangle is fine-grained equivalent to the famous 3SUM problem. As this variant is solvable in O(n 1.5 ) time and 3SUM is in O(n 2 ) time (and is conjectured to require n 2−o(1) time), our result gives the first fine-grained equivalence between natural problems of different running times. We also relate 3SUM to monochromatic triangle, and a coin change problem to monochromatic convolution, and thus to 3SUM.

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An Equivalence Class for Orthogonal Vectors

Cited by 18 publications

References 51 publications

Fine-Grained Completeness for Optimization in P

Fine-Grained Completeness for Optimization in P

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Monochromatic Triangles, Intermediate Matrix Products, and Convolutions

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