More consequences of falsifying SETH and the orthogonal vectors conjecture

Abboud, Amir; Bringmann, Karl; Dell, Holger; Nederlof, Jesper

doi:10.1145/3188745.3188938

Cited by 33 publications

(107 citation statements)

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“…Abboud et al [ABDN17] have shown (using techniques from [ALW14]) that if the ( , 4)-Hyperclique Hypothesis is false for some , then the Exact Weight -Clique Hypothesis is also false. Thus, the Hyperclique Hypothesis should be very believable even for k = 4.…”

Section: Hypothesis 22 (Max-k-sat Hypothesis)mentioning

confidence: 99%

Tight Hardness for Shortest Cycles and Paths in Sparse Graphs

Lincoln

Williams²,

Williams³

2018

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

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Fine-grained reductions have established equivalences between many core problems withÕ(n 3 )-time algorithms on n-node weighted graphs, such as Shortest Cycle, All-Pairs Shortest Paths (APSP), Radius, Replacement Paths, Second Shortest Paths, and so on. These problems also haveÕ(mn)-time algorithms on m-edge n-node weighted graphs, and such algorithms have wider applicability. Are these mn bounds optimal when m n 2 ?Starting from the hypothesis that the minimum weight (2 + 1)-Clique problem in edge weighted graphs requires n 2 +1−o(1) time, we prove that for all sparsities of the form m = Θ(n 1+1/ ), there is no O(n 2 + mn 1−ε ) time algorithm for ε > 0 for any of the below problems• Minimum Weight (2 + 1)-Cycle in a directed weighted graph, • Shortest Cycle in a directed weighted graph, • APSP in a directed or undirected weighted graph, • Radius (or Eccentricities) in a directed or undirected weighted graph, • Wiener index of a directed or undirected weighted graph, • Replacement Paths in a directed weighted graph, • Second Shortest Path in a directed weighted graph, • Betweenness Centrality of a given node in a directed weighted graph. That is, we prove hardness for a variety of sparse graph problems from the hardness of a dense graph problem. Our results also lead to new conditional lower bounds from several related hypothesis for unweighted sparse graph problems including k-cycle, shortest cycle, Radius, Wiener index and APSP. * andreali@mit.edu. Supported by the EECS Merrill Lynch Fellowship.†

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Section: Hypothesis 22 (Max-k-sat Hypothesis)mentioning

confidence: 99%

Tight Hardness for Shortest Cycles and Paths in Sparse Graphs

Lincoln

Williams²,

Williams³

2018

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

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“…All columns of B a,b differ entrywise from the first column B a,b (1) by at most Qn δ . Thus, if we consider the rank one matrix that has n δ columns identical to B a,b (1), we see that B a,b has Qn δ -approximate rank one. Hence by Theorem 1.1, we get that for any Q = O(n 3−ω−ε ) for ε > 0, we can pick δ = ε/2 and we'll get a truly subcubic time algorithm to Min-Plus multiply an arbitrary integer matrix A by B.…”

Section: New Subcubic Min-plus Productsmentioning

confidence: 99%

“…The hardness assumption we use is the ℓ-Uniform Hyperclique assumption used in prior works (see e.g. [18,1]):…”

Section: Conditional Lower Bounds For D-dimensionalmentioning

confidence: 99%

“…It is known (see [18]) that the natural extension of the techniques used to solve k-clique (in graphs) will not solve k-hyperclique in ℓ-uniform hypergraphs faster than n k . Moreover, there are known reductions from notoriously difficult problems such as Exact Weight k-Clique (a problem harder than Max Weight k-Clique) [1], Max ℓ-SAT and even harder Constrained Satisfaction Problems (CSPs) [27,18] to k-hyperclique in ℓ-uniform hypergraphs so that if the hypothesis is false, then all of these problems have surprisingly improved algorithms.…”

Section: Conditional Lower Bounds For D-dimensionalmentioning

confidence: 99%

“…Let A, B be two given n × n matrices whose entries are polylog n bit integers. Let W be a nonnegative integer and let d ≥ 1 be an integer with d = O (1). Suppose that for all k ′ , j ′ multiples of ∆, we can find two d by ∆ integer matrices X k ′ ,j ′ and Y k ′ ,j ′ , such that for any (k, j)…”

Section: Main Algorithmmentioning

confidence: 99%

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Truly Subcubic Min-Plus Product for Less Structured Matrices, with Applications

Williams¹,

Xu²

2020

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

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The All-Pairs Shortest Paths (APSP) problem is one of the most basic problems in computer science. The fastest known algorithms for APSP in n-node graphs run in n 3−o(1) time, and it is a big open problem whether a truly subcubic, O(n 3−ε ) for ε > 0 time algorithm exists for APSP. The Min-Plus product of two n × n matrices is known to be equivalent to APSP, where the optimal running times of the two problems differ by at most a constant factor. A natural way to approach understanding the complexity of APSP is thus understanding what structure (if any) is needed to solve Min-Plus product in truly subcubic time. The goal of this paper is to get truly subcubic algorithms for Min-Plus product for less structured inputs than what was previously known, and to apply them to versions of APSP and other problems. The results are as follows:(1) Our main result is the first truly subcubic algorithm for the Min-Plus product of two n × n matrices A and B with polylog n bit integer entries, where B has a partitioning into n ε × n ε blocks (for any ε > 0) where each block is at most n δ -far (for δ < 3 − ω, where 2 ≤ ω < 2.373) in ℓ ∞ norm from a constant rank integer matrix. This result presents the most general case to date of Min-Plus product that is still solvable in truly subcubic time.(2) The first application of our main result is a truly subcubic algorithm for APSP in a new type of geometric graph. Chan'10 solved APSP in truly subcubic time in geometric graphs whose edges have weights that are a function of the identities of the edge's end-points. Our result extends Chan's result in the case of integer edge weights by allowing the weights to differ from a function of the end-point identities by at most n δ for small δ.(3) In a second application we consider a batch version of the range mode problem in which one is given a sequence of numbers a 1 , . . . , a n and n intervals defining contiguous subsequences, and one is asked to compute the range mode of each subsequence. Chan et al.'14 showed that any O(n 1.5−ε ) time combinatorial algorithm for ε > 0 for this problem can be used to solve Boolean matrix multiplication combinatorially in truly subcubic time. We give the first O(n 1.5−ε ) time for ε > 0 algorithm for this batch range mode problem, showing that the hardness is indeed constrained to combinatorial algorithms.(4) Our final application is to the Maximum Subarray problem: given an n × n integer matrix, find the contiguous subarray of maximum entry sum. We show that Maximum Subarray can be solved in truly subcubic, O(n 3−ε ) (for ε > 0) time, as long as every entry of the input matrix is no larger than O(n 0.62 ) in absolute value. This is the first truly subcubic algorithm for an interesting case of Maximum Subarray. The Maximum Subarray problem with arbitrary integer entries is known to be subcubically equivalent to APSP, in that a truly subcubic, O(n 3−ε ) time algorithm for ε > 0 for one problem would imply a truly subcubic algorithm for the other. Because of this it is believed that Maximum Subarray does no...

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An Efficient Elastic-Degenerate Text Index? Not Likely

Gibney

2020

String Processing and Information Retrieval

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More consequences of falsifying SETH and the orthogonal vectors conjecture

Cited by 33 publications

References 55 publications

Tight Hardness for Shortest Cycles and Paths in Sparse Graphs

Tight Hardness for Shortest Cycles and Paths in Sparse Graphs

Truly Subcubic Min-Plus Product for Less Structured Matrices, with Applications

An Efficient Elastic-Degenerate Text Index? Not Likely

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