Tight Hardness for Shortest Cycles and Paths in Sparse Graphs

Lincoln, Andrea; Williams, Virginia Vassilevska; Williams, Ryan

doi:10.1137/1.9781611975031.80

Cited by 79 publications

(85 citation statements)

References 52 publications

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“…The O(mn) time bound is widely believed to be the optimal combinatorial running time up to subpolynomial factors 1 . In fact, even improving the running time for approximate APSP for multiplicative stretch (1 + ) and additive stretch (2 − ) for any ∈ [0, 1] by a truly polynomial factor implies major breakthroughs for several long-standing problems [DHZ00, WW18,LWW18].…”

Section: Related Workmentioning

confidence: 99%

Deterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler

Gutenberg¹,

Wulff-Nilsen²

2020

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

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In the decremental (1 + )-approximate Single-Source Shortest Path (SSSP) problem, we are given a graph G = (V, E) with n = |V |, m = |E|, undergoing edge deletions, and a distinguished source s ∈ V , and we are asked to process edge deletions efficiently and answer queries for distance estimates dist G (s, v) for each v ∈ V , at any stage, such that s, v). In the decremental (1 + )-approximate All-Pairs Shortest Path (APSP) problem, we are asked to answer queries for distance estimates dist G (u, v) for every u, v ∈ V . In this article, we consider the problems for undirected, unweighted graphs.We present a new deterministic algorithm for the decremental (1 + )-approximate SSSP problem that takes total update time O(mn 0.5+o (1) ). Our algorithm improves on the currently best algorithm for dense graphs by Chechik and Bernstein [STOC 2016] with total update timeÕ(n 2 ) and the best existing algorithm for sparse graphs with running timeÕ(n 1.25 √ m) [SODA 2017] whenever m = O(n 1.5−o(1) ). In order to obtain this new algorithm, we develop several new techniques including improved decremental cover data structures for graphs, a more efficient notion of the heavy/light decomposition framework introduced by Chechik and Bernstein and the first clustering technique to maintain a dynamic sparse emulator in the deterministic setting.As a by-product, we also obtain a new simple deterministic algorithm for the decremental (1+ )-approximate APSP problem with near-optimal total running timeÕ(mn/ ) matching the time complexity of the sophisticated but rather involved algorithm by Henzinger, Forster and Nanongkai [FOCS 2013].

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Section: Related Workmentioning

confidence: 99%

Deterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler

Gutenberg¹,

Wulff-Nilsen²

2020

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

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“…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%

“…The hypothesis is very believable for a variety of reasons. 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%

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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“…4 : it is not possible to determine the existence of a 4-clique in a graph with n nodes in time O(n 3 ). This is a special case of the k-Clique Hypothesis [14], which states that detecting a clique in a graph with n nodes requires n Enumerating Answers to UCQs. Given a UCQ Q over some schema S, we denote by E Q the enumeration problem E R , where R is the binary relation between instances I over S and sets of mappings Q(I ).…”

Section: Hypergraphsmentioning

confidence: 99%

“…A non-free-connex union of two CQs is intractable in the following cases: both CQs are intractable, or they both represent the same CQ up to a different projection. The hardness results presented here use problems with well-established assumptions on the lower bounds, such as Boolean matrix-multiplication [13] or finding a clique or a hyperclique in a graph [14].…”

Section: Introductionmentioning

confidence: 99%

On the Enumeration Complexity of Unions of Conjunctive Queries

Carmeli

Kröll

2019

Proceedings of the 38th ACM SIGMOD-SIGACT-SIGAI Symposium on Principles of Database Systems

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We study the enumeration complexity of Unions of Conjunctive Queries (UCQs). We aim to identify the UCQs that are tractable in the sense that the answer tuples can be enumerated with a linear preprocessing phase and a constant delay between every successive tuples. It has been established that, in the absence of self joins and under conventional complexity assumptions, the CQs that admit such an evaluation are precisely the free-connex ones. A union of tractable CQs is always tractable. We generalize the notion of freeconnexity from CQs to UCQs, thus showing that some unions containing intractable CQs are, in fact, tractable. Interestingly, some unions consisting of only intractable CQs are tractable too. The question of a finding a full characterization of the tractability of UCQs remains open. Nevertheless, we prove that for several classes of queries, free-connexity fully captures the tractable UCQs.According to the dichotomy of Bagan et al. [2], the enumeration problem for Q 2 is in DelayC lin , while Q 1 is intractable.

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Tight Hardness for Shortest Cycles and Paths in Sparse Graphs

Cited by 79 publications

References 52 publications

Deterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler

Deterministic Algorithms for Decremental Approximate Shortest Paths: Faster and Simpler

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

On the Enumeration Complexity of Unions of Conjunctive Queries

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