Subquadratic time approximation algorithms for the girth

Roditty, Liam; Williams, Virginia Vassilevska

doi:10.1137/1.9781611973099.67

Cited by 12 publications

(22 citation statements)

References 22 publications

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“…If it is not the case then, what is the best approximation factor we can get in subquadratic time? We note that in [17], Roditty and Vassilevska Williams conjectured that we cannot achieve a (2 − ε)-approximation already for unweighted graphs 5 .…”

Section: Proof Of Proposition 16mentioning

confidence: 79%

“…This was later completed by Lingas and Lundell [10], who proposed a randomized quasi 2-approximation algorithm for this problem that runs in time O(n 3/2 √ log n). In [17], Roditty and Vassilevska Williams presented the first deterministic approximation algorithm for the girth of unweighted graphs. Specifically, they obtained a 2-approximation algorithm in timeÕ(n 5/3 ), and they conjectured that there does not exist any subquadratic (2 − ε)-approximation for Girth.…”

Section: Related Workmentioning

confidence: 99%

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Faster Approximation Algorithms for Computing Shortest Cycles on Weighted Graphs

Ducoffe¹

2021

SIAM J. Discrete Math.

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Given an n-vertex m-edge graph G with non-negative edge-weights, a shortest cycle of G is one minimizing the sum of the weights on its edges. The girth of G is the weight of such a shortest cycle. We obtain several new approximation algorithms for computing the girth of weighted graphs:For any graph G with polynomially bounded integer weights, we present a deterministic algorithm that computes, inÕ(n 5/3 + m)-time 1 , a cycle of weight at most twice the girth of G. This matches both the approximation factor and -almost -the running time of the best known subquadratic-time approximation algorithm for the girth of unweighted graphs.Then, we turn our algorithm into a deterministic (2 + ε)-approximation for graphs with arbitrary non-negative edge-weights, at the price of a slightly worse running-time inÕ(n 5/3 polylog(1/ε) + m). For that, we introduce a generic method in order to obtain a polynomial-factor approximation of the girth in subquadratic time, that may be of independent interest. Finally, if we assume that the adjacency lists are sorted then we can get rid off the dependency in the number m of edges. Namely, we can transform our algorithms into anÕ(n 5/3 )-time randomized 4-approximation for graphs with non-negative edge-weights. This can be derandomized, thereby leading to anÕ(n 5/3 )-time deterministic 4-approximation for graphs with polynomially bounded integer weights, and anÕ(n 5/3 polylog(1/ε))-time deterministic (4 + ε)-approximation for graphs with non-negative edge-weights. To the best of our knowledge, these are the first known subquadratic-time approximation algorithms for computing the girth of weighted graphs. ACM Subject ClassificationTheory of computation → Design and analysis of algorithms; Theory of computation → Graph algorithms analysis; Theory of computation → Approximation algorithms analysis

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Section: Proof Of Proposition 16mentioning

confidence: 79%

Section: Related Workmentioning

confidence: 99%

Faster Approximation Algorithms for Computing Shortest Cycles on Weighted Graphs

Ducoffe¹

2021

SIAM J. Discrete Math.

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“…This is formalized in the next probably folklore theorem. A formal proof of it appears in [47]. Theorem 7.2.…”

Section: Lower Boundsmentioning

confidence: 99%

Approximating Cycles in Directed Graphs: Fast Algorithms for Girth and Roundtrip Spanners

Pachocki

Roditty

Sidford

et al. 2018

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

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The girth of a graph, i.e. the length of its shortest cycle, is a fundamental graph parameter. Unfortunately all known algorithms for computing, even approximately, the girth and girth-related structures in directed weighted m-edge and n-node graphs require Ω(min{n ω , mn}) time (for 2 ≤ ω < 2.373). In this paper, we drastically improve these runtimes as follows:• Multiplicative Approximations in Nearly Linear Time: We give an algorithm that in O(m) time computes an O(1)-multiplicative approximation of the girth as well as an O(1)-multiplicative roundtrip spanner with O(n) edges with high probability (w.h.p).• Nearly Tight Additive Approximations: For unweighted graphs and any a ∈ (0, 1) we give an algorithm that in O(mn 1−a ) time computes an O(n a )-additive approximation of the girth, w.h.p. We show that the runtime of our algorithm cannot be significantly improved without a breakthrough in combinatorial boolean matrix multiplication. We also show that if the girth is O(n a ), then the same guarantee can be achieved via a deterministic algorithm.Our main technical contribution to achieve these results is the first nearly linear time algorithm for computing roundtrip covers, a directed graph decomposition concept key to previous roundtrip spanner constructions. Previously it was not known how to compute these significantly faster than Ω(mn) time. Given the traditional difficulty in efficiently processing directed graphs, we hope our techniques may find further applications.

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“…This was later completed by Lingas and Lundell [19], who proposed a randomized quasi 2-approximation algorithm for this problem that runs in time O(n 3/2 √ log n). In [27], Roditty and Vassilevska Williams presented the first deterministic approximation algorithm for the girth of unweighted graphs. Specifically, they obtained a 2-approximation algorithm in time Õ(n 5/3 ), and they conjectured that there does not exist any subquadratic (2 − ε)approximation for Girth.…”

Section: Related Workmentioning

confidence: 99%

“…Overall, all the edges of E(P 1 ) ∪ E(P 2 ) are relaxed, and we claim that it contradicts our assumption that no cycle has been detected. Indeed, since P 1 , P 2 are different and they have the same ends, there exists a cycle C such that E(C) ⊆ E(P 1 ) ∪ E(P 2 ) (e.g., see [27,Lemma 2.5.]). Let e = xy ∈ E(C) be the last edge relaxed on the cycle.…”

Section: Reporting a Close Short Cyclementioning

confidence: 99%

Faster approximation algorithms for computing shortest cycles on weighted graphs

Ducoffe¹

2018

Preprint

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Given an n-vertex m-edge graph G with non negative edge-weights, a shortest cycle of G is one minimizing the sum of the weights on its edges. The girth of G is the weight of such a shortest cycle. We obtain several new approximation algorithms for computing the girth of weighted graphs:• For any graph G with polynomially bounded integer weights, we present a deterministic algorithm that computes, in Õ(n 5/3 + m)-time, a cycle of weight at most twice the girth of G. This matches both the approximation factor and -almost -the running time of the best known subquadratic-time approximation algorithm for the girth of unweighted graphs (Roditty and Vassilevska Williams, SODA'12). Our approach combines some new insights on the previous approximation algorithms for this problem (Lingas and Lundell, IPL'09; Roditty and Tov, TALG'13) with Hitting Set based methods that are used for approximate distance oracles and date back from (Thorup and Zwick, JACM'05).• Then, we turn our algorithm into a deterministic (2 + ε)-approximation for graphs with arbitrary non negative edge-weights, at the price of a slightly worse running-time in Õ(n 5/3 polylog(1/ε) + m). For that we introduce a novel polynomial-factor approximation of the girth, that makes more amenable the passing from the graphs with bounded integer edge-weights to the general case and is of independent interest.• Finally, if we insist in removing the dependency in the number m of edges, we can transform our algorithms into an Õ(n 5/3 )-time randomized 4-approximation for the graphs with non negative edge-weights -assuming the adjacency lists are sorted. Combined with the aforementioned Hitting Set based methods, this algorithm can be derandomized, thereby yielding an Õ(n 5/3 )-time deterministic 4-approximation for the graphs with polynomially bounded integer weights, and an Õ(n 5/3 polylog(1/ε))-time deterministic (4 + ε)approximation for the graphs with non negative edge-weights.To the best of our knowledge, these are the first known subquadratic-time approximation algorithms for computing the girth of weighted graphs.

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Subquadratic time approximation algorithms for the girth

Cited by 12 publications

References 22 publications

Faster Approximation Algorithms for Computing Shortest Cycles on Weighted Graphs

Faster Approximation Algorithms for Computing Shortest Cycles on Weighted Graphs

Approximating Cycles in Directed Graphs: Fast Algorithms for Girth and Roundtrip Spanners

Faster approximation algorithms for computing shortest cycles on weighted graphs

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