Efficient Approximation Algorithms for Shortest Cycles in Undirected Graphs

Lingas, Andrzej; Lundell, Eva-Marta

doi:10.1007/978-3-540-78773-0_63

Cited by 11 publications

(46 citation statements)

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“…It is widely believed that ω = 2 [13], and if true, that would imply we can compute the exact value of the girth in quasi quadratic time -at least when edge-weights are bounded. So far, all the approximation algorithms for Girth on weighted graphs run inΩ(n 2 )-time [10,15]. This leads us to the following, natural research question:…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

Faster Approximation Algorithms for Computing Shortest Cycles on Weighted Graphs

Ducoffe¹

2021

SIAM J. Discrete Math.

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“…Itai and Rodeh's additive 1-approximation, for instance, is at worst a multiplicative 4/3-approximation for the girth. More than 30 years after Itai and Rodeh's paper, Lingas and Lundell [16] presented the first algorithm that breaks the quadratic time bound, at the price of a much weaker approximation: their algorithm runs inÕ(n 3/2 ) time and returns a multiplicative 8/3-approximation. In fact, Lingas and Lundell's algorithm returns a cycle of length at most 2g + 2: this is almost a 2-approximation, but not quite.…”

Section: Introductionmentioning

confidence: 99%

“…We are able to answer Lingas and Lundell's open question by giving the first subquadratic time, multiplicative 2-approximation for the girth in undirected unweighted graphs. Furthermore, unlike the previous subquadratic time approximation algorithm for the girth in [16], our algorithm is deterministic! Theorem 1.2.…”

Section: Introductionmentioning

confidence: 99%

“…Lingas and Lundell [16] also presented an algorithm for undirected graphs with integer edge weights in the range {1, . .…”

Section: Introductionmentioning

confidence: 99%

“…1. ([13],[16]) For every vertex u ∈ V , BFScycle(u) runs in O(n) time. If a vertex v is at distance ℓ from a vertex u, and there is a simple cycle of length k going through v then BFS-cycle(u) reports a simple cycle of length at most to k+2ℓ if k is even, and k+2ℓ+1 if k is odd.…”

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confidence: 99%

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Subquadratic time approximation algorithms for the girth

Roditty¹,

Williams²

2012

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

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We study the problem of determining the girth of an unweighted undirected graph. We obtain several new efficient approximation algorithms for graphs with n nodes and m edges and unknown girth g. We consider additive and multiplicative approximations. Additive Approximations. We present: • anÕ(n 3 /m)-time algorithm which returns a cycle of length at most g + 2 if g is even and g + 3 if g is odd. This complements the seminal work of Itai and Rodeh [SIAM J. Computing'78] who gave an algorithm that in O(n 2) time finds a cycle of length g if g is even, and g + 1 if g is odd. • anÕ(n 3 /m)-time algorithm which returns a cycle of length at most g ′ + 2, where g ′ is the length of the shortest even cycle in G. This result complements the work of Yuster and Zwick [SIAM J. Discrete Math'97] who showed how to compute g ′ in O(n 2) time. Multiplicative Approximations. We present: • anÕ(n 5/3)-time algorithm which returns a cycle of length at most 3g/2 + z/2 when g is even and 3g/2 + z/2 + 1 when g is odd, where z = −g mod 4, z ∈ {0, 1, 2, 3}. This gives anÕ(n 5/3)-time 2approximation for the girth, the first subquadratic 2approximation algorithm, resolving an open question of Lingas and Lundell [IPL'09]. • an O(n 1.968)-time (8/5)-approximation algorithm for the girth in graphs with girth at least 4 (i.e., trianglefree graphs). This is the first subquadratic time (2 − ε)approximation algorithm for the girth for triangle-free graphs, for any ε > 0. We prove that a deterministic algorithm of this kind is not possible for directed graphs, thus showing a strong separation between undirected and directed graphs for girth approximation.

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Minimum Weight Cycles and Triangles: Equivalences and Algorithms

Roditty

Williams

2011

2011 IEEE 52nd Annual Symposium on Foundations of Computer Science

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We consider the fundamental algorithmic problem of finding a cycle of minimum weight in a weighted graph. In particular, we show that the minimum weight cycle problem in an undirected n-node graph with edge weights in {1, . . . , M } or in a directed n-node graph with edge weights in {−M, . . . , M } and no negative cycles can be efficiently reduced to finding a minimum weight triangle in an Θ(n)−node undirected graph with weights in {1, . . . , O(M )}. Roughly speaking, our reductions imply the following surprising phenomenon: a minimum cycle with an arbitrary number of weighted edges can be "encoded" using only three edges within roughly the same weight interval! This resolves a longstanding open problem posed in a seminal work by Itai and Rodeh [SIAM J. Computing 1978 and STOC'77] on minimum cycle in unweighted graphs. A direct consequence of our efficient reductions areÕ(M n ω ) ≤Õ(M n 2.376 )-time algorithms using fast matrix multiplication (FMM) for finding a minimum weight cycle in both undirected graphs with integral weights from the interval [1, M ] and directed graphs with integral weights from the interval [−M, M ]. The latter seems to reveal a strong separation between the all pairs shortest paths (APSP) problem and the minimum weight cycle problem in directed graphs as the fastest known APSP algorithm has a running time of O(M 0.681 n 2.575 ) by Zwick [J. ACM 2002].In contrast, when only combinatorial algorithms are allowed (that is, without FMM) the only known solution to minimum weight cycle is by computing APSP. Interestingly, any separation between the two problems in this case would be an amazing breakthrough as by a recent paper by Vassilevska W. and Williams [FOCS'10], any O(n 3−ε )-time algorithm (ε > 0) for minimum weight cycle immediately implies a O(n 3−δ )-time algorithm (δ > 0) for APSP.

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Efficient Approximation Algorithms for Shortest Cycles in Undirected Graphs

Cited by 11 publications

References 25 publications

Faster Approximation Algorithms for Computing Shortest Cycles on Weighted Graphs

Faster Approximation Algorithms for Computing Shortest Cycles on Weighted Graphs

Subquadratic time approximation algorithms for the girth

Minimum Weight Cycles and Triangles: Equivalences and Algorithms

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