Finding Small Sparse Cuts by Random Walk

Kwok, Tsz Chiu; Lau, Lap Chi

doi:10.1007/978-3-642-32512-0_52

Cited by 13 publications

(16 citation statements)

References 23 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We show that, however, the evolving set algorithm will only explore the Hamming balls, and the expansion is at least 1 − for all Hamming balls of size O(n/k). We note that this example also shows that the random walk algorithm [ST13,KL12] and the pagerank algorithm [ACL06,ZLM13] fail to disprove the small-set expansion hypothesis; see Section 3.…”

Section: Vertex Expansionmentioning

confidence: 86%

“…The local graph partitioning algorithms provide bicriteria approximation algorithms for computing small set expansion φ δ (G). It is observed in [OT12,KL12] that if the output size guarantee of the above local graph partitioning algorithm is improved from O(|S * | 1+ ) to O(|S * |), then the small-set expansion hypothesis is false. Oveis Gharan [Ove13,AOPT16] suggested a plan to prove such an output size guarantee using the evolving set process.…”

Section: Theorem 11 ([Aopt16])mentioning

confidence: 99%

“…The bound in (2.2) follows from an inductive argument using (2.3); see [LS90,ST13,KL12] and also a slightly more general version in Lemma 2.7 in Section 2.4. We remark that their proof of (2.3) crucially relies on the assumption that there is a self-loop of weight 1/2 on each vertex and is a bit magical.…”

Section: Lovász-simonovits Curvementioning

confidence: 99%

“…Random Walks, Personal Pagerank, and Heat Kernels The random walk local graph partitioning algorithm [ST13,ABS10,KL12] works by computing the vector p t := A t χ v for every vertex v for 1 t O(log n), sorting the vertices so that p t (1) p t (2) . .…”

Section: Limitationsmentioning

confidence: 99%

See 3 more Smart Citations

Random Walks and Evolving Sets: Faster Convergences and Limitations

Chan

Kwok

Lau

2017

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

Self Cite

View full text Add to dashboard Cite

Analyzing the mixing time of random walks is a wellstudied problem with applications in random sampling and more recently in graph partitioning. In this work, we present new analysis of random walks and evolving sets using more combinatorial graph structures, and show some implications in approximating small-set expansion. On the other hand, we provide examples showing the limitations of using random walks and evolving sets in disproving the small-set expansion hypothesis.1. We define a combinatorial analog of the spectral gap, and use it to prove the convergence of nonlazy random walks. A corollary is a tight lower bound on the small-set expansion of graph powers for any graph.2. We prove that random walks converge faster when the robust vertex expansion of the graph is larger. This provides an improved analysis of the local graph partitioning algorithm using the evolving set process, and also derives an alternative proof of an improved Cheeger's inequality.3. We give an example showing that the evolving set process fails to disprove the small-set expansion hypothesis. This refutes a conjecture of Oveis Gharan and shows the limitations of all existing local graph partitioning algorithms in approximating small-set expansion.

show abstract

Section: Vertex Expansionmentioning

confidence: 86%

Section: Theorem 11 ([Aopt16])mentioning

confidence: 99%

Section: Lovász-simonovits Curvementioning

confidence: 99%

Section: Limitationsmentioning

confidence: 99%

See 2 more Smart Citations

Random Walks and Evolving Sets: Faster Convergences and Limitations

Chan

Kwok

Lau

2017

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

Self Cite

View full text Add to dashboard Cite

show abstract

“…The local graph clustering algorithm of Spielman and Teng [39] has been improved, both in terms of running time and the quality of the cuts discovered; see, e.g., [3,4,29,40].…”

Section: Additional Related Workmentioning

confidence: 99%

Distributed Triangle Detection via Expander Decomposition

Chang

Pettie

Zhang

2019

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

View full text Add to dashboard Cite

We present improved distributed algorithms for triangle detection and its variants in the CONGEST model. We show that Triangle Detection, Counting, and Enumeration can be solved inÕ(n 1/2 ) rounds. In contrast, the previous state-of-the-art bounds for Triangle Detection and Enumeration wereÕ(n 2/3 ) andÕ(n 3/4 ), respectively, due to Izumi and LeGall (PODC 2017).The main technical novelty in this work is a distributed graph partitioning algorithm. We show that inÕ(n 1−δ ) rounds we can partition the edge set of the network G• Each connected component induced by E m has minimum degree Ω(n δ ) and conductance Ω(1/polylog(n)). As a consequence the mixing time of a random walk within the component is O(polylog(n)).• The subgraph induced by E s has arboricity at most n δ .• |E r | ≤ |E|/6.All of our algorithms are based on the following generic framework, which we believe is of interest beyond this work. Roughly, we deal with the set E s by an algorithm that is efficient for low-arboricity graphs, and deal with the set E r using recursive calls. For each connected component induced by E m , we are able to simulate CONGESTED-CLIQUE algorithms with small overhead by applying a routing algorithm due to Ghaffari, Kuhn, and Su (PODC 2017) for high conductance graphs.We consider Triangle Detection problems in distributed networks. In the LOCAL model [34], which has no limit on bandwidth, all variants of Triangle Detection can be solved in exactly one round of communication: every vertex v simply announces its neighborhood N (v) to all neighbors. However, in models that take bandwidth into account, e.g., CONGEST, Triangle Detection becomes significantly more complicated. Whereas many graph optimization problems studied in the CONGEST model are intrinsically "global" (i.e., require at least diameter time) [2,11,12,14,15,20,26], Triangle Detection is somewhat unusual in that it can, in principle, be solved using only locally available information.The CONGEST Model. The underlying distributed network is represented as an undirected graph G = (V, E), where each vertex corresponds to a computational device, and each edge corresponds to a bi-directional communication link. We assume each v ∈ V initially knows some global parameters such as n = |V |, ∆ = max v∈V deg(v), and D = diameter(G). Each vertex v has a distinct Θ(log n)-bit identifier ID(v). The computation proceeds according to synchronized rounds.In each round, each vertex v can perform unlimited local computation, and may send a distinct O(log n)-bit message to each of its neighbors. Throughout the paper we only consider the randomized variant of CONGEST. Each vertex is allowed to generate unlimited local random bits, but there is no global randomness.The Congested Clique Model. The CONGESTED-CLIQUE model is a variant of CONGEST that allows all-to-all communication. Each vertex initially knows its adjacent edges and the set of vertex IDs, which we can assume w.l.o.g. is {1, . . . , |V |}. In each round, each vertex transmits n − 1 O(log n)-bit messages, one addressed to ...

show abstract

Detecting and Characterizing Small Dense Bipartite-Like Subgraphs by the Bipartiteness Ratio Measure

Pan

2013

Algorithms and Computation

View full text Add to dashboard Cite

We study the problem of finding and characterizing subgraphs with small bipartiteness ratio. We give a bicriteria approximation algorithm SwpDB such that if there exists a subset S of volume at most k and bipartiteness ratio θ, then for any 0 < ǫ < 1/2, it finds a set S ′ of volume at most 2k 1+ǫ and bipartiteness ratio at most 4 θ/ǫ. By combining a truncation operation, we give a local algorithm LocDB, which has asymptotically the same approximation guarantee as the algorithm SwpDB on both the volume and bipartiteness ratio of the output set, and runs in time O(ǫ 2 θ −2 k 1+ǫ ln 3 k), independent of the size of the graph. Finally, we give a spectral characterization of the small dense bipartite-like subgraphs by using the kth largest eigenvalue of the Laplacian of the graph.

show abstract

Finding Small Sparse Cuts by Random Walk

Cited by 13 publications

References 23 publications

Random Walks and Evolving Sets: Faster Convergences and Limitations

Random Walks and Evolving Sets: Faster Convergences and Limitations

Distributed Triangle Detection via Expander Decomposition

Detecting and Characterizing Small Dense Bipartite-Like Subgraphs by the Bipartiteness Ratio Measure

Contact Info

Product

Resources

About