Cut Sparsifiers for Balanced Digraphs

Ikeda, Motoki; Tanigawa, Shin‐ichi

doi:10.1007/978-3-030-04693-4_17

Cited by 6 publications

(12 citation statements)

References 15 publications

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“…Our lower bound construction also refutes a conjecture of Ikeda and Tanigawa [20] that replacing β/λ e with the tighter 1/γ e in the sampling probabilities p e , where γ e is the directed edge connectivity of e, produces a better sparsifier. We give an example to show that sampling with probabilities proportional to 1/γ e may not even produce a sparsifier, i.e., does not preserve the values of all cuts!…”

Section: Our Resultssupporting

confidence: 83%

“…"For-All" Sparsification. Recently, Ikeda and Tanigawa [20] showed that there is a sparsifier for β-balanced directed graphs that uses O(βn/ 2 ) edges. 1 Indeed, this result can be recovered by using the following simple process: (a) remove all edge directions, (b) sample the resulting undirected graph using a standard (undirected) sparsification algorithm that samples every edge e at probability p e that is inversely proportional to the (undirected) edge connectivity λ e (see [17]), but after boosting the sampling probabilities by a factor of β, and (c) use a simple modification of the analysis from undirected sparsification to argue that the directed cuts are preserved because of the higher sampling probabilities.…”

Section: Our Resultsmentioning

confidence: 99%

“…As they note, their proof is an adaption of that of Fung et al [17], who gave a general framework that unifies the results of sampling according to various edge parameters (e.g., strength, effective resistance). Ikeda and Tanigawa [20] also showed how their directed sparsification algorithm can be used to obtain a faster approximation for the directed minimum cut problem (for some values of β), which was also studied by Ene et al [16].…”

Section: For-all Cut Sketchmentioning

confidence: 90%

“…4.1 For-All Cut Sparsifier: O(βn/ 2 ) Upper Bound For any β-balanced graph G on n vertices, Ikeda and Tanigawa [20] showed that G can be sparsified by a graph H with O(βn/ 2 ) edges. More specifically, they proved the following theorem.…”

Section: For-all Cut Sketchmentioning

confidence: 99%

“…More specifically, they proved the following theorem. Theorem 4.1 (Ikeda and Tanigawa [20]). Let G be a β-balanced graph on n vertices and m edges with maximum edge weight W .…”

Section: For-all Cut Sketchmentioning

confidence: 99%

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Sparsification of Directed Graphs via Cut Balance

Cen,

Cheng,

Panigrahi

et al. 2020

Preprint

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Sparsification, where the cut values of an input graph are approximately preserved by a sparse graph (called a cut sparsifier) or a succinct data structure (called a cut sketch), has been an influential tool in graph algorithms. But, this tool is restricted to undirected graphs, because some directed graphs are known to not admit sparsification. Such examples, however, are structurally very dissimilar to undirected graphs in that they exhibit highly unbalanced cuts. This motivates us to ask: can we sparsify a balanced digraph?To make this question concrete, we define balance β of a digraph as the maximum ratio of the cut value in the two directions (Ene et al., STOC 2016). We show the following results: * Part of this work was done when the author was a postdoctoral researcher at Duke University and when he was visiting the Institute of Advanced Study.

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Section: Our Resultssupporting

confidence: 83%

Section: Our Resultsmentioning

confidence: 99%

Section: For-all Cut Sketchmentioning

confidence: 90%

Section: For-all Cut Sketchmentioning

confidence: 99%

“…More specifically, they proved the following theorem. Theorem 4.1 (Ikeda and Tanigawa [20]). Let G be a β-balanced graph on n vertices and m edges with maximum edge weight W .…”

Section: For-all Cut Sketchmentioning

confidence: 99%

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Sparsification of Directed Graphs via Cut Balance

Cen,

Cheng,

Panigrahi

et al. 2020

Preprint

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Towards tight bounds for spectral sparsification of hypergraphs

Kapralov

Krauthgamer

Tardos

et al. 2021

Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing

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Tight Lower Bounds for Directed Cut Sparsification and Distributed Min-Cut

Cheng,

Li,

Lin

et al. 2024

Proc. ACM Manag. Data

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In this paper, we consider two fundamental cut approximation problems on large graphs. We prove new lower bounds for both problems that are optimal up to logarithmic factors. The first problem is approximating cuts in balanced directed graphs. In this problem, we want to build a data structure that can provide (1 ± ε)-approximation of cut values on a graph with n vertices. For arbitrary directed graphs, such a data structure requires Ω(n 2 ) bits even for constant ε. To circumvent this, recent works study β-balanced graphs, meaning that for every directed cut, the total weight of edges in one direction is at most β times the total weight in the other direction. We consider the for-each model, where the goal is to approximate each cut with constant probability, and the for-all model, where all cuts must be preserved simultaneously. We improve the previous Ømega(n √β/ε) lower bound in the for-each model to ~Ω (n √β /ε) and we improve the previous Ω(n β/ε) lower bound in the for-all model to Ω(n β/ε 2 ). This resolves the main open questions of (Cen et al., ICALP, 2021). The second problem is approximating the global minimum cut in a local query model, where we can only access the graph via degree, edge, and adjacency queries. We prove an ΩL(min m, m/ε 2 k R) lower bound for this problem, which improves the previous ΩL(m/k R) lower bound, where m is the number of edges, k is the minimum cut size, and we seek a (1+ε)-approximation. In addition, we show that existing upper bounds with minor modifications match our lower bound up to logarithmic factors.

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Cut Sparsifiers for Balanced Digraphs

Cited by 6 publications

References 15 publications

Sparsification of Directed Graphs via Cut Balance

Sparsification of Directed Graphs via Cut Balance

Towards tight bounds for spectral sparsification of hypergraphs

Tight Lower Bounds for Directed Cut Sparsification and Distributed Min-Cut

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