Improved Guarantees for Vertex Sparsification in Planar Graphs

Goranci, Gramoz; Henzinger, Monika; Pan, Peng

doi:10.1137/17m1163153

Cited by 21 publications

(21 citation statements)

References 45 publications

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“…For q = 1, every graph admits DAM of size s = O(k 4 ) [KNZ14], and there is a lower bound of s = Ω(k 2 ) even for planar graphs [KNZ14]. Goranci, Henzinger and Peng [GHP17] (independently of our work), recently constructed, for planar graphs with γ(G) = 1, a (1, O(k 2 ))-distance sparsifier that is planar but not a minor of the original graph. Proceeding to quality q = O(1), planar graphs admit a DAM with q = 1 + and s = O(k log k/ ) 2 [CGH16], and certain graph families, such as trees and outerplanar graphs, admit a DAM with q = O(1) and s = O(k) [Gup01, BG08, CXKR06, KNZ14].…”

Section: Distance Sparsificationmentioning

confidence: 99%

Refined Vertex Sparsifiers of Planar Graphs

Krauthgamer¹,

Rika²

2020

SIAM J. Discrete Math.

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We study the following version of cut sparsification. Given a large edge-weighted network G with k terminal vertices, compress it into a smaller network H with the same terminals, such that every minimum terminal cut in H (i.e., the minimum cut separating a given subset of terminals from all other terminals) approximates the corresponding one in G, up to a factor q ≥ 1 that is called the quality. (The case q = 1 is known also as a mimicking network). We provide new insights about the structure of minimum terminal cuts, leading to new results for cut sparsifiers of planar graphs.Our first contribution identifies a subset of the minimum terminal cuts, which we call elementary, that generates all the others. Consequently, H is a cut sparsifier if and only if it preserves all the elementary terminal cuts (up to this factor q). This structural characterization can reduce the number of requirements, and thus lead to simpler proofs and to improved bounds on the size of H. For example, it leads to a small improvement in the mimicking-network size for planar graphs, which is actually near-optimal by a very recent lower bound [Karpov, Pilipczuk, and Zych-Pawlewicz, arXiv:1706.06086].Our second and main contribution is to refine the known bounds in terms of γ = γ(G), which is defined as the minimum number of faces that are incident to all the terminals in a planar graph G. We prove that the number of elementary terminal cuts is O((2k/γ) 2γ ) (compared to O(2 k ) terminal cuts), and furthermore obtain a mimicking-network of size O(γ2 2γ k 4 ), which is near-optimal as a function of γ by the aforementioned recent lower bound. The main challenge here is that the mimicking-network size is smaller than the number of elementary terminal cuts (requirements), and indeed our analysis breaks the elementary terminal cuts even further, and carefully counting these fragments yields a smaller number of requirements.Our third contribution is a duality between cut sparsification and distance sparsification for certain planar graphs, when the sparsifier H is required to be a minor of G. This duality connects problems that were previously studied separately, implying new results, new proofs of known results, and equivalences between open gaps.H instead of on G, using less resources like runtime and memory, or achieving better accuracy when the solution is approximate. This paradigm has lead to remarkable successes, such as faster runtimes for fundamental problems, and the introduction of important concepts, from spanners [PU89] to cut and spectral sparsifiers [BK15, ST11]. In these examples, H is a subgraph of G with the same vertex set but sparse, and is sometimes called an edge sparsifier. In contrast, we aim to reduce the number of vertices in G, using so-called vertex sparsifiers.In the vertex-sparsification scenario, G has k designated vertices called terminals, and the goal is to construct a small graph H that contains these terminals, and maintains some of their features inside G, like distances or cuts. Throughout, a k-terminal netw...

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Section: Distance Sparsificationmentioning

confidence: 99%

Refined Vertex Sparsifiers of Planar Graphs

Krauthgamer¹,

Rika²

2020

SIAM J. Discrete Math.

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“…Electrical transformations have been widely applied to graph algorithms and network optimizations [1,28,37,45,47] and other fields of science and engineering [16,44,49,53,59,67]. For a history of electrical transformations and other related work, see [10].…”

Section: Our Resultsmentioning

confidence: 99%

Tightening Curves on Surfaces Monotonically with Applications

Chang¹,

Mesmay²

2020

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

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We prove the first polynomial bound on the number of monotonic homotopy moves required to tighten a collection of closed curves on any compact orientable surface, where the number of crossings in the curve is not allowed to increase at any time during the process. The best known upper bound before was exponential, which can be obtained by combining the algorithm of de Graaf and Schrijver [J. Comb. Theory Ser. B, 1997] together with an exponential upper bound on the number of possible surface maps. To obtain the new upper bound we apply tools from hyperbolic geometry, as well as operations in graph drawing algorithms-the cluster and pipe expansions-to the study of curves on surfaces.As corollaries, we present two efficient algorithms for curves and graphs on surfaces. First, we provide a polynomial-time algorithm to convert any given multicurve on a surface into minimal position. Such an algorithm only existed for single closed curves, and it is known that previous techniques do not generalize to the multicurve case. Second, we provide a polynomial-time algorithm to reduce any k-terminal plane graph (and more generally, surface graph) using degree-1 reductions, series-parallel reductions, and ∆Y -transformations for arbitrary integer k. Previous algorithms only existed in the planar setting when k ≤ 4, and all of them rely on extensive case-by-case analysis based on different values of k. Our algorithm makes use of the connection between electrical transformations and homotopy moves, and thus solves the problem in a unified fashion.

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“…Similarly, Krauthgamer and Rika [31] studied how to find minimum-size planar graphs which preserve terminal cuts. Goranci et al [22] studied how to find a minor of a directed graph with as few Steiner vertices and which preserves the reachability relationships between all k terminals, showing that O(k 3 ) vertices suffices for general graphs but O(log k ⋅ k 2 ) vertices suffices for planar graphs.…”

Section: Spr and Related Problemsmentioning

confidence: 99%

$O(1)$ Steiner Point Removal in Series-Parallel Graphs

Hershkowitz,

2021

Preprint

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We study how to vertex-sparsify a graph while preserving both the graph's metric and structure. Specifically, we study the Steiner point removal (SPR) problem where we are given a weighted graph G = (V, E, w) and terminal set V ′ ⊆ V and must compute a weighted minorA major open question in the area of metric embeddings is the existence of O(1) multiplicative distortion SPR solutions for every (non-trivial) minor-closed family of graphs. To this end prior work has studied SPR on trees, cactus and outerplanar graphs and showed that in these graphs such a minor exists with O(1) distortion.We give O(1) distortion SPR solutions for series-parallel graphs, extending the frontier of this line of work. The main engine of our approach is a new metric decomposition for series-parallel graphs which we call a hammock decomposition. Roughly, a hammock decomposition is a forest-like structure that preserves certain critical parts of the metric induced by a series-parallel graph.

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Improved Guarantees for Vertex Sparsification in Planar Graphs

Cited by 21 publications

References 45 publications

Refined Vertex Sparsifiers of Planar Graphs

Refined Vertex Sparsifiers of Planar Graphs

Tightening Curves on Surfaces Monotonically with Applications

$O(1)$ Steiner Point Removal in Series-Parallel Graphs

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