An Exponential Lower Bound for Cut Sparsifiers in Planar Graphs

Karpov, N. V.; Pilipczuk, Marcin; Zych-Pawlewicz, Anna

doi:10.1007/s00453-018-0504-8

Cited by 10 publications

(8 citation statements)

References 13 publications

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“…In comparison with our upper-bound which only considers the case γ = 1, the size of our sparsifiers from Theorem 1.4 is better by a Ω(k 2 ) factor. Subsequent to our work, Karpov et al [KPZP17] proved that there exists edge-weighted k-terminal planar graphs that require Ω(2 k ) edges in any exact cut sparsifier, which implies that it is necessary to have some additional assumption (e.g., γ = O(1)) to obtain a cut sparsifier of k O(1) size.…”

Section: Our Resultsmentioning

confidence: 85%

Improved Guarantees for Vertex Sparsification in Planar Graphs

Goranci¹,

Henzinger²,

Pan³

2020

SIAM J. Discrete Math.

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Graph Sparsification aims at compressing large graphs into smaller ones while preserving important characteristics of the input graph. In this work we study Vertex Sparsifiers, i.e., sparsifiers whose goal is to reduce the number of vertices. We focus on the following notions:(1) Given a digraph G = (V, E) and terminal vertices K ⊂ V with |K| = k, a (vertex) reachability sparsifier of G is a digraph H = (V H , E H ), K ⊂ V H that preserves all reachability information among terminal pairs. In this work we introduce the notion of reachability-preserving minors (RPMs) , i.e., we require H to be a minor of G. We show any directed graph G admits a RPM H of size O(k 3 ), and if G is planar, then the size of H improves to O(k 2 log k). We complement our upper-bound by showing that there exists an infinite family of grids such that any RPM must have Ω(k 2 ) vertices.(2) Given a weighted undirected graph G = (V, E) and terminal vertices K with |K| = k, an exact (vertex) cut sparsifier of G is a graph H with K ⊂ V H that preserves the value of minimum-cuts separating any bipartition of K. We show that planar graphs with all the k terminals lying on the same face admit exact cut sparsifiers of size O(k 2 ) that are also planar. Our result extends to flow and distance sparsifiers. It improves the previous best-known bound of O(k 2 2 2k ) for cut and flow sparsifiers by an exponential factor, and matches an Ω(k 2 ) lowerbound for this class of graphs. minors in undirected graphs by Krauthgamer et al. [KNZ14]. Theorem 1.1. Given a k-terminal digraph G, there exists a reachability-preserving minor H of G with size O(k 3 ).It might be interesting to compare the above result with the construction of reachability preserver by Abbound and Bodwin [AB18], where the reachability preserver for a pair-set P in a graph G is defined to be a subgraph of G that preserves the reachability of all pairs in P . The size (i.e., the number of edges) of such preservers is shown to be at least Ω(n 2/(d+1) |P | (d−1)/d ), for any integer d ≥ 2, which is in sharp contrast to our upper bound O(|P | 3/2 ) on the size of reachability-preserving minors by taking P to be the pair-set of all terminals.Furthermore, by exploiting a tight integration of our techniques with the compact distance oracles for planar graphs by Thorup [Tho04], we can show the following theorem regarding the size of reachability-preserving minors for planar digraphs.Theorem 1.2. Given a k-terminal planar digraph G, there exists a reachability-preserving minor H of G with size O(k 2 log k).We complement the above result by showing that there exist instances where the above upperbound is tight up to a O(log k) factor. Theorem 1.3. For infinitely many k ∈ N there exists a k-terminal acyclic directed grid G such that any reachability-preserving minor of G must use Ω(k 2 ) non-terminals.Cut, Flow and Distance Sparsifiers. We provide new constructions for quality-1 (exact) cut, flow and distance sparsifiers for k-terminal planar graphs, where all the terminals are assumed to lie on the s...

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Section: Our Resultsmentioning

confidence: 85%

Improved Guarantees for Vertex Sparsification in Planar Graphs

Goranci¹,

Henzinger²,

Pan³

2020

SIAM J. Discrete Math.

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“…Let us start with q = 1. Apart from the already mentioned work on a general graph G [HKNR98, KR14,KR13], there are also bounds for specific graph families, like bounded-treewidth or planar graphs [CSWZ00, KR13,KPZ17]. For planar G with γ(G) = 1, there is a recent tight upper bound s = O(k 2 ) [GHP17] (independently of our work), where the sparsifier is planar but is not a minor of the original graph.…”

Section: Related Workmentioning

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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“…The notion of face cover γ(G, K) was extensively studied in the context of Steiner tree problem [EMJ87, Ber90, KNvL19], cuts and (multicommodity) flows [MNS85,CW04], all pairs shortest path [Fre91, Fre95, CX00] and cut sparsifiers [KR17,KPZ17]. Given a drawing and a terminal set K, γ(G, K) can be found in 2 O(γ(G,K)) · poly(n) time, but generally it is known to be NP-hard [BM88].…”

Section: Related Workmentioning

confidence: 99%

A face cover perspective to ℓ₁ embeddings of planar graphs

Filtser¹

2020

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

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It was conjectured by Gupta et al. [Combinatorica04] that every planar graph can be embedded into 1 with constant distortion. However, given an n-vertex weighted planar graph, the best upper bound on the distortion is only O( √ log n), by Rao [SoCG99]. In this paper we study the case where there is a set K of terminals, and the goal is to embed only the terminals into 1 with low distortion. In a seminal paper, Okamura and Seymour [J.Comb.Theory81] showed that if all the terminals lie on a single face, they can be embedded isometrically into 1 . The more general case, where the set of terminals can be covered by γ faces, was studied by Lee and Sidiropoulos [STOC09] and Chekuri et al. [J.Comb.Theory13]. The state of the art is an upper bound of O(log γ) by Krauthgamer, Lee and Rika [SODA19]. Our contribution is a further improvement on the upper bound to O(√ log γ). Since every planar graph has at most O(n) faces, any further improvement on this result, will be a major breakthrough, directly improving upon Rao's long standing upper bound. Moreover, it is well known that the flow-cut gap equals to the distortion of the best embedding into 1 . Therefore, our result provides a polynomial time O( √ log γ)-approximation to the sparsest cut problem on planar graphs, for the case where all the demand pairs can be covered by γ faces.

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An Exponential Lower Bound for Cut Sparsifiers in Planar Graphs

Cited by 10 publications

References 13 publications

Improved Guarantees for Vertex Sparsification in Planar Graphs

Improved Guarantees for Vertex Sparsification in Planar Graphs

Refined Vertex Sparsifiers of Planar Graphs

A face cover perspective to ℓ₁ embeddings of planar graphs

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An Exponential Lower Bound for Cut Sparsifiers in Planar Graphs

Cited by 10 publications

References 13 publications

Improved Guarantees for Vertex Sparsification in Planar Graphs

Improved Guarantees for Vertex Sparsification in Planar Graphs

Refined Vertex Sparsifiers of Planar Graphs

A face cover perspective to ℓ1 embeddings of planar graphs

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Product

Resources

About

A face cover perspective to ℓ₁ embeddings of planar graphs