Approximation Algorithms and Hardness Results for Packing Element-Disjoint Steiner Trees in Planar Graphs

Aazami, Ashkan; Cheriyan, Joseph; Jampani, Krishnam Raju

doi:10.1007/978-3-642-03685-9_1

Cited by 6 publications

(24 citation statements)

References 18 publications

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“…Here (a) is by convexity: the quantity i: (1) . Balancing terms at α = kn 9/8 /m, this gives an upper bound of Õ m 1+o(1) n 3/8 U 1/4 , hence the claimed running time.…”

Section: Refined Running Times For Element Connectivitymentioning

confidence: 99%

“…The motivation for this work arose from the recent paper of Li and Panigrahi [34] that described a new algorithmic approach for finding the global minimum cut in an undirected graph. For a graph G = (V, E) with edge weights w : E → R >0 , the global minimum cut problem is to find the minimum weight subset of edges whose removal disconnects the graph; alternatively it is to find a set S, where ∅ S V , that minimizes w(δ(S)) 1 . When G is unweighted, this is called the edge connectivity of the graph.…”

Section: Introductionmentioning

confidence: 99%

“…Li and Panigrahi developed a new approach that improved this bound. Their algorithm runs in time O(m 1+o (1) ) plus the time to compute O(polylog(n)) (s, t)-minimum cut computations in a graph with m edges and n nodes. Their approach uses the (s, t)-minimum cut algorithm as a black box.…”

Section: Introductionmentioning

confidence: 99%

“…We refer the reader to surveys and related papers on network design [13-15, 22, 31] for extensive literature on this topic. It also plays an important role in packing vertex disjoint Steiner trees and forests among others [1,7,9,12]; [8] surveys this area. We now formally define element connectivity.…”

Section: Introductionmentioning

confidence: 99%

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Isolating Cuts, (Bi-)Submodularity, and Faster Algorithms for Global Connectivity Problems

Chekuri,

Quanrud

2021

Preprint

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Li and Panigrahi [34], in recent work, obtained the first deterministic algorithm for the global minimum cut of a weighted undirected graph that runs in time o(mn). They introduced an elegant and powerful technique to find isolating cuts for a terminal set in a graph via a small number of s-t minimum cut computations.In this paper we generalize their isolating cut approach to the abstract setting of symmetric bisubmodular functions (which also capture symmetric submodular functions). Our generalization to bisubmodularity is motivated by applications to element connectivity and vertex connectivity. Utilizing the general framework and other ideas we obtain significantly faster randomized algorithms for computing global (and subset) connectivity in a number of settings including hypergraphs, element connectivity and vertex connectivity in graphs, and for symmetric submodular functions.

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“…Here (a) is by convexity: the quantity i: (1) . Balancing terms at α = kn 9/8 /m, this gives an upper bound of Õ m 1+o(1) n 3/8 U 1/4 , hence the claimed running time.…”

Section: Refined Running Times For Element Connectivitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Isolating Cuts, (Bi-)Submodularity, and Faster Algorithms for Global Connectivity Problems

Chekuri,

Quanrud

2021

Preprint

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“…Note that in the special case of packing Steiner Trees, the paper of Aazami et al[1] shows that there are ⌊k/2⌋ − 1 element-disjoint Steiner Trees.…”

mentioning

confidence: 99%

A Graph Reduction Step Preserving Element-Connectivity and Applications

Chekuri

Korula

2009

Lecture Notes in Computer Science

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Given an undirected graph G = (V, E) and subset of terminals T ⊆ V , the element-connectivity κ ′ G (u, v) of two terminals u, v ∈ T is the maximum number of u-v paths that are pairwise disjoint in both edges and non-terminals V \ T (the paths need not be disjoint in terminals). Element-connectivity is more general than edge-connectivity and less general than vertex-connectivity. Hind and Oellermann [21] gave a graph reduction step that preserves the global element-connectivity of the graph. We show that this step also preserves local connectivity, that is, all the pairwise element-connectivities of the terminals. We give two applications of this reduction step to connectivity and network design problems.• Given a graph G and disjoint terminal sets T 1 , T 2 , . . . , T m , we seek a maximum number of elementdisjoint Steiner forests where each forest connects eachIf G is planar (or more generally, has fixed genus), we show that there exist Ω(k) Steiner forests. Our proofs are constructive, giving poly-time algorithms to find these forests; these are the first non-trivial algorithms for packing element-disjoint Steiner Forests.• We give a very short and intuitive proof of a spider-decomposition theorem of Chuzhoy and Khanna [12] in the context of the single-sink k-vertex-connectivity problem; this yields a simple and alternative analysis of an O(k log n) approximation.Our results highlight the effectiveness of the element-connectivity reduction step; we believe it will find more applications in the future. A Graph ReductionStep Preserving Element Connectivity: The well-known splitting-off operation introduced by Lovász [34] is a standard tool in the study of (primarily) edge-connectivity problems. Given an undirected multi-graph G and two edges su and sv incident to s, the splitting-off operation replaces su and sv by the single edge uv. Lovász proved the following theorem on splitting-off to preserve global edge-connectivity.Theorem 1.1 (Lovász). Let G = (V ∪ {s}, E) be an undirected multi-graph in which V is k-edge-connected for some k ≥ 2 and degree of s is even. Then for every edge su there is another edge sv such that V is k-edge-connected after splitting-off su and sv.Mader strengthened the above theorem to show the existence of a pair of edges incident to s that when split-off preserve the local edge-connectivity of the graph.

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Upper Bounds on the Semitotal Forcing Number of Graphs

Liang

Chen²,

Xu³

2023

Bull. Aust. Math. Soc.

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Let G be a graph with no isolated vertex. A semitotal forcing set of G is a (zero) forcing set S such that every vertex in S is within distance 2 of another vertex of S. The semitotal forcing number $F_{t2}(G)$ is the minimum cardinality of a semitotal forcing set in G. In this paper, we prove that it is NP-complete to determine the semitotal forcing number of a graph. We also prove that if $G\neq K_n$ is a connected graph of order $n\geq 4$ with maximum degree $\Delta \geq 2$ , then $F_{t2}(G)\leq (\Delta-1)n/\Delta$ , with equality if and only if either $G=C_{4}$ or $G=P_{4}$ or $G=K_{\Delta ,\Delta }$ .

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Approximation Algorithms and Hardness Results for Packing Element-Disjoint Steiner Trees in Planar Graphs

Cited by 6 publications

References 18 publications

Isolating Cuts, (Bi-)Submodularity, and Faster Algorithms for Global Connectivity Problems

Isolating Cuts, (Bi-)Submodularity, and Faster Algorithms for Global Connectivity Problems

A Graph Reduction Step Preserving Element-Connectivity and Applications

Upper Bounds on the Semitotal Forcing Number of Graphs

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