research-article

Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs

Authors:

Marcin Pilipczuk,

Michał Pilipczuk,

Piotr Sankowski,

Erik Jan Van LeeuwenAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 14, Issue 4

Article No.: 53, Pages 1 - 73

https://doi.org/10.1145/3239560

Published: 17 September 2018 Publication History

Abstract

We propose polynomial-time algorithms that sparsify planar and bounded-genus graphs while preserving optimal or near-optimal solutions to Steiner problems. Our main contribution is a polynomial-time algorithm that, given an unweighted undirected graph G embedded on a surface of genus g and a designated face f bounded by a simple cycle of length k, uncovers a set F ⊆ E(G) of size polynomial in g and k that contains an optimal Steiner tree for any set of terminals that is a subset of the vertices of f.

We apply this general theorem to prove that:

— Given an unweighted graph G embedded on a surface of genus g and a terminal set S⊆ V(G), one can in polynomial time find a set F ⊆ E(G) that contains an optimal Steiner tree T for S and that has size polynomial in g and |E(T)|.

— An analogous result holds for an optimal Steiner forest for a set S of terminal pairs.

— Given an unweighted planar graph G and a terminal set S⊆ V(G), one can in polynomial time find a set F ⊆ E(G) that contains an optimal (edge) multiway cut C separating S (i.e., a cutset that intersects any path with endpoints in different terminals from S) and that has size polynomial in |C|.

In the language of parameterized complexity, these results imply the first polynomial kernels for Steiner Tree and Steiner Forest on planar and bounded-genus graphs (parameterized by the size of the tree and forest, respectively) and for (Edge) Multiway Cut on planar graphs (parameterized by the size of the cutset).

Additionally, we obtain a weighted variant of our main contribution: a polynomial-time algorithm that, given an undirected plane graph G with positive edge weights, a designated face f bounded by a simple cycle of weight w(f), and an accuracy parameter ε > 0, uncovers a set F ⊆ E(G) of total weight at most poly(ε^-1 ) w(f) that, for any set of terminal pairs that lie on f, contains a Steiner forest within additive error ε w(f) from the optimal Steiner forest.

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Włodarczyk MZehavi M(2023)Planar Disjoint Paths, Treewidth, and Kernels2023 IEEE 64th Annual Symposium on Foundations of Computer Science (FOCS)10.1109/FOCS57990.2023.00044(649-662)Online publication date: 6-Nov-2023
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Index Terms

Network Sparsification for Steiner Problems on Planar and Bounded-Genus Graphs
1. Theory of computation
  1. Design and analysis of algorithms
    1. Graph algorithms analysis
      1. Sparsification and spanners
    2. Parameterized complexity and exact algorithms
      1. Fixed parameter tractability

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cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 14, Issue 4

October 2018

445 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/3266298

Editor:
Aravind Srinivasan
University of Maryland, USA

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Publication History

Published: 17 September 2018

Accepted: 01 July 2018

Revised: 01 April 2018

Received: 01 July 2017

Published in TALG Volume 14, Issue 4

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