Covering Vectors by Spaces in Perturbed Graphic Matroids and Their Duals
Perturbed graphic matroids are binary matroids that can be obtained from a graphic matroid by adding a noise of small rank. More precisely, an r-rank perturbed graphic matroid M is a binary matroid that can be represented in the form I +P, where I is the incidence matrix of some graph and P is a binary matrix of rank at most r. Such matroids naturally appear in a number of theoretical and applied settings. The main motivation behind our work is an attempt to understand which parameterized algorithms for various problems on graphs could be lifted to perturbed graphic matroids.
We study the parameterized complexity of a natural generalization (for matroids) of the following fundamental problems on graphs: Steiner Tree and Multiway Cut. In this generalization, called the Space Cover problem, we are given a binary matroid M with a ground set E, a set of terminals T subseteq E, and a non-negative integer k. The task is to decide whether T can be spanned by a subset of E \ T of size at most k.
We prove that on graphic matroid perturbations, for every fixed r, Space Cover is fixed-parameter tractable parameterized by k. On the other hand, the problem becomes W[1]-hard when parameterized by r+k+|T| and it is NP-complete for r <= 2 and |T|<= 2.
On cographic matroids, that are the duals of graphic matroids, Space Cover generalizes another fundamental and well-studied problem, namely Multiway Cut. We show that on the duals of perturbed graphic matroids the Space Cover problem is fixed-parameter tractable parameterized by r+k.
Binary matroids
perturbed graphic matroids
spanning set
parameterized complexity
Mathematics of computing~Combinatorial algorithms
Theory of computation~Parameterized complexity and exact algorithms
59:1-59:13
Track A: Algorithms, Complexity and Games
The research leading to these results has received funding from the Research Council of Norway via the projects "CLASSIS" and "MULTIVAL".
A full version of the paper is available at http://arxiv.org/abs/1902.06957.
We thank Jim Geelen for valuable insights regarding matroid minors.
Fedor V.
Fomin
Fedor V. Fomin
Department of Informatics, University of Bergen, Norway
Petr A.
Golovach
Petr A. Golovach
Department of Informatics, University of Bergen, Norway
Daniel
Lokshtanov
Daniel Lokshtanov
Department of Computer Science, University of California Santa Barbara, USA
Saket
Saurabh
Saket Saurabh
The Institute of Mathematical Sciences, HBNI, Chennai, India
Meirav
Zehavi
Meirav Zehavi
Ben-Gurion University, Israel
10.4230/LIPIcs.ICALP.2019.59
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Fedor V. Fomin, Petr A. Golovach, Daniel Lokshtanov, Saket Saurabh, and Meirav Zehavi
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