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Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs

Authors Fedor V. Fomin , Petr A. Golovach

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Author Details

Fedor V. Fomin
  • Department of Informatics, University of Bergen, Norway
Petr A. Golovach
  • Department of Informatics, University of Bergen, Norway

Funding

The research leading to these results has received funding from the Research Council of Norway via the project "MULTIVAL" (grant no. 263317).

Acknowledgements

We thank Torstein Strømme, Daniel Lokshtanov, and Pranabendu Misra for fruitful discussions on the topic of this paper. We also grateful to Saket Saurabh for helpful suggestions that allowed us to improve our results.

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Fedor V. Fomin and Petr A. Golovach. Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 49:1-49:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ESA.2020.49

Abstract

We study algorithmic properties of the graph class Chordal-ke, that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fill-in at most k. We discover that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from Chordal-ke. While various parameterized algorithms on graphs for many structural parameters like vertex cover or treewidth can be found in the literature, up to the Exponential Time Hypothesis (ETH), the existence of subexponential parameterized algorithms for most of the structural parameters and optimization problems is highly unlikely. This is why we find the algorithmic behavior of the "fill-in parameterization" very unusual. 
Being intrigued by this behaviour, we identify a large class of optimization problems on Chordal-ke that admit algorithms with the typical running time 2^𝒪(√k log k) ⋅ n^𝒪(1). Examples of the problems from this class are finding an independent set of maximum weight, finding a feedback vertex set or an odd cycle transversal of minimum weight, or the problem of finding a maximum induced planar subgraph. On the other hand, we show that for some fundamental optimization problems, like finding an optimal graph coloring or finding a maximum clique, are FPT on Chordal-ke when parameterized by k but do not admit subexponential in k algorithms unless ETH fails. 
Besides subexponential time algorithms, the class of Chordal-ke graphs appears to be appealing from the perspective of kernelization (with parameter k). While it is possible to show that most of the weighted variants of optimization problems do not admit polynomial in k kernels on Chordal-ke graphs, this does not exclude the existence of Turing kernelization and kernelization for unweighted graphs. In particular, we construct a polynomial Turing kernel for Weighted Clique on Chordal-ke graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of Chordal-ke, namely, Interval-ke and Split-ke graphs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Parameterized complexity
  • structural parameterization
  • subexponential algorithms
  • kernelization
  • chordal graphs
  • fill-in
  • independent set
  • clique
  • coloring

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