Abstract
We study algorithmic properties of the graph class Chordalke, that is, graphs that can be turned into a chordal graph by adding at most k edges or, equivalently, the class of graphs of fillin at most k. We discover that a number of fundamental intractable optimization problems being parameterized by k admit subexponential algorithms on graphs from Chordalke. 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 "fillin parameterization" very unusual.
Being intrigued by this behaviour, we identify a large class of optimization problems on Chordalke 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 Chordalke when parameterized by k but do not admit subexponential in k algorithms unless ETH fails.
Besides subexponential time algorithms, the class of Chordalke 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 Chordalke 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 Chordalke graphs. For (unweighted) Independent Set we design polynomial kernels on two interesting subclasses of Chordalke, namely, Intervalke and Splitke graphs.
BibTeX  Entry
@InProceedings{fomin_et_al:LIPIcs:2020:12915,
author = {Fedor V. Fomin and Petr A. Golovach},
title = {{Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs}},
booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)},
pages = {49:149:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771627},
ISSN = {18688969},
year = {2020},
volume = {173},
editor = {Fabrizio Grandoni and Grzegorz Herman and Peter Sanders},
publisher = {Schloss DagstuhlLeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2020/12915},
URN = {urn:nbn:de:0030drops129157},
doi = {10.4230/LIPIcs.ESA.2020.49},
annote = {Keywords: Parameterized complexity, structural parameterization, subexponential algorithms, kernelization, chordal graphs, fillin, independent set, clique, coloring}
}
Keywords: 

Parameterized complexity, structural parameterization, subexponential algorithms, kernelization, chordal graphs, fillin, independent set, clique, coloring 
Collection: 

28th Annual European Symposium on Algorithms (ESA 2020) 
Issue Date: 

2020 
Date of publication: 

26.08.2020 