Author
Petr A. Golovach
Other affiliations: Durham University, Steklov Mathematical Institute, Syktyvkar State University ...read more
Bio: Petr A. Golovach is an academic researcher from University of Bergen. The author has contributed to research in topics: Parameterized complexity & Chordal graph. The author has an hindex of 31, co-authored 319 publications receiving 3719 citations. Previous affiliations of Petr A. Golovach include Durham University & Steklov Mathematical Institute.
Papers published on a yearly basis
Papers
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TL;DR: In this article, the authors survey known results on the computational complexity of k-coloring and k-COLORING for graph classes that are characterized by one or two forbidden induced subgraphs, and also consider a number of variants: for example, where the problem is to extend a partial coloring, or where lists of permissible colors are given for each vertex.
Abstract: For a positive integer k, a k-coloring of a graph inline image is a mapping inline image such that inline image whenever inline image. The COLORING problem is to decide, for a given G and k, whether a k-coloring of G exists. If k is fixed (i.e., it is not part of the input), we have the decision problem k-COLORING instead. We survey known results on the computational complexity of COLORING and k-COLORING for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial coloring, or where lists of permissible colors are given for each vertex.
128 citations
Posted Content•
TL;DR: In this article, the authors survey known results on the computational complexity of coloring and coloring for graph classes that are characterized by one or two forbidden induced subgraphs, and also consider a number of variants: for example, where the problem is to extend a partial colouring, or where lists of permissible colours are given for each vertex.
Abstract: For a positive integer $k$, a $k$-colouring of a graph $G=(V,E)$ is a mapping $c: V\rightarrow\{1,2,...,k\}$ such that $c(u)
eq c(v)$ whenever $uv\in E$. The Colouring problem is to decide, for a given $G$ and $k$, whether a $k$-colouring of $G$ exists. If $k$ is fixed (that is, it is not part of the input), we have the decision problem $k$-Colouring instead. We survey known results on the computational complexity of Colouring and $k$-Colouring for graph classes that are characterized by one or two forbidden induced subgraphs. We also consider a number of variants: for example, where the problem is to extend a partial colouring, or where lists of permissible colours are given for each vertex.
109 citations
TL;DR: It is proved that computing the minimum number of cops that are guaranteed to catch a robber on a given graph is NP-hard and that the parameterized version of the problem is W[2]-hard; the proof extends to the case where the robber moves s time faster than the cops.
95 citations
TL;DR: This work compares the fixed parameter complexity of various variants of coloring problems when parameterized by treewidth and by vertex cover number and concludes that parametrization by the vertex covernumber provides a significant drop in the complexity of the problems.
91 citations
TL;DR: The results imply that the running time of many clique-width-based algorithms is essentially the best the authors can hope for (up to a widely believed assumption from parameterized complexity, namely $FPT
eq W[1]$).
Abstract: We show that Edge Dominating Set, Hamiltonian Cycle, and Graph Coloring are $W[1]$-hard parameterized by clique-width. It was an open problem, explicitly mentioned in several papers, whether any of these problems is fixed parameter tractable when parameterized by the clique-width, that is, solvable in time $g(k)\cdot n^{O(1)}$ on $n$-vertex graphs of clique-width $k$, where $g$ is some function of $k$ only. Our results imply that the running time $O(n^{f(k)})$ of many clique-width-based algorithms is essentially the best we can hope for (up to a widely believed assumption from parameterized complexity, namely $FPT
eq W[1]$).
90 citations
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Book•
27 Jul 2015
TL;DR: This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area, providing a toolbox of algorithmic techniques.
Abstract: This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area. The book covers many of the recent developments of the field, including application of important separators, branching based on linear programming, Cut & Count to obtain faster algorithms on tree decompositions, algorithms based on representative families of matroids, and use of the Strong Exponential Time Hypothesis. A number of older results are revisited and explained in a modern and didactic way. The book provides a toolbox of algorithmic techniques. Part I is an overview of basic techniques, each chapter discussing a certain algorithmic paradigm. The material covered in this part can be used for an introductory course on fixed-parameter tractability. Part II discusses more advanced and specialized algorithmic ideas, bringing the reader to the cutting edge of current research. Part III presents complexity results and lower bounds, giving negative evidence by way of W[1]-hardness, the Exponential Time Hypothesis, and kernelization lower bounds. All the results and concepts are introduced at a level accessible to graduate students and advanced undergraduate students. Every chapter is accompanied by exercises, many with hints, while the bibliographic notes point to original publications and related work.
1,544 citations
TL;DR: A complete view of the current state of the art with respect to layout problems from an algorithmic point of view is presented.
Abstract: Graph layout problems are a particular class of combinatorial optimization problems whose goal is to find a linear layout of an input graph in such way that a certain objective cost is optimized. This survey considers their motivation, complexity, approximation properties, upper and lower bounds, heuristics and probabilistic analysis on random graphs. The result is a complete view of the current state of the art with respect to layout problems from an algorithmic point of view.
665 citations
Book•
20 Mar 2013TL;DR: This book builds on the author’s prior research in this area, and also his practical experience of developing algorithms for matching kidney patients to donors in the UK, for assigning medical students to hospitals in Scotland, and for allocating students to elective courses and projects.
Abstract: Preliminary Definitions, Results and Motivation Stable Matching Problems: The Stable Marriage Problem: An Update SM and HR with Indifference The Stable Roommates Problem Further Stable Matching Problems Other Optimal Matching Problems: Pareto Optimal Matchings Popular Matchings Profile-Based Optimal Matchings.
488 citations
TL;DR: This chapter proves lower bounds based on ETH for the time needed to solve various problems, and in many cases these lower bounds match the running time of the best known algorithms for the problem.
Abstract: The Exponential Time Hypothesis (ETH) is a conjecture stating that, roughly speaking, n-variable 3-SAT cannot be solved in time 2o(n). In this chapter, we prove lower bounds based on ETH for the time needed to solve various problems. In many cases, these lower bounds match (up to small factors) the running time of the best known algorithms for the problem.
396 citations
Book•
02 Sep 2011TL;DR: One of the main goals of the book is to bring together the key results in the field; as such, it presents structural, probabilistic, and algorithmic results on Cops and Robbers games.
Abstract: This book is the first and only one of its kind on the topic of Cops and Robbers games, and more generally, on the field of vertex pursuit games on graphs The book is written in a lively and highly readable fashion, which should appeal to both senior undergraduates and experts in the field (and everyone in between) One of the main goals of the book is to bring together the key results in the field; as such, it presents structural, probabilistic, and algorithmic results on Cops and Robbers games Several recent and new results are discussed, along with a comprehensive set of references The book is suitable for self-study or as a textbook, owing in part to the over 200 exercises The reader will gain insight into all the main directions of research in the field and will be exposed to a number of open problems
368 citations