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.

Parameterized Algorithms

Wittgenstein's work remains, undeniably, now, that off one of those few philosophers who will be read by all future generations.

Remarks on the Foundations of Mathematics. By L. Wittgenstein. Edited by G. H. Von Wright, R. Rhees and G. E. M. Anscombe. Translated by G. E. M. Anscombe. Pp. 400. 37s. 6d. 1956. (Basil Blackwell, Oxford)

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.

/pdf/algorithmics-of-matching-under-preferences-2l3a9z5euq.pdf

Algorithmics of Matching Under Preferences

The reliance on multi/many-core systems to satisfy the high performance requirement of complex embedded software applications is increasing. This necessitates the need to realize efficient mapping methodologies for such complex computing platforms. This paper provides an extensive survey and categorization of state-of-the-art mapping methodologies and highlights the emerging trends for multi/many-core systems. The methodologies aim at optimizing system's resource usage, performance, power consumption, temperature distribution and reliability for varying application models. The methodologies perform design-time and run-time optimization for static and dynamic workload scenarios, respectively. These optimizations are necessary to fulfill the end-user demands. Comparison of the methodologies based on their optimization aim has been provided. The trend followed by the methodologies and open research challenges have also been discussed.

/pdf/mapping-on-multi-many-core-systems-survey-of-current-and-31ms8lelxo.pdf

Mapping on multi/many-core systems: survey of current and emerging trends

The theory of cooperative games provides a rich mathematical framework with which to understand the interactions between self-interested agents in settings where they can benefit from cooperation, and where binding agreements between agents can be made. Our aim in this talk is to describe the issues that arise when we consider cooperative game theory through a computational lens. We begin by introducing basic concepts from cooperative game theory, and in particular the key solution concepts: the core and the Shapley value. We then introduce the key issues that arise if one is to consider the cooperative games in a computational setting: in particular, the issue of representing games, and the computational complexity of cooperative solution concepts.

/pdf/computational-aspects-of-cooperative-game-theory-22sw4t9fz8.pdf

Computational Aspects of Cooperative Game Theory

The Surjective Homomorphism problem is to test whether a given graph G called the guest graph allows a vertex-surjective homomorphism to some other given graph H called the host graph. The bijective and injective homomorphism problems can be formulated in terms of spanning subgraphs and subgraphs, and as such their computational complexity has been extensively studied. What about the surjective variant? Because this problem is NP-complete in general, we restrict the guest and the host graph to belong to graph classes G and H, respectively. We determine to what extent a certain choice of G and H influences its computational complexity. We observe that the problem is polynomial-time solvable if H is the class of paths, whereas it is NP-complete if G is the class of paths. Moreover, we show that the problem is even NP-complete on many other elementary graph classes, namely linear forests, unions of complete graphs, cographs, proper interval graphs, split graphs and trees of pathwidth at most 2. In contrast, we prove that the problem is fixed-parameter tractable in k if G is the class of trees and H is the class of trees with at most k leaves, or if G and H are equal to the class of graphs with vertex cover number at most k.

/pdf/finding-vertex-surjective-graph-homomorphisms-2abecv7cke.pdf

Finding vertex-surjective graph homomorphisms

For a fixed graph H, the H-Minor Edit problem takes as input a graph G and an integer k and asks whether G can be modified into H by a total of at most k edge contractions, edge deletions and vertex deletions Replacing edge contractions by vertex dissolutions yields the H-Topological Minor Edit problem For each problem we show polynomial-time solvable and NP-complete cases depending on the choice of H Moreover, when G is AT-free, chordal or planar, we show that H-Minor Edit is polynomial-time solvable for all graphs H

/pdf/graph-editing-to-a-fixed-target-2ddthqljez.pdf

Graph editing to a fixed target.

The (Perfect) Matching Cut problem is to decide if a connected graph has a (perfect) matching that is also an edge cut. The Disconnected Perfect Matching problem is to decide if a connected graph has a perfect matching that contains a matching cut. Both Matching Cut and Disconnected Perfect Matching are NP-complete for planar graphs of girth $5$, whereas Perfect Matching Cut is known to be NP-complete even for subcubic bipartite graphs of arbitrarily large fixed girth. We prove that Matching Cut and Disconnected Perfect Matching are also NP-complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Our result for Matching Cut resolves a 20-year old open problem. We also show that the more general problem $d$-Cut, for every fixed $d\geq 1$, is NP-complete for graphs of arbitrarily large fixed girth. Furthermore, we show that Matching Cut, Perfect Matching Cut and Disconnected Perfect Matching are NP-complete for $H$-free graphs whenever $H$ contains a connected component with two vertices of degree at least $3$. Afterwards, we update the state-of-the-art summaries for $H$-free graphs and compare them not only with each other, but also with a known and full classification of the Maximum Matching Cut problem, which is to determine a largest matching cut of a given graph $G$. Finally, by combining existing results, we obtain a complete complexity classification of Perfect Matching Cut for $H$-subgraph-free graphs where $H$ is any finite set of graphs.

Matching Cuts in Graphs of High Girth and H-Free Graphs

A disconnected cut of a connected graph is a vertex cut that itself also induces a disconnected subgraph. The corresponding decision problem is called Disconnected Cut. It is known that Disconnected Cut is NP-hard on general graphs, while polynomial-time algorithms exist for several graph classes. However, the complexity of the problem on claw-free graphs remained an open question. Its connection to the complexity of the problem to contract a claw-free graph to the 4-vertex cycle C_4 led Ito et al. (TCS 2011) to explicitly ask to resolve this open question. We prove that Disconnected Cut is polynomial-time solvable on claw-free graphs, answering the question of Ito et al. The basis for our result is a decomposition theorem for claw-free graphs of diameter 2, which we believe is of independent interest and builds on the research line initiated by Chudnovsky and Seymour (JCTB 2007-2012) and Hermelin et al. (ICALP 2011). On our way to exploit this decomposition theorem, we characterize how disconnected cuts interact with certain cobipartite subgraphs, and prove two further algorithmic results, namely that Disconnected Cut is polynomial-time solvable on circular-arc graphs and line graphs.

https://drops.dagstuhl.de/opus/volltexte/2018/9524/pdf/LIPIcs-ESA-2018-61.pdf/

Disconnected Cuts in Claw-free Graphs

A graph is (H1,H2)-free for a pair of graphs H1,H2 if it contains no induced subgraph isomorphic to H1 or H2. We continue a study into the complexity of Colouring and Clique Cover for (H1,H2)-free graphs by focusing on the case where their complexities coincide. The complement H of a graph H has the same vertices as H and an edge between two distinct vertices if and only if these two vertices are not adjacent in H . By combining new and known results we classify the computational complexity of Colouring and Clique Cover for (H,H)-free graphs except when H = sP1 +P3 for s ≥ 3 or H = sP1 +P4 for s ≥ 2. We also show that these are the only open cases when considering all bigenic graph classes closed under complementation.

Daniël Paulusma

Papers

Finding vertex-surjective graph homomorphisms

Graph editing to a fixed target.

Matching Cuts in Graphs of High Girth and H-Free Graphs

Disconnected Cuts in Claw-free Graphs

Hereditary Graph Classes: When Colouring and Clique Cover Coincide