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

We prove that the problem of determining whether there exists a path of length at most l between two given k-colourings in the reconfiguration graph for k-Colouring is fixed-parameter tractable for all fixed k � 1, when parameterized by l. This addresses an open problem of Mouawad, Nishimura, Raman, Simjour and Suzuki (IPEC 2013). We also show that the problem, when parameterized by l, even has a linear vertex kernel for k = 3 and no polynomial kernel for all k � 4 unless the polynomial hierarchy collapses.

Colouring Reconfiguration Is Fixed-Parameter Tractable ⋆

Disconnected cuts in claw-free graphs

We almost completely resolve the computational complexity of Graph Isomorphism for classes of graphs characterized by two forbidden induced subgraphs $H_1$ and $H_2$. Schweitzer settled the complexity of this problem restricted to $(H_1,H_2)$-free graphs for all but a finite number of pairs $(H_1,H_2)$, but without explicitly giving the number of open cases. Grohe and Schweitzer proved that Graph Isomorphism is polynomial-time solvable on graph classes of bounded clique-width. By combining known results with a number of new results, we reduce the number of open cases to seven. By exploiting the strong relationship between Graph Isomorphism and clique-width, we simultaneously reduce the number of open cases for boundedness of clique-width for $(H_1,H_2)$-free graphs to five.

/pdf/graph-isomorphism-for-h-1-h-2-free-graphs-an-almost-complete-3ybt1o3h85.pdf

Graph Isomorphism for $(H_1,H_2)$-Free Graphs: An Almost Complete Dichotomy

A graph class is hereditary if it is closed under vertex deletion. We give examples of NP-hard, PSPACE-complete and NEXPTIME-complete problems that become constant-time solvable for every hereditary graph class that is not equal to the class of all graphs.

Hard Problems That Quickly Become Very Easy

We examine the effect of bounding the diameter for well-studied variants of the Colouring problem. A colouring is acyclic, star, or injective if any two colour classes induce a forest, star forest or disjoint union of vertices and edges, respectively. The corresponding decision problems are Acyclic Colouring, Star Colouring and Injective Colouring. The last problem is also known as $L(1,1)$-Labelling and we also consider the framework of $L(a,b)$-Labelling. We prove a number of (almost-)complete complexity classifications. In particular, we show that for graphs of diameter at most $d$, Acyclic $3$-Colouring is polynomial-time solvable if $d\leq 2$ but NP-complete if $d\geq 4$, and Star $3$-Colouring is polynomial-time solvable if $d\leq 3$ but NP-complete for $d\geq 8$. As far as we are aware, Star $3$-Colouring is the first problem that exhibits a complexity jump for some $d\geq 3$. Our third main result is that $L(1,2)$-Labelling is NP-complete for graphs of diameter $2$; we relate the latter problem to a special case of Hamiltonian Path.

Daniël Paulusma

Papers

Colouring Reconfiguration Is Fixed-Parameter Tractable ⋆

Disconnected cuts in claw-free graphs

Graph Isomorphism for \((H_1,H_2)\)-Free Graphs: An Almost Complete Dichotomy

Hard Problems That Quickly Become Very Easy

Acyclic, Star, and Injective Colouring: Bounding the Diameter