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

Hierarchical decompositions of graphs are interesting for algorithmic purposes There are several types of hierarchical decompositions Tree decompositions are the best known ones On graphs of tree-width at most k , ie, that have tree decompositions of width at most k , where k is fixed, every decision or optimization problem expressible in monadic second-order logic has a linear algorithm We prove that this is also the case for graphs of clique-width at most k , where this complexity measure is associated with hierarchical decompositions of another type, and where logical formulas are no longer allowed to use edge set quantifications We develop applications to several classes of graphs that include cographs and are, like cographs, defined by forbidding subgraphs with ``too many'' induced paths with four vertices

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Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width

Approximating clique-width and branch-width

The study of graph structure has advanced in recent years with great strides: finite graphs can be described algebraically, enabling them to be constructed out of more basic elements. Separately the properties of graphs can be studied in a logical language called monadic second-order logic. In this book, these two features of graph structure are brought together for the first time in a presentation that unifies and synthesizes research over the last 25 years. The author not only provides a thorough description of the theory, but also details its applications, on the one hand to the construction of graph algorithms, and, on the other to the extension of formal language theory to finite graphs. Consequently the book will be of interest to graduate students and researchers in graph theory, finite model theory, formal language theory, and complexity theory.

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Graph Structure and Monadic Second-Order Logic: A Language-Theoretic Approach

: Significant progress has been made with solution of location problems and in preprocessing and decomposition for discrete optimization. There has also been research on the application of techniques from combinational optimization to nonlinear problems.

http://www.dtic.mil/dtic/tr/fulltext/u2/a273552.pdf

Discrete Applied Mathematics

Graphs of clique-width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we show that the (q,q-4) graphs are of clique width at most q and P4-tidy graphs are of clique-width at most 4. Furthermore, the k-expression (for k=4 or k=q) associated with such a graph can be found in linear time.

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Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width

Graphs of clique–width at most k were introduced by Courcelle, Engelfriet and Rozenberg (1993) as graphs which can be defined by k-expressions based on graph operations which use k vertex labels. In this paper we study the clique–width of perfect graph classes. On one hand, we show that every distance–hereditary graph, has clique–width at most 3, and a 3–expression defining it can be obtained in linear time. On the other hand, we show that the classes of unit interval and permutation graphs are not of bounded clique–width. More precisely, we show that for every $n\in {\mathcal N}$ there is a unit interval graph In and a permutation graph Hn having n2 vertices, each of whose clique–width is at least n. These results allow us to see the border within the hierarchy of perfect graphs between classes whose clique–width is bounded and classes whose clique–width is unbounded. Finally we show that every n×n square grid, $n\in {\mathcal N}$, n ≥ 3, has clique–width exactly n+1.

On the clique-width of some perfect graph classes

We discuss the parametrized complexity of counting and evaluation problems on graphs where the range of counting is denable in monadic second-order logic (MSOL). We show that for bounded tree-width these problems are solvable in polynomial time. The same holds for bounded clique width in the cases, where the decomposition, which establishes the bound on the clique-width, can be computed in polynomial time and for problems expressible by monadic second-order formulas without edge set quantication. Such quantications are allowed in the case of graphs with bounded tree-width. As applications we discuss in detail how this aects the parametrized complexity of the permanent and the hamiltonian of a matrix, and more generally, various generating functions of MSOL denable graph properties. Finally, our results are also applicable to SAT and ]SAT. ? 2001 Elsevier Science B.V. All rights reserved.

On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic

Treewidth is generally regarded as one of the most useful parameterizations of a graph's construction. Clique-width is a similar parameterization that shares one of the powerful properties of treewidth, namely: if a graph is of bounded treewidth (or clique-width), then there is a polynomial time algorithm for any graph problem expressible in monadic second order logic, using quantifiers on vertices (in the case of clique-width you must assume a clique-width parse expression is given). In studying the relationship between treewidth and clique-width, Courcelle and Olariu [Discrete Appl. Math., 101 (2000), pp. 77--114] showed that any graph of bounded treewidth is also of bounded clique-width; in particular, for any graph G with treewidth k, the clique-width of G is at most 4 * 2k - 1 + 1.
In this paper, we improve this result by showing that the clique-width of G is at most 3 * 2k - 1 and, more importantly, that there is an exponential lower bound on this relationship. In particular, for any k, there is a graph G with treewidth equal to k, where the clique-width of G is at least $2^{\lfloor k/2\rfloor - 1}$.

Udi Rotics

Papers

Linear Time Solvable Optimization Problems on Graphs of Bounded Clique-Width

Linear Time Solvable Optimization Problems on Graphs of Bounded Clique Width

On the clique-width of some perfect graph classes

On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic

On the Relationship Between Clique-Width and Treewidth