Fully polynomial FPT algorithms for some classes of bounded clique-width graphs

Parameterized complexity theory has enabled a refined classification of the difficulty of NP-hard optimization problems on graphs with respect to key structural properties, and so to a better understanding of their true difficulties. More recently, hardness results for problems in P were achieved using reasonable complexity theoretic assumptions such as: Strong Exponential Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to these assumptions, many graph theoretic problems do not admit truly subquadratic algorithms, nor even truly subcubic algorithms (Williams and Williams, FOCS 2010 and Abboud, Grandoni, Williams, SODA 2015). A central technique used to tackle the difficulty of the above mentioned problems is fixed-parameter algorithms for polynomial-time problems with polynomial dependency in the fixed parameter (P-FPT). This technique was introduced by Abboud, Williams and Wang in SODA 2016 and continued by Husfeldt (IPEC 2016) and Fomin et al. (SODA 2017), using the treewidth as a parameter. Applying this technique to clique-width, another important graph parameter, remained to be done. In this paper we study several graph theoretic problems for which hardness results exist such as cycle problems (triangle detection, triangle counting, girth, diameter), distance problems (diameter, eccentricities, Gromov hyperbolicity, betweenness centrality) and maximum matching. We provide hardness results and fully polynomial FPT algorithms, using clique-width and some of its upper-bounds as parameters (split-width, modular-width and $P\_4$-sparseness). We believe that our most important result is an ${\cal O}(k^4 \cdot n + m)$-time algorithm for computing a maximum matching where $k$ is either the modular-width or the $P\_4$-sparseness. The latter generalizes many algorithms that have been introduced so far for specific subclasses such as cographs, $P\_4$-lite graphs, $P\_4$-extendible graphs and $P\_4$-tidy graphs. Our algorithms are based on preprocessing methods using modular decomposition, split decomposition and primeval decomposition. Thus they can also be generalized to some graph classes with unbounded clique-width

05301 Abstracts Collection -- Exact Algorithms and Fixed-Parameter Tractability

From 24.07.05 to 29.07.05, the Dagstuhl Seminar 05301 ``Exact Algorithms and Fixed-Parameter Tractability\u27\u27 was held in the International Conference and Research Center (IBFI), Schloss Dagstuhl. This is a collection of abstracts of the presentations given during the seminar

Efficient parameterized algorithms on structured graphs

In der klassischen Komplexitätstheorie werden worst-case Laufzeiten von Algorithmen typischerweise einzig abhängig von der Eingabegröße angegeben. In dem Kontext der parametrisierten Komplexitätstheorie versucht man die Analyse der Laufzeit dahingehend zu verfeinern, dass man zusätzlich zu der Eingabengröße noch einen Parameter berücksichtigt, welcher angibt, wie strukturiert die Eingabe bezüglich einer gewissen Eigenschaft ist. Ein parametrisierter Algorithmus nutzt dann diese beschriebene Struktur aus und erreicht so eine Laufzeit, welche schneller ist als die eines besten unparametrisierten Algorithmus, falls der Parameter klein ist. Der erste Hauptteil dieser Arbeit führt die Forschung in diese Richtung weiter aus und untersucht den Einfluss von verschieden Parametern auf die Laufzeit von bekannten effizient lösbaren Problemen. Einige vorgestellte Algorithmen sind dabei adaptive Algorithmen, was bedeutet, dass die Laufzeit von diesen Algorithmen mit der Laufzeit des besten unparametrisierten Algorithm für den größtmöglichen Parameterwert übereinstimmt und damit theoretisch niemals schlechter als die besten unparametrisierten Algorithmen und übertreffen diese bereits für leicht nichttriviale Parameterwerte. Motiviert durch den allgemeinen Erfolg und der Vielzahl solcher parametrisierten Algorithmen, welche eine vielzahl verschiedener Strukturen ausnutzen, untersuchen wir im zweiten Hauptteil dieser Arbeit, wie man solche unterschiedliche homogene Strukturen zu mehr heterogenen Strukturen vereinen kann. Ausgehend von algebraischen Ausdrücken, welche benutzt werden können, um von Parametern beschriebene Strukturen zu definieren, charakterisieren wir klar und robust heterogene Strukturen und zeigen exemplarisch, wie sich die Parameter tree-depth und modular-width heterogen verbinden lassen. Wir beschreiben dazu effiziente Algorithmen auf heterogenen Strukturen mit Laufzeiten, welche im Spezialfall mit den homogenen Algorithmen übereinstimmen.In classical complexity theory, the worst-case running times of algorithms depend solely on the size of the input. In parameterized complexity the goal is to refine the analysis of the running time of an algorithm by additionally considering a parameter that measures some kind of structure in the input. A parameterized algorithm then utilizes the structure described by the parameter and achieves a running time that is faster than the best general (unparameterized) algorithm for instances of low parameter value. In the first part of this thesis, we carry forward in this direction and investigate the influence of several parameters on the running times of well-known tractable problems. Several presented algorithms are adaptive algorithms, meaning that they match the running time of a best unparameterized algorithm for worst-case parameter values. Thus, an adaptive parameterized algorithm is asymptotically never worse than the best unparameterized algorithm, while it outperforms the best general algorithm already for slightly non-trivial parameter values. As illustrated in the first part of this thesis, for many problems there exist efficient parameterized algorithms regarding multiple parameters, each describing a different kind of structure. In the second part of this thesis, we explore how to combine such homogeneous structures to more general and heterogeneous structures. Using algebraic expressions, we define new combined graph classes of heterogeneous structure in a clean and robust way, and we showcase this for the heterogeneous merge of the parameters tree-depth and modular-width, by presenting parameterized algorithms on such heterogeneous graph classes and getting running times that match the homogeneous cases throughout

Counting Euler Tours in Undirected Bounded Treewidth Graphs

We show that counting Euler tours in undirected bounded tree-width graphs is tractable even in parallel - by proving a $\#SAC^1$ upper bound. This is in stark contrast to #P-completeness of the same problem in general graphs. Our main technical contribution is to show how (an instance of) dynamic programming on bounded \emph{clique-width} graphs can be performed efficiently in parallel. Thus we show that the sequential result of Espelage, Gurski and Wanke for efficiently computing Hamiltonian paths in bounded clique-width graphs can be adapted in the parallel setting to count the number of Hamiltonian paths which in turn is a tool for counting the number of Euler tours in bounded tree-width graphs. Our technique also yields parallel algorithms for counting longest paths and bipartite perfect matchings in bounded-clique width graphs. While establishing that counting Euler tours in bounded tree-width graphs can be computed by non-uniform monotone arithmetic circuits of polynomial degree (which characterize $\#SAC^1$) is relatively easy, establishing a uniform $\#SAC^1$ bound needs a careful use of polynomial interpolation.Comment: 17 pages; There was an error in the proof of the GapL upper bound claimed in the previous version which has been subsequently remove