Parameterized complexity theory has lead to a wide range of algorithmic breakthroughs within the last decades, but the practicability of these methods for real-world problems is still not well understood. We investigate the practicability of one of the fundamental approaches of this field: dynamic programming on tree decompositions. Indisputably, this is a key technique in parameterized algorithms and modern algorithm design. Despite the enormous impact of this approach in theory, it still has very little influence on practical implementations. The reasons for this phenomenon are manifold. One of them is the simple fact that such an implementation requires a long chain of non-trivial tasks (as computing the decomposition, preparing it,. . . ). We provide an easy way to implement such dynamic programs that only requires the definition of the update rules. With this interface, dynamic programs for various problems, such as 3-coloring, can be implemented easily in about 100 lines of structured Java code.The theoretical foundation of the success of dynamic programming on tree decompositions is well understood due to Courcelle's celebrated theorem, which states that every MSO-definable problem can be efficiently solved if a tree decomposition of small width is given. We seek to provide practical access to this theorem as well, by presenting a lightweight model-checker for a small fragment of MSO. This fragment is powerful enough to describe many natural problems, and our model-checker turns out to be very competitive against similar state-of-the-art tools. ACM Subject Classification Theory of computation → Design and analysis of algorithmsBy combining this result (known as Courcelle's Theorem) with the f (tw(G)) · |G| algorithm of Bodlaender [7] to compute an optimal tree decomposition in fpt-time, a wide range of graph-theoretic problems is known to be solvable on these tree-like graphs. Unfortunately, both ingredients of this approach are very expensive in practice.One of the major achievements concerning practical parameterized algorithms was the discovery of a practically fast algorithm for treewidth due to Tamaki [19]. Concerning Courcelle's Theorem, there are currently two contenders concerning efficient implementations of it: D-Flat, an Answer Set Programming (ASP) solver for problems on tree decompositions [1]; and Sequoia, an MSO solver based on model checking games [17]. Both solvers allow to solve very general problems and the corresponding overhead might, thus, be large compared to a straightforward implementation of the dynamic programs for specific problems. Our ContributionsIn order to study the practicability of dynamic programs on tree decompositions, we expand our tree decomposition library Jdrasil with an easy to use interface for such programs: The user only needs to specify the update rules for the different kind of nodes within the tree decomposition. The remaining work -computing a suitable optimized tree decomposition and performing the actual run of the dynamic program -are done by Jdrasil. Th...

Counting and uniform sampling of directed acyclic graphs (DAGs) from a Markov equivalence class are fundamental tasks in graphical causal analysis. In this paper, we show that these tasks can be performed in polynomial time, solving a long-standing open problem in this area. Our algorithms are effective and easily implementable. Experimental results show that the algorithms significantly outperform state-of-the-art methods.

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