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It is known that a number of natural graph problems which are FPT parameterized by treewidth become W-hard when parameterized by clique-width. It is therefore desirable to find a different structural graph parameter which is as general as possible, covers dense graphs but does not incur such a heavy algorithmic penalty.The main contribution of this paper is to consider a parameter called modular-width, defined using the well-known notion of modular decompositions. Using a combination of ILPs and dynamic programming we manage to design FPT algorithms for Coloring and Partitioning into paths (and hence Hamiltonian path and Hamiltonian cycle), which are W-hard for both clique-width and its recently introduced restriction, shrub-depth. We thus argue that modular-width occupies a sweet spot as a graph parameter, generalizing several simpler notions on dense graphs but still evading the "price of generality" paid by clique-width.

Possibly the most famous algorithmic meta-theorem is Courcelle's theorem, which states that all MSO-expressible graph properties are decidable in linear time for graphs of bounded treewidth. Unfortunately, the running time's dependence on the formula describing the problem is in general a tower of exponentials of unbounded height, and there exist lower bounds proving that this cannot be improved even if we restrict ourselves to deciding FO logic on trees.We investigate whether this parameter dependence can be improved by focusing on two proper subclasses of the class of bounded treewidth graphs: graphs of bounded vertex cover and graphs of bounded max-leaf number. We prove stronger algorithmic meta-theorems for these more restricted classes of graphs. More specifically, we show it is possible to decide any FO property in both of these classes with a singly exponential parameter dependence and that it is possible to decide MSO logic on graphs of bounded vertex cover with a doubly exponential parameter dependence. We also prove lower bound results which show that our upper bounds cannot be improved significantly, under widely believed complexity assumptions. Our work addresses an open problem posed by Michael Fellows.

Combining the techniques of approximation algorithms and parameterized complexity has long been considered a promising research area, but relatively few results are currently known. In this paper we study the parameterized approximability of a number of problems which are known to be hard to solve exactly when parameterized by treewidth or clique-width. Our main contribution is to present a natural randomized rounding technique that extends well-known ideas and can be used for both of these widths. Applying this very generic technique we obtain approximation schemes for a number of problems, evading both polynomial-time inapproximability and parameterized intractability bounds. log n ǫ O(w) n O(1) , where w is the input graph's clique-width.Max Cut is of course a problem of central importance in the contexts of both approximability and parameterized complexity. It is APX-hard (so an approximation ratio of 1 + ǫ is probably impossible in polynomial time) and W-hard parameterized by clique-width (so the fastest exact algorithm probably needs time roughly n w ). Our main point here is that using a parameterized ⋆

In (k, r)-Center we are given a (possibly edge-weighted) graph and are asked to select at most k vertices (centers), so that all other vertices are at distance at most r from a center. In this paper we provide a number of tight fine-grained bounds on the complexity of this problem with respect to various standard graph parameters. Specifically:• For any r ≥ 1, we show an algorithm that solves the problem in O * ((3r + 1) cw ) time, where cw is the clique-width of the input graph, as well as a tight SETH lower bound matching this algorithm's performance. As a corollary, for r = 1, this closes the gap that previously existed on the complexity of Dominating Set parameterized by cw.• We strengthen previously known FPT lower bounds, by showing that (k, r)-Center is W[1]-hard parameterized by the input graph's vertex cover (if edge weights are allowed), or feedback vertex set, even if k is an additional parameter. Our reductions imply tight ETH-based lower bounds. Finally, we devise an algorithm parameterized by vertex cover for unweighted graphs.• We show that the complexity of the problem parameterized by tree-depth is 2 Θ(td 2 ) , by showing an algorithm of this complexity and a tight ETH-based lower bound.We complement these mostly negative results by providing FPT approximation schemes parameterized by clique-width or treewidth, which work efficiently independently of the values of k, r. In particular, we give algorithms which, for any > 0, run in time O * ((tw/ ) O(tw) ), O * ((cw/ ) O(cw) ) and return a (k, (1 + )r)-center if a (k, r)-center exists, thus circumventing the problem's W-hardness.

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Abstract. We study a recently introduced generalization of the Vertex Cover (VC) problem, called Power Vertex Cover (PVC). In this problem, each edge of the input graph is supplied with a positive integer demand. A solution is an assignment of (power) values to the vertices, so that for each edge one of its endpoints has value as high as the demand, and the total sum of power values assigned is minimized. We investigate how this generalization affects the parameterized complexity of Vertex Cover. On the positive side, when parameterized by the value of the optimal P , we give an O * (1.274 P )-time branching algorithm (O * is used to hide factors polynomial in the input size), and also an O * (1.325 P )-time algorithm for the more general asymmetric case of the problem, where the demand of each edge may differ for its two endpoints. When the parameter is the number of vertices k that receive positive value, we give O * (1.619 k ) and O * (k k )-time algorithms for the symmetric and asymmetric cases respectively, as well as a simple quadratic kernel for the asymmetric case. We also show that PVC becomes significantly harder than classical VC when parameterized by the graph's treewidth t. More specifically, we prove that unless the ETH is false, there is no n o(t) -time algorithm for PVC. We give a method to overcome this hardness by designing an FPT approximation scheme which gives a (1+ )-approximation to the optimal solution in time FPT in parameters t and 1/ .