Between Treewidth and Clique-width

Many hard graph problems can be solved efficiently when restricted to graphs of bounded treewidth, and more generally to graphs of bounded clique-width. But there is a price to be paid for this generality, exemplified by the four problems MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set that are all FPT parameterized by treewidth but none of which can be FPT parameterized by clique-width unless FPT = W[1], as shown by Fomin et al [7, 8]. We therefore seek a structural graph parameter that shares some of the generality of clique-width without paying this price. Based on splits, branch decompositions and the work of Vatshelle [18] on Maximum Matching-width, we consider the graph parameter sm-width which lies between treewidth and clique-width. Some graph classes of unbounded treewidth, like distance-hereditary graphs, have bounded sm-width. We show that MaxCut, Graph Coloring, Hamiltonian Cycle and Edge Dominating Set are all FPT parameterized by sm-width

Towards Exact Structural Thresholds for Parameterized Complexity

Parameterized complexity seeks to optimally use input structure to obtain faster algorithms for NP-hard problems. This has been most successful for graphs of low treewidth, i.e., graphs decomposable by small separators: Many problems admit fast algorithms relative to treewidth and many of them are optimal under the Strong Exponential-Time Hypothesis (SETH). Fewer such results are known for more general structure such as low clique-width (decomposition by large and dense but structured separators) and more restrictive structure such as low deletion distance to some sparse graph class. Despite these successes, such results remain "islands" within the realm of possible structure. Rather than adding more islands, we seek to determine the transitions between them, that is, we aim for structural thresholds where the complexity increases as input structure becomes more general. Going from deletion distance to treewidth, is a single deletion set to a graph with simple components enough to yield the same lower bound as for treewidth or does it take many disjoint separators? Going from treewidth to clique-width, how much more density entails the same complexity as clique-width? Conversely, what is the most restrictive structure that yields the same lower bound? For treewidth, we obtain both refined and new lower bounds that apply already to graphs with a single separator X such that G-X has treewidth at most r = ?(1), while G has treewidth |X|+?(1). We rule out algorithms running in time ?^*((r+1-?)^k) for Deletion to r-Colorable parameterized by k = |X|; this implies the same lower bound relative to treedepth and (hence) also to treewidth. It specializes to ?^*((3-?)^k) for Odd Cycle Transversal where tw(G-X) ? r = 2 is best possible. For clique-width, an extended version of the above reduction rules out time ?^*((4-?)^k), where X is allowed to be a possibly large separator consisting of k (true) twinclasses, while the treewidth of G - X remains r; this is proved also for the more general Deletion to r-Colorable and it implies the same lower bound relative to clique-width. Further results complement what is known for Vertex Cover, Dominating Set and Maximum Cut. All lower bounds are matched by existing and newly designed algorithms

Parameterized Edge Hamiltonicity

We study the parameterized complexity of the classical Edge Hamiltonian Path problem and give several fixed-parameter tractability results. First, we settle an open question of Demaine et al. by showing that Edge Hamiltonian Path is FPT parameterized by vertex cover, and that it also admits a cubic kernel. We then show fixed-parameter tractability even for a generalization of the problem to arbitrary hypergraphs, parameterized by the size of a (supplied) hitting set. We also consider the problem parameterized by treewidth or clique-width. Surprisingly, we show that the problem is FPT for both of these standard parameters, in contrast to its vertex version, which is W-hard for clique-width. Our technique, which may be of independent interest, relies on a structural characterization of clique-width in terms of treewidth and complete bipartite subgraphs due to Gurski and Wanke

Are there any good digraph width measures?

Several different measures for digraph width have appeared in the last few years. However, none of them shares all the "nice" properties of treewidth: First, being \emph{algorithmically useful} i.e. admitting polynomial-time algorithms for all \MS1-definable problems on digraphs of bounded width. And, second, having nice \emph{structural properties} i.e. being monotone under taking subdigraphs and some form of arc contractions. As for the former, (undirected) \MS1 seems to be the least common denominator of all reasonably expressive logical languages on digraphs that can speak about the edge/arc relation on the vertex set.The latter property is a necessary condition for a width measure to be characterizable by some version of the cops-and-robber game characterizing the ordinary treewidth. Our main result is that \emph{any reasonable} algorithmically useful and structurally nice digraph measure cannot be substantially different from the treewidth of the underlying undirected graph. Moreover, we introduce \emph{directed topological minors} and argue that they are the weakest useful notion of minors for digraphs

Fixed-parameter tractable canonization and isomorphism test for graphs of bounded treewidth

We give a fixed-parameter tractable algorithm that, given a parameter $k$ and two graphs $G_1,G_2$, either concludes that one of these graphs has treewidth at least $k$, or determines whether $G_1$ and $G_2$ are isomorphic. The running time of the algorithm on an $n$-vertex graph is $2^{O(k^5\log k)}\cdot n^5$, and this is the first fixed-parameter algorithm for Graph Isomorphism parameterized by treewidth. Our algorithm in fact solves the more general canonization problem. We namely design a procedure working in $2^{O(k^5\log k)}\cdot n^5$ time that, for a given graph $G$ on $n$ vertices, either concludes that the treewidth of $G$ is at least $k$, or: * finds in an isomorphic-invariant way a graph $\mathfrak{c}(G)$ that is isomorphic to $G$; * finds an isomorphism-invariant construction term --- an algebraic expression that encodes $G$ together with a tree decomposition of $G$ of width $O(k^4)$. Hence, the isomorphism test reduces to verifying whether the computed isomorphic copies or the construction terms for $G_1$ and $G_2$ are equal.Comment: Full version of a paper presented at FOCS 201

Kernel Bounds for Structural Parameterizations of Pathwidth

Assuming the AND-distillation conjecture, the Pathwidth problem of determining whether a given graph G has pathwidth at most k admits no polynomial kernelization with respect to k. The present work studies the existence of polynomial kernels for Pathwidth with respect to other, structural, parameters. Our main result is that, unless NP is in coNP/poly, Pathwidth admits no polynomial kernelization even when parameterized by the vertex deletion distance to a clique, by giving a cross-composition from Cutwidth. The cross-composition works also for Treewidth, improving over previous lower bounds by the present authors. For Pathwidth, our result rules out polynomial kernels with respect to the distance to various classes of polynomial-time solvable inputs, like interval or cluster graphs. This leads to the question whether there are nontrivial structural parameters for which Pathwidth does admit a polynomial kernelization. To answer this, we give a collection of graph reduction rules that are safe for Pathwidth. We analyze the success of these results and obtain polynomial kernelizations with respect to the following parameters: the size of a vertex cover of the graph, the vertex deletion distance to a graph where each connected component is a star, and the vertex deletion distance to a graph where each connected component has at most c vertices.Comment: This paper contains the proofs omitted from the extended abstract published in the proceedings of Algorithm Theory - SWAT 2012 - 13th Scandinavian Symposium and Workshops, Helsinki, Finland, July 4-6, 201

Approximating acyclicity parameters of sparse hypergraphs

The notions of hypertree width and generalized hypertree width were introduced by Gottlob, Leone, and Scarcello in order to extend the concept of hypergraph acyclicity. These notions were further generalized by Grohe and Marx, who introduced the fractional hypertree width of a hypergraph. All these width parameters on hypergraphs are useful for extending tractability of many problems in database theory and artificial intelligence. In this paper, we study the approximability of (generalized, fractional) hyper treewidth of sparse hypergraphs where the criterion of sparsity reflects the sparsity of their incidence graphs. Our first step is to prove that the (generalized, fractional) hypertree width of a hypergraph H is constant-factor sandwiched by the treewidth of its incidence graph, when the incidence graph belongs to some apex-minor-free graph class. This determines the combinatorial borderline above which the notion of (generalized, fractional) hypertree width becomes essentially more general than treewidth, justifying that way its functionality as a hypergraph acyclicity measure. While for more general sparse families of hypergraphs treewidth of incidence graphs and all hypertree width parameters may differ arbitrarily, there are sparse families where a constant factor approximation algorithm is possible. In particular, we give a constant factor approximation polynomial time algorithm for (generalized, fractional) hypertree width on hypergraphs whose incidence graphs belong to some H-minor-free graph class