Winning Cores in Parity Games

We introduce the novel notion of winning cores in parity games and develop a deterministic polynomial-time under-approximation algorithm for solving parity games based on winning core approximation. Underlying this algorithm are a number properties about winning cores which are interesting in their own right. In particular, we show that the winning core and the winning region for a player in a parity game are equivalently empty. Moreover, the winning core contains all fatal attractors but is not necessarily a dominion itself. Experimental results are very positive both with respect to quality of approximation and running time. It outperforms existing state-of-the-art algorithms significantly on most benchmarks

New Deterministic Algorithms for Solving Parity Games

We study parity games in which one of the two players controls only a small number $k$ of nodes and the other player controls the $n-k$ other nodes of the game. Our main result is a fixed-parameter algorithm that solves bipartite parity games in time $k^{O(\sqrt{k})}\cdot O(n^3)$, and general parity games in time $(p+k)^{O(\sqrt{k})} \cdot O(pnm)$, where $p$ is the number of distinct priorities and $m$ is the number of edges. For all games with $k = o(n)$ this improves the previously fastest algorithm by Jurdzi{\'n}ski, Paterson, and Zwick (SICOMP 2008). We also obtain novel kernelization results and an improved deterministic algorithm for graphs with small average degree

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

Solving Problems on Graphs of High Rank-Width

A modulator of a graph G to a specified graph class H is a set of vertices whose deletion puts G into H. The cardinality of a modulator to various tractable graph classes has long been used as a structural parameter which can be exploited to obtain FPT algorithms for a range of hard problems. Here we investigate what happens when a graph contains a modulator which is large but "well-structured" (in the sense of having bounded rank-width). Can such modulators still be exploited to obtain efficient algorithms? And is it even possible to find such modulators efficiently? We first show that the parameters derived from such well-structured modulators are strictly more general than the cardinality of modulators and rank-width itself. Then, we develop an FPT algorithm for finding such well-structured modulators to any graph class which can be characterized by a finite set of forbidden induced subgraphs. We proceed by showing how well-structured modulators can be used to obtain efficient parameterized algorithms for Minimum Vertex Cover and Maximum Clique. Finally, we use well-structured modulators to develop an algorithmic meta-theorem for deciding problems expressible in Monadic Second Order (MSO) logic, and prove that this result is tight in the sense that it cannot be generalized to LinEMSO problems.Comment: Accepted at WADS 201

On Structural Parameterizations of the Bounded-Degree Vertex Deletion Problem

We study the parameterized complexity of the Bounded-Degree Vertex Deletion problem (BDD), where the aim is to find a maximum induced subgraph whose maximum degree is below a given degree bound. Our focus lies on parameters that measure the structural properties of the input instance. We first show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treedepth, and even the size of a minimum vertex deletion set into graphs of pathwidth and treedepth at most three. We thereby resolve an open question stated in Betzler, Bredereck, Niedermeier and Uhlmann (2012) concerning the complexity of BDD parameterized by the feedback vertex set number. On the positive side, we obtain fixed-parameter algorithms for the problem with respect to the decompositional parameter treecut width and a novel problem-specific parameter called the core fracture number

Uniform Kernelization Complexity of Hitting Forbidden Minors

The F-Minor-Free Deletion problem asks, for a fixed set F and an input consisting of a graph G and integer k, whether k vertices can be removed from G such that the resulting graph does not contain any member of F as a minor. This paper analyzes to what extent provably effective and efficient preprocessing is possible for F-Minor-Free Deletion. Fomin et al. (FOCS 2012) showed that the special case Planar F-Deletion (when F contains at least one planar graph) has a kernel of size f(F) * k^{g(F)} for some functions f and g. The degree g of the polynomial grows very quickly; it is not even known to be computable. Fomin et al. left open whether Planar F-Deletion has kernels whose size is uniformly polynomial, i.e., of the form f(F) * k^c for some universal constant c that does not depend on F. Our results in this paper are twofold. (1) We prove that some Planar F-Deletion problems do not have uniformly polynomial kernels (unless NP is in coNP/poly). In particular, we prove that Treewidth-Eta Deletion does not have a kernel with O(k^{eta/4} - eps) vertices for any eps > 0, unless NP is in coNP/poly. In fact, we even prove the kernelization lower bound for the larger parameter vertex cover number. This resolves an open problem of Cygan et al. (IPEC 2011). It is a natural question whether further restrictions on F lead to uniformly polynomial kernels. However, we prove that even when F contains a path, the degree of the polynomial must, in general, depend on the set F. (2) A canonical F-Minor-Free Deletion problem when F contains a path is Treedepth-eta Deletion: can k vertices be removed to obtain a graph of treedepth at most eta? We prove that Treedepth-eta Deletion admits uniformly polynomial kernels with O(k^6) vertices for every fixed eta. In order to develop the kernelization we prove several new results about the structure of optimal treedepth-decompositions

New Deterministic Algorithms for Solving Parity Games

We study parity games in which one of the two players controls only a small number k of nodes and the other player controls the n-k n-kn-k other nodes of the game. Our main result is a fixed-parameter algorithm that solves bipartite parity games in time k o(k v ) ·o(n 3 ) ko(k)·o(n3)k^{o(\sqrt{k})}\cdot o(n^3) and general parity games in time (p+k) o(k v ) ·o(pnm) (p+k)o(k)·o(pnm)(p+k)^{o(\sqrt{k})} \cdot o(pnm), where p denotes the number of distinct priorities and m denotes the number of edges. For all games with k=o(n) k=o(n)k = o(n) this improves the previously fastest algorithm by jurdzinski, paterson, and zwick (sicomp 2008).we also obtain novel kernelization results and an improved deterministic algorithm for graphs with small average degree