Faster approximation schemes and parameterized algorithms on (odd-)H-minor-free graphs

AbstractWe improve the running time of the general algorithmic technique known as Baker’s approach (1994) [1] on H-minor-free graphs from O(nf(|H|)) to O(f(|H|)nO(1)). The numerous applications include, e.g. a 2-approximation for coloring and PTASes for various problems such as dominating set and max-cut, where we obtain similar improvements.On classes of odd-minor-free graphs, which have gained significant attention in recent time, we obtain a similar acceleration for a variant of the structural decomposition theorem proved by Demaine et al. (2010) [20]. We use these algorithms to derive faster 2-approximations; furthermore, we present the first PTASes and subexponential FPT-algorithms for independent set and vertex cover on these graph classes using a novel dynamic programming technique.We also introduce a technique to derive (nearly) subexponential parameterized algorithms on H-minor-free graphs. Our technique applies, in particular, to problems such as Steiner tree, (directed) subgraph with a property, (directed) longest path, and (connected/independent) dominating set, on some or all proper minor-closed graph classes. We obtain as a corollary that all problems with a minor-monotone subexponential kernel and amenable to our technique can be solved in subexponential FPT-time onH-minor free graphs. This results in a general methodology for subexponential parameterized algorithms outside the framework of bidimensionality

A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions

Hitting and Harvesting Pumpkins

The "c-pumpkin" is the graph with two vertices linked by c>0 parallel edges. A c-pumpkin-model in a graph G is a pair A,B of disjoint subsets of vertices of G, each inducing a connected subgraph of G, such that there are at least c edges in G between A and B. We focus on covering and packing c-pumpkin-models in a given graph: On the one hand, we provide an FPT algorithm running in time 2^O(k) n^O(1) deciding, for any fixed c>0, whether all c-pumpkin-models can be covered by at most k vertices. This generalizes known single-exponential FPT algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the cases c=1,2 respectively. On the other hand, we present a O(log n)-approximation algorithm for both the problems of covering all c-pumpkin-models with a smallest number of vertices, and packing a maximum number of vertex-disjoint c-pumpkin-models.Comment: v2: several minor change

Subexponential Parameterized Odd Cycle Transversal on Planar Graphs

In the Odd Cycle Transversal (OCT) problem we are given a graph G on n vertices and an integer k, the objective is to determine whether there exists a vertex set O in G of size at most k such that G - O is bipartite. Reed, Smith and Vetta [Oper. Res. Lett., 2004] gave an algorithm for OCT with running time 3^kn^{O(1)}. Assuming the exponential time hypothesis of Impagliazzo, Paturi and Zane, the running time can not be improved to 2^{o(k)}n^{O(1)}. We show that OCT admits a randomized algorithm running in O(n^{O(1)} + 2^{O(sqrt{k} log k)}n) time when the input graph is planar. As a byproduct we also obtain a linear time algorithm for OCT on planar graphs with running time O(n^O(1) + 2O( sqrt(k) log k) n) time. This improves over an algorithm of Fiorini et al. [Disc. Appl. Math., 2008]

Structurally Parameterized d-Scattered Set

In $d$-Scattered Set we are given an (edge-weighted) graph and are asked to select at least $k$ vertices, so that the distance between any pair is at least $d$, thus generalizing Independent Set. We provide upper and lower bounds on the complexity of this problem with respect to various standard graph parameters. In particular, we show the following: - For any $d\ge2$, an $O^*(d^{\textrm{tw}})$-time algorithm, where $\textrm{tw}$ is the treewidth of the input graph. - A tight SETH-based lower bound matching this algorithm's performance. These generalize known results for Independent Set. - $d$-Scattered Set is W[1]-hard parameterized by vertex cover (for edge-weighted graphs), or feedback vertex set (for unweighted graphs), even if $k$ is an additional parameter. - A single-exponential algorithm parameterized by vertex cover for unweighted graphs, complementing the above-mentioned hardness. - A $2^{O(\textrm{td}^2)}$-time algorithm parameterized by tree-depth ($\textrm{td}$), as well as a matching ETH-based lower bound, both for unweighted graphs. We complement these mostly negative results by providing an FPT approximation scheme parameterized by treewidth. In particular, we give an algorithm which, for any error parameter $\epsilon > 0$, runs in time $O^*((\textrm{tw}/\epsilon)^{O(\textrm{tw})})$ and returns a $d/(1+\epsilon)$-scattered set of size $k$, if a $d$-scattered set of the same size exists

Subexponential-time parameterized algorithm for Steiner tree on planar graphs

The well-known bidimensionality theory provides a method for designing fast, subexponential-time parameterized algorithms for a vast number of NP-hard problems on sparse graph classes such as planar graphs, bounded genus graphs, or, more generally, graphs with a fixed excluded minor. However, in order to apply the bidimensionality framework the considered problem needs to fulfill a special density property. Some well-known problems do not have this property, unfortunately, with probably the most prominent and important example being the Steiner Tree problem. Hence the question whether a subexponential-time parameterized algorithm for Steiner Tree on planar graphs exists has remained open. In this paper, we answer this question positively and develop an algorithm running in O(2^{O((k log k)^{2/3})}n) time and polynomial space, where k is the size of the Steiner tree and n is the number of vertices of the graph. Our algorithm does not rely on tools from bidimensionality theory or graph minors theory, apart from Baker's classical approach. Instead, we introduce new tools and concepts to the study of the parameterized complexity of problems on sparse graphs.publishedVersio

Fixed-Parameter Algorithms for Rectilinear Steiner tree and Rectilinear Traveling Salesman Problem in the plane

Given a set $P$ of $n$ points with their pairwise distances, the traveling salesman problem (TSP) asks for a shortest tour that visits each point exactly once. A TSP instance is rectilinear when the points lie in the plane and the distance considered between two points is the $l_1$ distance. In this paper, a fixed-parameter algorithm for the Rectilinear TSP is presented and relies on techniques for solving TSP on bounded-treewidth graphs. It proves that the problem can be solved in $O\left(nh7^h\right)$ where $h \leq n$ denotes the number of horizontal lines containing the points of $P$. The same technique can be directly applied to the problem of finding a shortest rectilinear Steiner tree that interconnects the points of $P$ providing a $O\left(nh5^h\right)$ time complexity. Both bounds improve over the best time bounds known for these problems.Comment: 24 pages, 13 figures, 6 table