348 research outputs found

    Bidimensionality and Geometric Graphs

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    In this paper we use several of the key ideas from Bidimensionality to give a new generic approach to design EPTASs and subexponential time parameterized algorithms for problems on classes of graphs which are not minor closed, but instead exhibit a geometric structure. In particular we present EPTASs and subexponential time parameterized algorithms for Feedback Vertex Set, Vertex Cover, Connected Vertex Cover, Diamond Hitting Set, on map graphs and unit disk graphs, and for Cycle Packing and Minimum-Vertex Feedback Edge Set on unit disk graphs. Our results are based on the recent decomposition theorems proved by Fomin et al [SODA 2011], and our algorithms work directly on the input graph. Thus it is not necessary to compute the geometric representations of the input graph. To the best of our knowledge, these results are previously unknown, with the exception of the EPTAS and a subexponential time parameterized algorithm on unit disk graphs for Vertex Cover, which were obtained by Marx [ESA 2005] and Alber and Fiala [J. Algorithms 2004], respectively. We proceed to show that our approach can not be extended in its full generality to more general classes of geometric graphs, such as intersection graphs of unit balls in R^d, d >= 3. Specifically we prove that Feedback Vertex Set on unit-ball graphs in R^3 neither admits PTASs unless P=NP, nor subexponential time algorithms unless the Exponential Time Hypothesis fails. Additionally, we show that the decomposition theorems which our approach is based on fail for disk graphs and that therefore any extension of our results to disk graphs would require new algorithmic ideas. On the other hand, we prove that our EPTASs and subexponential time algorithms for Vertex Cover and Connected Vertex Cover carry over both to disk graphs and to unit-ball graphs in R^d for every fixed d

    Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs

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    We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, {\sc kk-Leaf Out-Branching}, which is to find an oriented spanning tree with at least kk leaves, we obtain an algorithm solving the problem in time 2O(klogk)n+nO(1)2^{O(\sqrt{k} \log k)} n+ n^{O(1)} on directed graphs whose underlying undirected graph excludes some fixed graph HH as a minor. For the special case when the input directed graph is planar, the running time can be improved to 2O(k)n+nO(1)2^{O(\sqrt{k})}n + n^{O(1)}. The second example is a generalization of the {\sc Directed Hamiltonian Path} problem, namely {\sc kk-Internal Out-Branching}, which is to find an oriented spanning tree with at least kk internal vertices. We obtain an algorithm solving the problem in time 2O(klogk)+nO(1)2^{O(\sqrt{k} \log k)} + n^{O(1)} on directed graphs whose underlying undirected graph excludes some fixed apex graph HH as a minor. Finally, we observe that for any ϵ>0\epsilon>0, the {\sc kk-Directed Path} problem is solvable in time O((1+ϵ)knf(ϵ))O((1+\epsilon)^k n^{f(\epsilon)}), where ff is some function of \ve. Our methods are based on non-trivial combinations of obstruction theorems for undirected graphs, kernelization, problem specific combinatorial structures and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs

    Linear kernels for outbranching problems in sparse digraphs

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    In the kk-Leaf Out-Branching and kk-Internal Out-Branching problems we are given a directed graph DD with a designated root rr and a nonnegative integer kk. The question is to determine the existence of an outbranching rooted at rr that has at least kk leaves, or at least kk internal vertices, respectively. Both these problems were intensively studied from the points of view of parameterized complexity and kernelization, and in particular for both of them kernels with O(k2)O(k^2) vertices are known on general graphs. In this work we show that kk-Leaf Out-Branching admits a kernel with O(k)O(k) vertices on H\mathcal{H}-minor-free graphs, for any fixed family of graphs H\mathcal{H}, whereas kk-Internal Out-Branching admits a kernel with O(k)O(k) vertices on any graph class of bounded expansion.Comment: Extended abstract accepted for IPEC'15, 27 page

    Bidimensionality and EPTAS

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    Bidimensionality theory is a powerful framework for the development of metaalgorithmic techniques. It was introduced by Demaine et al. as a tool to obtain sub-exponential time parameterized algorithms for problems on H-minor free graphs. Demaine and Hajiaghayi extended the theory to obtain PTASs for bidimensional problems, and subsequently improved these results to EPTASs. Fomin et. al related the theory to the existence of linear kernels for parameterized problems. In this paper we revisit bidimensionality theory from the perspective of approximation algorithms and redesign the framework for obtaining EPTASs to be more powerful, easier to apply and easier to understand. Two of the most widely used approaches to obtain PTASs on planar graphs are the Lipton-Tarjan separator based approach, and Baker's approach. Demaine and Hajiaghayi strengthened both approaches using bidimensionality and obtained EPTASs for a multitude of problems. We unify the two strenghtened approaches to combine the best of both worlds. At the heart of our framework is a decomposition lemma which states that for "most" bidimensional problems, there is a polynomial time algorithm which given an H-minor-free graph G as input and an e > 0 outputs a vertex set X of size e * OPT such that the treewidth of G n X is f(e). Here, OPT is the objective function value of the problem in question and f is a function depending only on e. This allows us to obtain EPTASs on (apex)-minor-free graphs for all problems covered by the previous framework, as well as for a wide range of packing problems, partial covering problems and problems that are neither closed under taking minors, nor contractions. To the best of our knowledge for many of these problems including cycle packing, vertex-h-packing, maximum leaf spanning tree, and partial r-dominating set no EPTASs on planar graphs were previously known

    Subexponential Parameterized Algorithms for Planar and Apex-Minor-Free Graphs via Low Treewidth Pattern Covering

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    We prove the following theorem. Given a planar graph GG and an integer kk, it is possible in polynomial time to randomly sample a subset AA of vertices of GG with the following properties: (i) AA induces a subgraph of GG of treewidth O(klogk)\mathcal{O}(\sqrt{k}\log k), and (ii) for every connected subgraph HH of GG on at most kk vertices, the probability that AA covers the whole vertex set of HH is at least (2O(klog2k)nO(1))1(2^{\mathcal{O}(\sqrt{k}\log^2 k)}\cdot n^{\mathcal{O}(1)})^{-1}, where nn is the number of vertices of GG. Together with standard dynamic programming techniques for graphs of bounded treewidth, this result gives a versatile technique for obtaining (randomized) subexponential parameterized algorithms for problems on planar graphs, usually with running time bound 2O(klog2k)nO(1)2^{\mathcal{O}(\sqrt{k} \log^2 k)} n^{\mathcal{O}(1)}. The technique can be applied to problems expressible as searching for a small, connected pattern with a prescribed property in a large host graph, examples of such problems include Directed kk-Path, Weighted kk-Path, Vertex Cover Local Search, and Subgraph Isomorphism, among others. Up to this point, it was open whether these problems can be solved in subexponential parameterized time on planar graphs, because they are not amenable to the classic technique of bidimensionality. Furthermore, all our results hold in fact on any class of graphs that exclude a fixed apex graph as a minor, in particular on graphs embeddable in any fixed surface
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