617 research outputs found

    Complexity of Grundy coloring and its variants

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    The Grundy number of a graph is the maximum number of colors used by the greedy coloring algorithm over all vertex orderings. In this paper, we study the computational complexity of GRUNDY COLORING, the problem of determining whether a given graph has Grundy number at least kk. We also study the variants WEAK GRUNDY COLORING (where the coloring is not necessarily proper) and CONNECTED GRUNDY COLORING (where at each step of the greedy coloring algorithm, the subgraph induced by the colored vertices must be connected). We show that GRUNDY COLORING can be solved in time O(2.443n)O^*(2.443^n) and WEAK GRUNDY COLORING in time O(2.716n)O^*(2.716^n) on graphs of order nn. While GRUNDY COLORING and WEAK GRUNDY COLORING are known to be solvable in time O(2O(wk))O^*(2^{O(wk)}) for graphs of treewidth ww (where kk is the number of colors), we prove that under the Exponential Time Hypothesis (ETH), they cannot be solved in time O(2o(wlogw))O^*(2^{o(w\log w)}). We also describe an O(22O(k))O^*(2^{2^{O(k)}}) algorithm for WEAK GRUNDY COLORING, which is therefore \fpt for the parameter kk. Moreover, under the ETH, we prove that such a running time is essentially optimal (this lower bound also holds for GRUNDY COLORING). Although we do not know whether GRUNDY COLORING is in \fpt, we show that this is the case for graphs belonging to a number of standard graph classes including chordal graphs, claw-free graphs, and graphs excluding a fixed minor. We also describe a quasi-polynomial time algorithm for GRUNDY COLORING and WEAK GRUNDY COLORING on apex-minor graphs. In stark contrast with the two other problems, we show that CONNECTED GRUNDY COLORING is \np-complete already for k=7k=7 colors.Comment: 24 pages, 7 figures. This version contains some new results and improvements. A short paper based on version v2 appeared in COCOON'1

    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

    Parameterized Approximation Schemes using Graph Widths

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    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

    Between Treewidth and Clique-width

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    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

    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
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