At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a clique-sum of pieces almost-embeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2-approximation to graph coloring, constant-factor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to half-integral multicommodity flow, subexponential fixed-parameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor. 1.

Abstract. A divide-and-conquer strategy based on variations of the Lipton-Tarjan planar separator theorem has been one of the most common approaches for solving planar graph problems for more than 20 years. We present a new framework for designing fast subexponential exact and parameterized algorithms on planar graphs. Our approach is based on geometric properties of planar branch decompositions obtained by Seymour & Thomas, combined with refined techniques of dynamic programming on planar graphs based on properties of non-crossing partitions. Compared to divide-and-conquer algorithms, the main advantages of our method are a) it is a generic method which allows to attack broad classes of problems; b) the obtained algorithms provide a better worst case analysis. To exemplify our approach we show how to obtain an O(26.903pn) time algorithm solving weighted Hamiltonian Cycle. We observe how our technique can be used to solve Planar Graph TSP in time O(29.8594pn). Our approach can be used to design parameterized algorithms as well. For example we introduce the first 2O(pk)nO(1) time algorithm for parameterized Planar k-cycle by showing that for a given k we can decide if a planar graph on n vertices has a cycle of length at least k in time O(213.6pkn + n3). 1 Introduction The celebrated Lipton-Tarjan planar separator theorem [20] is one of the most common approaches to obtain algorithms with subexponential running time for many problems on planar graphs [21]. The? This work was partially supported by Norges forskningsr*ad project 160778/V30,

This paper surveys the theory of bidimensionality. This theory characterizes a broad range of graph problems (‘bidimensional’) that admit efficient approximate or fixed-parameter solutions in a broad range of graphs. These graph classes include planar graphs, map graphs, bounded-genus graphs and graphs excluding any fixed minor. In particular, bidimensionality theory builds on the Graph Minor Theory of Robertson and Seymour by extending the mathematical results and building new algorithmic tools. Here, we summarize the known combinatorial and algorithmic results of bidimensionality theory with the high-level ideas involved in their proof; we describe the previous work on which the theory is based and/or extends; and we mention several remaining open problems. 1.

We present empirical results on computing optimal dominating sets in networks by means of data reduction through preprocessing rules. Thus, we demonstrate the usefulness of so far only theoretically considered reduction techniques for practically solving one of the most important network problems in combinatorial optimization.

Bidimensionality theory appears to be a powerful framework in the development of meta-algorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain sub-exponential time parameterized algorithms for bidimensional problems on H-minor free graphs. Demaine and Hajiaghayi [SODA 2005] extended the theory to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this paper, we establish a third meta-algorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In parameterized complexity, each problem instance comes with a parameter k and the parameterized problem is said to admit a linear kernel if there is a polynomial time algorithm, called

We prove that any H-minor-free graph, for a fixed graph H, of treewidth w has an \Omega (w) *\Omega ( w) grid graph as a minor. Thus grid minors suffice to certify that H-minor-free graphs havelarge treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and bounded-genus graphs, and is known not to hold for generalgraphs. The approach of this paper can be viewed more generally as a framework for extending combinatorial results on planar graphs to hold on H-minor-free graphs for any fixed H. Ourresult has many combinatorial consequences on bidimensionality theory, parameter-treewidth bounds, separator theorems, and bounded local treewidth; each of these combinatorial resultshas several algorithmic consequences including subexponential fixed-parameter algorithms and approximation algorithms.

Abstract. This paper surveys the theory of bidimensional graph problems. We summarize the known combinatorial and algorithmic results of this theory, the foundational Graph Minor results on which this theory is based, and the remaining open problems. 1

Abstract. The Power Dominating Set problem is a variant of the classical domination problem in graphs: Given an undirected graph G = (V, E), find a minimum P ⊆ V such that all vertices in V are “observed” by vertices in P. Herein, a vertex observes itself and all its neighbors, and if an observed vertex has all but one of its neighbors observed, then the remaining neighbor becomes observed as well. We show that Power Dominating Set can be solved by “bounded-treewidth dynamic programs.” Moreover, we simplify and extend several NP-completeness results, particularly showing that Power Dominating Set remains NP-complete for planar graphs, for circle graphs, and for split graphs. Specifically, our improved reductions imply that Power Dominating Set parameterized by |P | is W[2]-hard and cannot be better approximated than Dominating Set. 1

Abstract. We give a novel general approach for solving NP-hard optimization problems that combines dynamic programming and fast matrix multiplication. The technique is based on reducing much of the computation involved to matrix multiplication. We exemplify our approach on problems like Vertex Cover, Dominating Set and Longest Path. Our approach works faster than the usual dynamic programming solution for any vertex subset problem on graphs of bounded branchwidth. In particular, we obtain the currently fastest algorithms for Planar Vertex Cover of runtime O(2 2.52 √ n), for Planar Dominating Set of runtime exact O(2 3.99 √ n) and parameterized O(2 11.98 √ k) · n O(1) , and for Planar Longest Path of runtime O(2 5.58 √ n). The exponent of the running time is depending heavily on the running time of the fastest matrix multiplication algorithm that is currently o(n 2.376). 1

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