Frederic Dorn

Efficient exact algorithms on planar graphs: exploiting sphere cut branch decompositions

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 new 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

Subexponential parameterized algorithms

O(2^{6.903\sqrt{n}}n^{3/2}+n^{3})

Dynamic programming and fast matrix multiplication

time algorithm solving weighted Hamiltonian Cycle. We observe how our technique can be used to solve Planar Graph TSP in time

Survey: Subexponential parameterized algorithms

O(2^{10.8224\sqrt{n}}n^{3/2}+n^{3})

Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs

. Our approach can be used to design parameterized algorithms as well. For example we introduce the first

Efficient Algorithms for Eulerian Extension and Rural Postman

2^{O\sqrt{k}}k^{O(1)}.n^{O(1)}

Catalan structures and dynamic programming in H-minor-free graphs

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 ≥ k in time

Tight bounds and a fast FPT algorithm for directed Max-Leaf Spanning Tree

O(2^{13.6\sqrt{k}}\sqrt{k}n+n^{3})

Two birds with one stone: the best of branchwidth and treewidth with one algorithm

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Tight Bounds and a Fast FPT Algorithm for Directed Max-Leaf Spanning Tree

We present a series of techniques for the design of subexponential parameterized algorithms for graph problems. The design of such algorithms usually consists of two main steps: first find a branch- (or tree-) decomposition of the input graph whose width is bounded by a sublinear function of the parameter and, second, use this decomposition to solve the problem in time that is single exponential to this bound. The main tool for the first step is Bidimensionality Theory. Here we present the potential, but also the boundaries, of this theory. For the second step, we describe recent techniques, associating the analysis of sub-exponential algorithms to combinatorial bounds related to Catalan numbers. As a result, we have 2O(√k) ˙ nO(1) time algorithms for a wide variety of parameterized problems on graphs, where n is the size of the graph and k is the parameter.