Sang-il Oum

Width Parameters Beyond Tree-width and their Applications

Besides the very successful concept of tree-width (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compared to trees have been born and studied over the past years. These concepts and parameters have proved to be useful tools in many applications, especially in the design of efficient algorithms. Our presented novel look at the contemporary developments of these ‘width’ parameters in combinatorial structures delivers—besides traditional tree-width and derived dynamic programming schemes—also a number of other useful parameters like branch-width, rank-width (clique-width) or hypertree-width. In this contribution, we demonstrate how ‘width’ parameters of graphs and generalized structures (such as matroids or hypergraphs), can be used to improve the design of parameterized algorithms and the structural analysis in other applications on an abstract level.

Finding Branch-Decompositions and Rank-Decompositions

We present a new algorithm that can output the rank-decomposition of width at most

Vertex-minors, monadic second-order logic, and a conjecture by Seese

k

Rank-Width and Well-Quasi-Ordering

of a graph if such exists. For that we use an algorithm that, for an input matroid represented over a fixed finite field, outputs its branch-decomposition of width at most

Approximating rank-width and clique-width quickly

k

Testing branch-width

if such exists. This algorithm works also for partitioned matroids. Both of these algorithms are fixed-parameter tractable, that is, they run in time

Rank-width of random graphs

O(n^3)

CLASSIFICATION OF REAL BOTT MANIFOLDS AND ACYCLIC DIGRAPHS

where

Finding branch-decompositions and rank-decompositions

n

A Relative of Hadwiger's Conjecture

is the number of vertices / elements of the input, for each constant value of