143 research outputs found
Parameterized Algorithms for Directed Maximum Leaf Problems
We prove that finding a rooted subtree with at least leaves in a digraph
is a fixed parameter tractable problem. A similar result holds for finding
rooted spanning trees with many leaves in digraphs from a wide family
that includes all strong and acyclic digraphs. This settles completely an open
question of Fellows and solves another one for digraphs in . Our
algorithms are based on the following combinatorial result which can be viewed
as a generalization of many results for a `spanning tree with many leaves' in
the undirected case, and which is interesting on its own: If a digraph of order with minimum in-degree at least 3 contains a rooted
spanning tree, then contains one with at least leaves
A tourist guide through treewidth
A short overview is given of many recent results in algorithmic graph theory that deal with the notions treewidth, and pathwidth. We discuss algorithms that find tree-decompositions, algorithms that use tree-decompositions to solve hard problems efficiently, graph minor theory, and some applications. The paper contains an extensive bibliography
Faster Algorithms For Vertex Partitioning Problems Parameterized by Clique-width
Many NP-hard problems, such as Dominating Set, are FPT parameterized by
clique-width. For graphs of clique-width given with a -expression,
Dominating Set can be solved in time. However, no FPT algorithm
is known for computing an optimal -expression. For a graph of clique-width
, if we rely on known algorithms to compute a -expression via
rank-width and then solving Dominating Set using the -expression,
the above algorithm will only give a runtime of . There
have been results which overcome this exponential jump; the best known
algorithm can solve Dominating Set in time by avoiding
constructing a -expression [Bui-Xuan, Telle, and Vatshelle. Fast dynamic
programming for locally checkable vertex subset and vertex partitioning
problems. Theoret. Comput. Sci., 2013. doi:10.1016/j.tcs.2013.01.009]. We
improve this to . Indeed, we show that for a graph of
clique-width , a large class of domination and partitioning problems
(LC-VSP), including Dominating Set, can be solved in . Our main tool is a variant of rank-width using the rank of a -
matrix over the rational field instead of the binary field.Comment: 13 pages, 5 figure
On the stable degree of graphs
We define the stable degree s(G) of a graph G by s(G)∈=∈ min max d (v), where the minimum is taken over all maximal independent sets U of G. For this new parameter we prove the following. Deciding whether a graph has stable degree at most k is NP-complete for every fixed k∈≥∈3; and the stable degree is hard to approximate. For asteroidal triple-free graphs and graphs of bounded asteroidal number the stable degree can be computed in polynomial time. For graphs in these classes the treewidth is bounded from below and above in terms of the stable degree
Graph and String Parameters: Connections Between Pathwidth, Cutwidth and the Locality Number
We investigate the locality number, a recently introduced structural parameter for strings (with applications in pattern matching with variables), and its connection to two important graph-parameters, cutwidth and pathwidth. These connections allow us to show that computing the locality number is NP-hard but fixed-parameter tractable (when the locality number or the alphabet size is treated as a parameter), and can be approximated with ratio O(sqrt{log{opt}} log n). As a by-product, we also relate cutwidth via the locality number to pathwidth, which is of independent interest, since it improves the best currently known approximation algorithm for cutwidth. In addition to these main results, we also consider the possibility of greedy-based approximation algorithms for the locality number
Counting Problems in Parameterized Complexity
This survey is an invitation to parameterized counting problems for readers with a background in parameterized algorithms and complexity. After an introduction to the peculiarities of counting complexity, we survey the parameterized approach to counting problems, with a focus on two topics of recent interest: Counting small patterns in large graphs, and counting perfect matchings and Hamiltonian cycles in well-structured graphs.
While this survey presupposes familiarity with parameterized algorithms and complexity, we aim at explaining all relevant notions from counting complexity in a self-contained way
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