13 research outputs found
Using Neighborhood Diversity to Solve Hard Problems
Parameterized algorithms are a very useful tool for dealing with NP-hard
problems on graphs. Yet, to properly utilize parameterized algorithms it is
necessary to choose the right parameter based on the type of problem and
properties of the target graph class. Tree-width is an example of a very
successful graph parameter, however it cannot be used on dense graph classes
and there also exist problems which are hard even on graphs of bounded
tree-width. Such problems can be tackled by using vertex cover as a parameter,
however this places severe restrictions on admissible graph classes.
Michael Lampis has recently introduced neighborhood diversity, a new graph
parameter which generalizes vertex cover to dense graphs. Among other results,
he has shown that simple parameterized algorithms exist for a few problems on
graphs of bounded neighborhood diversity. Our article further studies this area
and provides new algorithms parameterized by neighborhood diversity for the
p-Vertex-Disjoint Paths, Graph Motif and Precoloring Extension problems -- the
latter two being hard even on graphs of bounded tree-width
Line graphs of bounded clique-width
AbstractWe show that a set of graphs has bounded tree-width or bounded path-width if and only if the corresponding set of line graphs has bounded clique-width or bounded linear clique-width, respectively. This relationship implies some interesting algorithmic properties and re-proves already known results in a very simple way. It also shows that the minimization problem for NLC-width is NP-complete
Induced Disjoint Paths in Circular-Arc Graphs in Linear Time
The Induced Disjoint Paths problem is to test whether a graph G with k distinct pairs of vertices (si,ti) contains paths P1,…,Pk such that Pi connects si and ti for i=1,…,k, and Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their ends) for 1≤
Clique-width: When Hard Does Not Mean Impossible
In recent years, the parameterized complexity approach has lead to the introduction of many new algorithms and frameworks on graphs and digraphs of bounded clique-width and, equivalently, rank-width. However, despite intensive work on the subject, there still exist well-established hard problems where neither a parameterized algorithm nor a theoretical obstacle to its existence are known. Our article is interested mainly in the digraph case, targeting the well-known Minimum Leaf Out-Branching (cf. also Minimum Leaf Spanning Tree) and Edge Disjoint Paths problems on digraphs of bounded clique-width with non-standard new approaches.
The first part of the article deals with the Minimum Leaf Out-Branching problem and introduces a novel XP-time algorithm wrt. clique-width. We remark that this problem is known to be W[2]-hard, and that our algorithm does not resemble any of the previously published attempts solving special cases of it such as the Hamiltonian Path. The second part then looks at the Edge Disjoint Paths problem (both on graphs and digraphs) from a different perspective -- rather surprisingly showing that this problem has a definition in the MSO_1 logic of graphs. The linear-time FPT algorithm wrt. clique-width then follows as a direct consequence
Algorithmic Aspects of Switch Cographs
This paper introduces the notion of involution module, the first
generalization of the modular decomposition of 2-structure which has a unique
linear-sized decomposition tree. We derive an O(n^2) decomposition algorithm
and we take advantage of the involution modular decomposition tree to state
several algorithmic results. Cographs are the graphs that are totally
decomposable w.r.t modular decomposition. In a similar way, we introduce the
class of switch cographs, the class of graphs that are totally decomposable
w.r.t involution modular decomposition. This class generalizes the class of
cographs and is exactly the class of (Bull, Gem, Co-Gem, C_5)-free graphs. We
use our new decomposition tool to design three practical algorithms for the
maximum cut, vertex cover and vertex separator problems. The complexity of
these problems was still unknown for this class of graphs. This paper also
improves the complexity of the maximum clique, the maximum independant set, the
chromatic number and the maximum clique cover problems by giving efficient
algorithms, thanks to the decomposition tree. Eventually, we show that this
class of graphs has Clique-Width at most 4 and that a Clique-Width expression
can be computed in linear time
Induced disjoint paths in circular-arc graphs in linear time
The Induced Disjoint Paths problem is to test whether an graph G on n vertices with k distinct pairs of vertices (si,ti) contains paths P1,…,Pk such that Pi connects si and ti for i=1,…,k, and Pi and Pj have neither common vertices nor adjacent vertices (except perhaps their ends) for 1≤