928 research outputs found
Partitioning Perfect Graphs into Stars
The partition of graphs into "nice" subgraphs is a central algorithmic
problem with strong ties to matching theory. We study the partitioning of
undirected graphs into same-size stars, a problem known to be NP-complete even
for the case of stars on three vertices. We perform a thorough computational
complexity study of the problem on subclasses of perfect graphs and identify
several polynomial-time solvable cases, for example, on interval graphs and
bipartite permutation graphs, and also NP-complete cases, for example, on grid
graphs and chordal graphs.Comment: Manuscript accepted to Journal of Graph Theor
Popular Matchings in Complete Graphs
Our input is a complete graph on vertices where each vertex
has a strict ranking of all other vertices in . Our goal is to construct a
matching in that is popular. A matching is popular if does not lose
a head-to-head election against any matching , where each vertex casts a
vote for the matching in where it gets assigned a better partner.
The popular matching problem is to decide whether a popular matching exists or
not. The popular matching problem in is easy to solve for odd .
Surprisingly, the problem becomes NP-hard for even , as we show here.Comment: Appeared at FSTTCS 201
Log-space Algorithms for Paths and Matchings in k-trees
Reachability and shortest path problems are NL-complete for general graphs.
They are known to be in L for graphs of tree-width 2 [JT07]. However, for
graphs of tree-width larger than 2, no bound better than NL is known. In this
paper, we improve these bounds for k-trees, where k is a constant. In
particular, the main results of our paper are log-space algorithms for
reachability in directed k-trees, and for computation of shortest and longest
paths in directed acyclic k-trees.
Besides the path problems mentioned above, we also consider the problem of
deciding whether a k-tree has a perfect macthing (decision version), and if so,
finding a perfect match- ing (search version), and prove that these two
problems are L-complete. These problems are known to be in P and in RNC for
general graphs, and in SPL for planar bipartite graphs [DKR08].
Our results settle the complexity of these problems for the class of k-trees.
The results are also applicable for bounded tree-width graphs, when a
tree-decomposition is given as input. The technique central to our algorithms
is a careful implementation of divide-and-conquer approach in log-space, along
with some ideas from [JT07] and [LMR07].Comment: Accepted in STACS 201
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