94 research outputs found
Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs
A bipartite graph is convex if the vertices in can be
linearly ordered such that for each vertex , the neighbors of are
consecutive in the ordering of . An induced matching of is a
matching such that no edge of connects endpoints of two different edges of
. We show that in a convex bipartite graph with vertices and
weighted edges, an induced matching of maximum total weight can be computed in
time. An unweighted convex bipartite graph has a representation of
size that records for each vertex the first and last neighbor
in the ordering of . Given such a compact representation, we compute an
induced matching of maximum cardinality in time.
In convex bipartite graphs, maximum-cardinality induced matchings are dual to
minimum chain covers. A chain cover is a covering of the edge set by chain
subgraphs, that is, subgraphs that do not contain induced matchings of more
than one edge. Given a compact representation, we compute a representation of a
minimum chain cover in time. If no compact representation is given, the
cover can be computed in time.
All of our algorithms achieve optimal running time for the respective problem
and model. Previous algorithms considered only the unweighted case, and the
best algorithm for computing a maximum-cardinality induced matching or a
minimum chain cover in a convex bipartite graph had a running time of
On the approximability of the maximum induced matching problem
In this paper we consider the approximability of the maximum induced matching problem (MIM). We give an approximation algorithm with asymptotic performance ratio <i>d</i>-1 for MIM in <i>d</i>-regular graphs, for each <i>d</i>≥3. We also prove that MIM is APX-complete in <i>d</i>-regular graphs, for each <i>d</i>≥3
On Minimum Maximal Distance-k Matchings
We study the computational complexity of several problems connected with
finding a maximal distance- matching of minimum cardinality or minimum
weight in a given graph. We introduce the class of -equimatchable graphs
which is an edge analogue of -equipackable graphs. We prove that the
recognition of -equimatchable graphs is co-NP-complete for any fixed . We provide a simple characterization for the class of strongly chordal
graphs with equal -packing and -domination numbers. We also prove that
for any fixed integer the problem of finding a minimum weight
maximal distance- matching and the problem of finding a minimum weight
-independent dominating set cannot be approximated in polynomial
time in chordal graphs within a factor of unless
, where is a fixed constant (thereby
improving the NP-hardness result of Chang for the independent domination case).
Finally, we show the NP-hardness of the minimum maximal induced matching and
independent dominating set problems in large-girth planar graphs.Comment: 15 pages, 4 figure
On the Parameterized Complexity of the Acyclic Matching Problem
A matching is a set of edges in a graph with no common endpoint. A matching M
is called acyclic if the induced subgraph on the endpoints of the edges in M is
acyclic. Given a graph G and an integer k, Acyclic Matching Problem seeks for
an acyclic matching of size k in G. The problem is known to be NP-complete. In
this paper, we investigate the complexity of the problem in different aspects.
First, we prove that the problem remains NP-complete for the class of planar
bipartite graphs of maximum degree three and arbitrarily large girth. Also, the
problem remains NP-complete for the class of planar line graphs with maximum
degree four. Moreover, we study the parameterized complexity of the problem. In
particular, we prove that the problem is W[1]-hard on bipartite graphs with
respect to the parameter k. On the other hand, the problem is fixed parameter
tractable with respect to the parameters tw and (k, c4), where tw and c4 are
the treewidth and the number of cycles with length 4 of the input graph. We
also prove that the problem is fixed parameter tractable with respect to the
parameter k for the line graphs and every proper minor-closed class of graphs
(including planar graphs)
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