5,211 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
Polyhedral characteristics of balanced and unbalanced bipartite subgraph problems
We study the polyhedral properties of three problems of constructing an
optimal complete bipartite subgraph (a biclique) in a bipartite graph. In the
first problem we consider a balanced biclique with the same number of vertices
in both parts and arbitrary edge weights. In the other two problems we are
dealing with unbalanced subgraphs of maximum and minimum weight with
nonnegative edges. All three problems are established to be NP-hard. We study
the polytopes and the cone decompositions of these problems and their
1-skeletons. We describe the adjacency criterion in 1-skeleton of the polytope
of the balanced complete bipartite subgraph problem. The clique number of
1-skeleton is estimated from below by a superpolynomial function. For both
unbalanced biclique problems we establish the superpolynomial lower bounds on
the clique numbers of the graphs of nonnegative cone decompositions. These
values characterize the time complexity in a broad class of algorithms based on
linear comparisons
Hitting and Harvesting Pumpkins
The "c-pumpkin" is the graph with two vertices linked by c>0 parallel edges.
A c-pumpkin-model in a graph G is a pair A,B of disjoint subsets of vertices of
G, each inducing a connected subgraph of G, such that there are at least c
edges in G between A and B. We focus on covering and packing c-pumpkin-models
in a given graph: On the one hand, we provide an FPT algorithm running in time
2^O(k) n^O(1) deciding, for any fixed c>0, whether all c-pumpkin-models can be
covered by at most k vertices. This generalizes known single-exponential FPT
algorithms for Vertex Cover and Feedback Vertex Set, which correspond to the
cases c=1,2 respectively. On the other hand, we present a O(log
n)-approximation algorithm for both the problems of covering all
c-pumpkin-models with a smallest number of vertices, and packing a maximum
number of vertex-disjoint c-pumpkin-models.Comment: v2: several minor change
Mod/Resc Parsimony Inference
We address in this paper a new computational biology problem that aims at
understanding a mechanism that could potentially be used to genetically
manipulate natural insect populations infected by inherited, intra-cellular
parasitic bacteria. In this problem, that we denote by \textsc{Mod/Resc
Parsimony Inference}, we are given a boolean matrix and the goal is to find two
other boolean matrices with a minimum number of columns such that an
appropriately defined operation on these matrices gives back the input. We show
that this is formally equivalent to the \textsc{Bipartite Biclique Edge Cover}
problem and derive some complexity results for our problem using this
equivalence. We provide a new, fixed-parameter tractability approach for
solving both that slightly improves upon a previously published algorithm for
the \textsc{Bipartite Biclique Edge Cover}. Finally, we present experimental
results where we applied some of our techniques to a real-life data set.Comment: 11 pages, 3 figure
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