6,457 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 maximal chain subgraphs and covers of bipartite graphs
In this paper, we address three related problems. One is the enumeration of all the maximal edge induced chain subgraphs of a bipartite graph, for which we provide a polynomial delay algorithm. We give bounds on the number of maximal chain subgraphs for a bipartite graph and use them to establish the input-sensitive complexity of the enumeration problem.
The second problem we treat is the one of finding the minimum number of chain subgraphs needed to cover all the edges a bipartite graph. For this we provide an exact exponential algorithm with a non trivial complexity. Finally, we approach the problem of enumerating all minimal chain subgraph covers of a bipartite graph and show that it can be solved in quasi-polynomial time
Linear-Time Algorithms for Maximum-Weight Induced Matchings and Minimum Chain Covers in Convex Bipartite Graphs
A bipartite graph G=(U,V,E) is convex if the vertices in V can be linearly ordered such that for each vertex u∈U, the neighbors of u are consecutive in the ordering of V. An induced matching H of G is a matching for which no edge of E connects endpoints of two different edges of H. We show that in a convex bipartite graph with n vertices and m weighted edges, an induced matching of maximum total weight can be computed in O(n+m) time. An unweighted convex bipartite graph has a representation of size O(n) that records for each vertex u∈U the first and last neighbor in the ordering of V. Given such a compact representation, we compute an induced matching of maximum cardinality in O(n) 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 O(n) time. If no compact representation is given, the cover can be computed in O(n+m) time. All of our algorithms achieve optimal linear running time for the respective problem and model, and they improve and generalize the previous results in several ways: The best algorithms for the unweighted problem versions had a running time of O(n2) (Brandstädt et al. in Theor. Comput. Sci. 381(1–3):260–265, 2007. https://doi.org/10.1016/j.tcs.2007.04.006). The weighted case has not been considered before
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
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
Parameterized Edge Hamiltonicity
We study the parameterized complexity of the classical Edge Hamiltonian Path
problem and give several fixed-parameter tractability results. First, we settle
an open question of Demaine et al. by showing that Edge Hamiltonian Path is FPT
parameterized by vertex cover, and that it also admits a cubic kernel. We then
show fixed-parameter tractability even for a generalization of the problem to
arbitrary hypergraphs, parameterized by the size of a (supplied) hitting set.
We also consider the problem parameterized by treewidth or clique-width.
Surprisingly, we show that the problem is FPT for both of these standard
parameters, in contrast to its vertex version, which is W-hard for
clique-width. Our technique, which may be of independent interest, relies on a
structural characterization of clique-width in terms of treewidth and complete
bipartite subgraphs due to Gurski and Wanke
Decompositions into subgraphs of small diameter
We investigate decompositions of a graph into a small number of low diameter
subgraphs. Let P(n,\epsilon,d) be the smallest k such that every graph G=(V,E)
on n vertices has an edge partition E=E_0 \cup E_1 \cup ... \cup E_k such that
|E_0| \leq \epsilon n^2 and for all 1 \leq i \leq k the diameter of the
subgraph spanned by E_i is at most d. Using Szemer\'edi's regularity lemma,
Polcyn and Ruci\'nski showed that P(n,\epsilon,4) is bounded above by a
constant depending only \epsilon. This shows that every dense graph can be
partitioned into a small number of ``small worlds'' provided that few edges can
be ignored. Improving on their result, we determine P(n,\epsilon,d) within an
absolute constant factor, showing that P(n,\epsilon,2) = \Theta(n) is unbounded
for \epsilon
n^{-1/2} and P(n,\epsilon,4) = \Theta(1/\epsilon) for \epsilon > n^{-1}. We
also prove that if G has large minimum degree, all the edges of G can be
covered by a small number of low diameter subgraphs. Finally, we extend some of
these results to hypergraphs, improving earlier work of Polcyn, R\"odl,
Ruci\'nski, and Szemer\'edi.Comment: 18 page
- …