711 research outputs found
On Tree-Constrained Matchings and Generalizations
We consider the following \textsc{Tree-Constrained Bipartite Matching} problem: Given two rooted trees , and a weight function , find a maximum weight matching between nodes of the two trees, such that none of the matched nodes is an ancestor of another matched node in either of the trees. This generalization of the classical bipartite matching problem appears, for example, in the computational analysis of live cell video data. We show that the problem is -hard and thus, unless , disprove a previous claim that it is solvable in polynomial time. Furthermore, we give a -approximation algorithm based on a combination of the local ratio technique and a careful use of the structure of basic feasible solutions of a natural LP-relaxation, which we also show to have an integrality gap of .
In the second part of the paper, we consider a natural generalization of the problem, where trees are replaced by partially ordered sets (posets). We show that the local ratio technique gives a -approximation for the -dimensional matching generalization of the problem, in which the maximum number of incomparable elements below (or above) any given element in each poset is bounded by . We finally give an almost matching integrality gap example, and an inapproximability result showing that the dependence on is most likely unavoidable
On Tree-Constrained Matchings and Generalizations
International audienceWe consider the following Tree-Constrained Bipartite Matching problem: Given a bipartite graph G=(V1,V2,E) with edge weights w:E↦ℝ+w:E↦R+, a rooted tree T1 on the set V1 and a rooted tree T2 on the set V1, find a maximum weight matching M in G, such that none of the matched nodes is an ancestor of another matched node in either of the trees. This generalization of the classical bipartite matching problem appears, for example, in the computational analysis of live cell video data. We show that the problem is APX-hard and thus, unless =P=NP, disprove a previous claim that it is solvable in polynomial time. Furthermore, we give a 2-approximation algorithm based on a combination of the local ratio technique and a careful use of the structure of basic feasible solutions of a natural LP-relaxation, which we also show to have an integrality gap of 2−o(1).In the second part of the paper, we consider a natural generalization of the problem, where trees are replaced by partially ordered sets (posets). We show that the local ratio technique gives a 2kρ-approximation for the k-dimensional matching generalization of the problem, in which the maximum number of incomparable elements below (or above) any given element in each poset is bounded by ρ. We finally give an almost matching integrality gap example, and an inapproximability result showing that the dependence on ρ is most likely unavoidable
The generalized Robinson-Foulds metric
The Robinson-Foulds (RF) metric is arguably the most widely used measure of
phylogenetic tree similarity, despite its well-known shortcomings: For example,
moving a single taxon in a tree can result in a tree that has maximum distance
to the original one; but the two trees are identical if we remove the single
taxon. To this end, we propose a natural extension of the RF metric that does
not simply count identical clades but instead, also takes similar clades into
consideration. In contrast to previous approaches, our model requires the
matching between clades to respect the structure of the two trees, a property
that the classical RF metric exhibits, too. We show that computing this
generalized RF metric is, unfortunately, NP-hard. We then present a simple
Integer Linear Program for its computation, and evaluate it by an
all-against-all comparison of 100 trees from a benchmark data set. We find that
matchings that respect the tree structure differ significantly from those that
do not, underlining the importance of this natural condition.Comment: Peer-reviewed and presented as part of the 13th Workshop on
Algorithms in Bioinformatics (WABI2013
How many matchings cover the nodes of a graph?
Given an undirected graph, are there matchings whose union covers all of
its nodes, that is, a matching--cover? A first, easy polynomial solution
from matroid union is possible, as already observed by Wang, Song and Yuan
(Mathematical Programming, 2014). However, it was not satisfactory neither from
the algorithmic viewpoint nor for proving graphic theorems, since the
corresponding matroid ignores the edges of the graph.
We prove here, simply and algorithmically: all nodes of a graph can be
covered with matchings if and only if for every stable set we have
. When , an exception occurs: this condition is not
enough to guarantee the existence of a matching--cover, that is, the
existence of a perfect matching, in this case Tutte's famous matching theorem
(J. London Math. Soc., 1947) provides the right `good' characterization. The
condition above then guarantees only that a perfect -matching exists, as
known from another theorem of Tutte (Proc. Amer. Math. Soc., 1953).
Some results are then deduced as consequences with surprisingly simple
proofs, using only the level of difficulty of bipartite matchings. We give some
generalizations, as well as a solution for minimization if the edge-weights are
non-negative, while the edge-cardinality maximization of matching--covers
turns out to be already NP-hard.
We have arrived at this problem as the line graph special case of a model
arising for manufacturing integrated circuits with the technology called
`Directed Self Assembly'.Comment: 10 page
Matchings in metric spaces, the dual problem and calibrations modulo 2
We show that for a metric space with an even number of points there is a
1-Lipschitz map to a tree-like space with the same matching number. This result
gives the first basic version of an unoriented Kantorovich duality. The study
of the duality gives a version of global calibrations for 1-chains with
coefficients in . Finally we extend the results to infinite metric
spaces and present a notion of "matching dimension" which arises naturally.Comment: We corrected some typos and clarified some of the notations and
formulations. The new version uses the New York Journal of Mathematics
templat
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A survey of induction algorithms for machine learning
Central to all systems for machine learning from examples is an induction algorithm. The purpose of the algorithm is to generalize from a finite set of training examples a description consistent with the examples seen, and, hopefully, with the potentially infinite set of examples not seen. This paper surveys four machine learning induction algorithms. The knowledge representation schemes and a PDL description of algorithm control are emphasized. System characteristics that are peculiar to a domain of application are de-emphasized. Finally, a comparative summary of the learning algorithms is presented
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