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Competitive Weighted Matching In Transversal Matroids
"... Consider a bipartite graph with a set of leftvertices and a set of rightvertices. All the edges adjacent to the same leftvertex have the same weight. We present an algorithm that, given the set of rightvertices and the number of leftvertices, processes a uniformly random permutation of the left ..."
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of the leftvertices, one leftvertex at a time. In processing a particular leftvertex, the algorithm either permanently matches the leftvertex to a thusfar unmatched rightvertex, or decides never to match the leftvertex. The weight of the matching returned by our algorithm is within a constant factor
Matroids
, 2009
"... One of the primary goals of pure mathematics is to identify common patterns that occur in disparate circumstances, and to create unifying abstractions which identify commonalities and provide a useful framework for further theorems. For example the pattern of an associative operation with inverses a ..."
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One of the primary goals of pure mathematics is to identify common patterns that occur in disparate circumstances, and to create unifying abstractions which identify commonalities and provide a useful framework for further theorems. For example the pattern of an associative operation with inverses and an identity occurs frequently, and gives rise to the notion of an abstract group. On top of the basic axioms of a group, a vast
Local Equivalence of Transversals in Matroids
, 1996
"... Given any system of n subsets in a matroid M , a transversal of this system is an ntuple of elements of M , one from each set, which is independent. Two transversals differing by exactly one element are adjacent, and two transversals connected by a sequence of adjacencies are locally equivalent, th ..."
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Given any system of n subsets in a matroid M , a transversal of this system is an ntuple of elements of M , one from each set, which is independent. Two transversals differing by exactly one element are adjacent, and two transversals connected by a sequence of adjacencies are locally equivalent
Matchings, Matroids and Unimodular Matrices
, 1995
"... We focus on combinatorial problems arising from symmetric and skewsymmetric matrices. For much of the thesis we consider properties concerning the principal submatrices. In particular, we are interested in the property that every nonsingular principal submatrix is unimodular; matrices having this p ..."
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Cited by 13 (1 self)
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this property are called principally unimodular. Principal unimodularity is a generalization of total unimodularity, and we generalize key polyhedral and matroidal results on total unimodularity. Highlights include a generalization of Hoffman and Kruskal's result on integral polyhedra, a generalization
Matroid matching with Dilworth truncation
"... Let H = (V, E) be a hypergraph and let k ≥ 1 and l ≥ 0 be fixed integers. Let M be the matroid with groundset E s.t. a set F ⊆ E is independent if and only if each X ⊆ V with kX  − l ≥ 0 spans at most kX  − l hyperedges of F. We prove that if H is dense enough, then M satisfies the double cir ..."
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circuit property, thus the minmax formula of Dress and Lovász on the maximum matroid matching holds for M. Our result implies the BergeTutte formula on the maximum matching of graphs (k = 1, l = 0), generalizes Lovász ’ graphic matroid (cycle matroid) matching formula to hypergraphs (k = l = 1
Transversal and cotransversal matroids via their representations
 ELECTRON J. COMBIN
, 2007
"... It is known that the duals of transversal matroids are precisely the strict gammoids. We show that, by representing these two families of matroids geometrically, one obtains a simple proof of their duality. ..."
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It is known that the duals of transversal matroids are precisely the strict gammoids. We show that, by representing these two families of matroids geometrically, one obtains a simple proof of their duality.
Matchings, Matroids and Submodular Functions
, 2008
"... This thesis focuses on three fundamental problems in combinatorial optimization: nonbipartite matching, matroid intersection, and submodular function minimization. We develop simple, efficient, randomized algorithms for the first two problems, and prove new lower bounds for the last two problems. F ..."
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This thesis focuses on three fundamental problems in combinatorial optimization: nonbipartite matching, matroid intersection, and submodular function minimization. We develop simple, efficient, randomized algorithms for the first two problems, and prove new lower bounds for the last two problems
Matroid Secretary for Regular and Decomposable Matroids
"... In the matroid secretary problem we are given a stream of elements and asked to choose a set of elements that maximizes the total value of the set, subject to being an independent set of a matroid given in advance. The difficulty comes from the assumption that decisions are irrevocable: if we choose ..."
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, and conjectured that every matroid admits an O(1)competitive algorithm. However, most matroids that are known to admit an O(1)competitive algorithm can be easily represented using graphs (e.g. graphic, cographic, and transversal matroids). In particular, there is very little known about Frepresentable matroids
Lectures on matroids and oriented matroids
, 2005
"... These lecture notes were prepared for the Algebraic Combinatorics; in Europe (ACE) Summer School in Vienna, July 2005. ..."
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These lecture notes were prepared for the Algebraic Combinatorics; in Europe (ACE) Summer School in Vienna, July 2005.
WEIGHTED JOINSEMILATTICES AND TRANSVERSAL MATROIDS
, 1974
"... We investigate joinsemilattices in which each element is assigned a nonnegative weight in a strictly increasing way. A joinsubsemilattice of a Boolean lattice is weighted by cardinality, and we give a characterization of these in terms of the notion of a spread. The collection of flats with no co ..."
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with no coloops (isthmuses) of a matroid or pregeometry, partially ordered by settheoretic inclusion, forms a joinsemilattice which is weighted by rank. For transversal matroids these joinsemilattices are isomorphic to joinsubsemilattices of Boolean lattices. Using a previously obtained characterization
Results 1  10
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307,035