377 research outputs found
Valuations for matroid polytope subdivisions
We prove that the ranks of the subsets and the activities of the bases of a
matroid define valuations for the subdivisions of a matroid polytope into
smaller matroid polytopes.Comment: 19 pages. 2 figures; added section 6 + other correction
Simplicial and Cellular Trees
Much information about a graph can be obtained by studying its spanning
trees. On the other hand, a graph can be regarded as a 1-dimensional cell
complex, raising the question of developing a theory of trees in higher
dimension. As observed first by Bolker, Kalai and Adin, and more recently by
numerous authors, the fundamental topological properties of a tree --- namely
acyclicity and connectedness --- can be generalized to arbitrary dimension as
the vanishing of certain cellular homology groups. This point of view is
consistent with the matroid-theoretic approach to graphs, and yields
higher-dimensional analogues of classical enumerative results including
Cayley's formula and the matrix-tree theorem. A subtlety of the
higher-dimensional case is that enumeration must account for the possibility of
torsion homology in trees, which is always trivial for graphs. Cellular trees
are the starting point for further high-dimensional extensions of concepts from
algebraic graph theory including the critical group, cut and flow spaces, and
discrete dynamical systems such as the abelian sandpile model.Comment: 39 pages (including 5-page bibliography); 5 figures. Chapter for
forthcoming IMA volume "Recent Trends in Combinatorics
Matroids are Immune to Braess Paradox
The famous Braess paradox describes the following phenomenon: It might happen
that the improvement of resources, like building a new street within a
congested network, may in fact lead to larger costs for the players in an
equilibrium. In this paper we consider general nonatomic congestion games and
give a characterization of the maximal combinatorial property of strategy
spaces for which Braess paradox does not occur. In a nutshell, bases of
matroids are exactly this maximal structure. We prove our characterization by
two novel sensitivity results for convex separable optimization problems over
polymatroid base polyhedra which may be of independent interest.Comment: 21 page
Laminar Matroids
A laminar family is a collection of subsets of a set such
that, for any two intersecting sets, one is contained in the other. For a
capacity function on , let be \{I:|I\cap A|
\leq c(A)\text{ for all A\in\mathscr{A}}\}. Then is the
collection of independent sets of a (laminar) matroid on . We present a
method of compacting laminar presentations, characterize the class of laminar
matroids by their excluded minors, present a way to construct all laminar
matroids using basic operations, and compare the class of laminar matroids to
other well-known classes of matroids.Comment: 17 page
Differentially Private Decomposable Submodular Maximization
We study the problem of differentially private constrained maximization of
decomposable submodular functions. A submodular function is decomposable if it
takes the form of a sum of submodular functions. The special case of maximizing
a monotone, decomposable submodular function under cardinality constraints is
known as the Combinatorial Public Projects (CPP) problem [Papadimitriou et al.,
2008]. Previous work by Gupta et al. [2010] gave a differentially private
algorithm for the CPP problem. We extend this work by designing differentially
private algorithms for both monotone and non-monotone decomposable submodular
maximization under general matroid constraints, with competitive utility
guarantees. We complement our theoretical bounds with experiments demonstrating
empirical performance, which improves over the differentially private
algorithms for the general case of submodular maximization and is close to the
performance of non-private algorithms
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