78 research outputs found
Resource Competition on Integral Polymatroids
We study competitive resource allocation problems in which players distribute
their demands integrally on a set of resources subject to player-specific
submodular capacity constraints. Each player has to pay for each unit of demand
a cost that is a nondecreasing and convex function of the total allocation of
that resource. This general model of resource allocation generalizes both
singleton congestion games with integer-splittable demands and matroid
congestion games with player-specific costs. As our main result, we show that
in such general resource allocation problems a pure Nash equilibrium is
guaranteed to exist by giving a pseudo-polynomial algorithm computing a pure
Nash equilibrium.Comment: 17 page
Toric Ideals of Lattice Path Matroids and Polymatroids
We show that the toric ideal of a lattice path polymatroid is generated by
quadrics corresponding to symmetric exchanges, and give a monomial order under
which these quadrics form a Gr\"obner basis. We then obtain an analogous result
for lattice path matroids.Comment: 9 pages, 4 figure
Lattice polyhedra and submodular flows
Lattice polyhedra, as introduced by GrĂśflin and Hoffman, form a common framework for various discrete optimization problems. They are specified by a lattice structure on the underlying matrix satisfying certain sub- and supermodularity constraints. Lattice polyhedra provide one of the most general frameworks of total dual integral systems. So far no combinatorial algorithm has been found for the corresponding linear optimization problem. We show that the important class of lattice polyhedra in which the underlying lattice is of modular characteristic can be reduced to the EdmondsâGiles polyhedra. Thus, submodular flow algorithms can be applied to this class of lattice polyhedra. In contrast to a previous result of Schrijver, we do not explicitly require that the lattice is distributive. Moreover, our reduction is very simple in that it only uses an arbitrary maximal chain in the lattice
Polymatroid greedoids
AbstractThis paper discusses polymatroid greedoids, a superclass of them, called local poset greedoids, and their relations to other subclasses of greedoids. Polymatroid greedoids combine in a certain sense the different relaxation concepts of matroids as polymatroids and as greedoids. Some characterization results are given especially for local poset greedoids via excluded minors. General construction principles for intersection of matroids and polymatroid greedoids with shelling structures are given. Furthermore, relations among many subclasses of greedoids which are known so far, are demonstrated
A version of Tutte's polynomial for hypergraphs
Tutte's dichromate T(x,y) is a well known graph invariant. Using the original
definition in terms of internal and external activities as our point of
departure, we generalize the valuations T(x,1) and T(1,y) to hypergraphs. In
the definition, we associate activities to hypertrees, which are
generalizations of the indicator function of the edge set of a spanning tree.
We prove that hypertrees form a lattice polytope which is the set of bases in a
polymatroid. In fact, we extend our invariants to integer polymatroids as well.
We also examine hypergraphs that can be represented by planar bipartite graphs,
write their hypertree polytopes in the form of a determinant, and prove a
duality property that leads to an extension of Tutte's Tree Trinity Theorem.Comment: 49 page
An algebraic framework for the greedy algorithm with applications to the core and Weber set of cooperative games
An algebraic model generalizing submodular polytopes is presented, where modular functions on partially ordered sets take over the role of vectors in . This model unifies various generalizations of combinatorial models in which the greedy algorithm and the Monge algorithm are successful and generalizations of the notions of core and Weber set in cooperative game theory. As a further application, we show that an earlier model of ours as well as the algorithmic model of Queyranne, Spieksma and Tardella for the Monge algorithm can be treated within the framework of usual matroid theory (on unordered ground-sets), which permits also the efficient algorithmic solution of the intersection problem within this model. \u
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