403 research outputs found
On distinct distances in homogeneous sets in the Euclidean space
A homogeneous set of points in the -dimensional Euclidean space
determines at least distinct distances
for a constant . In three-space, we slightly improve our general bound
and show that a homogeneous set of points determines at least
distinct distances
Matching structure and bargaining outcomes in buyerâseller networks
We examine the relationship between the matching structure of a bipartite (buyer-seller) network and the (expected) shares of the unit surplus that each connected pair in this network can create. We show that in different bargaining environments, these shares are closely related to the Gallai-Edmonds Structure Theorem. This theorem characterizes the structure of maximum matchings in an undirected graph. We show that the relationship between the (expected) shares and the tructure Theorem is not an artefact of a particular bargaining mechanism or trade centralization. However, this relationship does not necessarily generalize to non-bipartite networks or to networks with heterogeneous link values
The Least-core and Nucleolus of Path Cooperative Games
Cooperative games provide an appropriate framework for fair and stable profit
distribution in multiagent systems. In this paper, we study the algorithmic
issues on path cooperative games that arise from the situations where some
commodity flows through a network. In these games, a coalition of edges or
vertices is successful if it enables a path from the source to the sink in the
network, and lose otherwise. Based on dual theory of linear programming and the
relationship with flow games, we provide the characterizations on the CS-core,
least-core and nucleolus of path cooperative games. Furthermore, we show that
the least-core and nucleolus are polynomially solvable for path cooperative
games defined on both directed and undirected network
The Szemeredi-Trotter Theorem in the Complex Plane
It is shown that points and lines in the complex Euclidean plane
determine point-line incidences. This
bound is the best possible, and it generalizes the celebrated theorem by
Szemer\'edi and Trotter about point-line incidences in the real Euclidean plane
.Comment: 24 pages, 5 figures, to appear in Combinatoric
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