Limited-combinatorial sets

A convex combinatorial property of compact sets in the plane and its roots in lattice theory

K. Adaricheva and M. Bolat have recently proved that if $,mathcal U_0$ and $,mathcal U_1$ are circles in a triangle with vertices $A_0,A_1,A_2$, then there exist $jin {0,1,2}$ and $kin{0,1}$ such that $,mathcal U_{1-k}$ is included in the convex hull of $,mathcal U_kcup({A_0,A_1, A_2}setminus{A_j})$. One could say disks instead of circles.Here we prove the existence of such a $j$ and $k$ ...

Combinatorial geometry of point sets with collinearities

In this thesis we study various combinatorial problems relating to the geometry of point sets in the Euclidean plane. The unifying theme is that all the problems involve point sets that are not in general position, but have some collinearities. As well as giving rise to natural and interesting problems, the study of point sets with collinearities has important connections to other areas of math...

Symmetric Sets of Curves and Combinatorial Arrays

Let V be an algebraic curve. P is a set of points on V . C is a set of curves each of which intersects V at some points of P. We denote IP (C, V ) as the intersection multiplicity of C ∈ C with V at the point P ∈ P. If (V,P, C) satisfies the following conditions, we call it a symmetric set of curves: (1) for any point p ∈ P, the number of C having intersection multiplicity a is λa, (2) for any ...

Generating Realistic Data Sets for Combinatorial Auctions

Combinatorial optimization over two random point sets

We analyze combinatorial optimization problems over a random pair of points (X ,Y) in R of equal cardinal. Typical examples include the matching of minimal length, the traveling salesperson tour constrained to alternate between points of each set, or the connected bipartite r-regular graph of minimal length. As the cardinal of the sets goes to infinity, we investigate the convergence of such bi...