Partition problems in high dimensional boxes

The Complexity of Combinatorial Optimization Problems on d-Dimensional Boxes

The Maximum Independent Set problem in d-box graphs, i.e., in intersection graphs of axis-parallel rectangles in Rd, is known to be NP-hard for any fixed d ≥ 2. A challenging open problem is, how close the solution can be approximated by a polynomial time algorithm. For the restricted case of d-boxes with bounded aspect ratio a PTAS exists [12]. In general case no polynomial time algorithm with...

Learning unions of high-dimensional boxes over the reals

In [4] an algorithm is presented that exactly learns (using membership queries and equivalence queries) several classes of unions of boxes in high dimension over finite discrete domains. The running time of the algorithm is polynomial in the logarithm of the size of the domain and other parameters of the target function (in particular, the dimension). We go one step further and present a PAC+MQ...

Coloring t-dimensional m-Boxes

Abstract. Call the set S1 × · · · × St t–dimensional m–box if |Si| = m for every i = 1, . . . , t. Let Rt(m, r) be the smallest integer R such that for every r–coloring of t–fold cartesian product of [R] one can find a monochromatic t–dimensional m–box. We give a lower and an upper bound for Rt(m, r). We also consider the discrepancy problem connected to this set-system. Among other bounds we p...

Learning Boxes in High Dimension

Open Problems in Partition Regularity

A finite or infinite matrix A with rational entries is called partition regular if whenever the natural numbers are finitely coloured there is a monochromatic vector x with Ax = 0. Many of the classical theorems of Ramsey Theory may naturally be interpreted as assertions that particular matrices are partition regular. While in the finite case partition regularity is well understood, very little...