#### Date of Graduation

8-2011

#### Document Type

Thesis

#### Degree Name

Master of Science in Computer Science (MS)

#### Degree Level

Graduate

#### Department

Computer Science & Computer Engineering

#### Advisor

Wing-Ning Li

#### Committee Member

Gordon Beavers

#### Second Committee Member

Russell Deaton

#### Keywords

Communication complexity, Computational complexity, High-performance computing, Ramsey theory

#### Abstract

This thesis studies problems at the intersection of Ramsey-theoretic mathematics, computational complexity, and communication complexity. The prototypical example of such a problem is Monochromatic-Rectangle-Free Grid Coloring. In an instance of Monochromatic-Rectangle-Free Grid Coloring, we are given a chessboard-like grid graph of dimensions n and m, where the vertices of the graph correspond to squares in the chessboard, and a number of allowed colors, c. The goal is to assign one of the allowed colors to each vertex of the grid graph so that no four vertices arranged in an axis-parallel rectangle are colored monochromatically. Our results include: 1. A conditional, graph-theoretic proof that deciding Monochromatic-Rectangle-Free Grid Coloring requires time superpolynomial in the input size. 2. A natural interpretation of Monochromatic-Rectangle-Free Grid Coloring as a lower bound on the communication complexity of a cluster of related predicates. 3. Original, best-yet, monochromatic-square-free grid colorings: a 2-coloring of the 13 x 13 grid, and a 3-coloring of the 39 x 39 grid. 4. An empirically-validated computational plan to decide a particular instance of Monochromatic-Rectangle-Free Grid Coloring that has been heavily studied by the broader theory community, but remains unsolved: whether the 17 x 17 grid can be 4-colored without monochromatic rectangles. Our plan is based in high-performance computing and is expected to take one year to complete.

#### Recommended Citation

Apon, Daniel Christopher, "On the Complexity of Grid Coloring" (2011). *Theses and Dissertations*. 108.

https://scholarworks.uark.edu/etd/108