An NP-Complete Problem in Grid Coloring

A c-coloring of G(n,m)=n x m is a mapping of G(n,m) into {1,...,c} such that no four corners forming a rectangle have the same color. In 2009 a challenge was proposed via the internet to find a 4-coloring of G(17,17). This attracted considerable attention from the popular mathematics community. A coloring was produced; however, finding it proved to be difficult. The question arises: is the problem of grid coloring is difficult in general? We present three results that support this conjecture, (1) an NP completeness result, (2) a lower bound on Tree-resolution, (3) a lower bound on Tree-CP proofs. Note that items (2) and (3) yield statements from Ramsey Theory which are of size polynomial in their parameters and require exponential size in various proof systems.Comment: 25 page

Classifying Problems into Complexity Classes

A fundamental problem in computer science is, stated informally: Given a problem, how hard is it?. We measure hardness by looking at the following question: Given a set A whats is the fastest algorithm to determine if “x ∈ A? ” We measure the speed of an algorithm by how long it takes to run on inputs of length n, as a function of n. For example, sorting a list of length n can be done in roughly n log n steps. Obtaining a fast algorithm is only half of the problem. Can you prove that there is no better algorithm? This is notoriously difficult; however, we can classify problems into complexity classes where those in the same class are roughly equally hard. In this chapter we define many complexity classes and describing natural problems that are in them. Our classes go all the way from regular languages to various shades of undecidable. We then summarize all that is known about these classes.