Solving Some Instances of the 2-Color Problem

Abstract

Abstract. In the field of Discrete Tomography, the 2-color problem consists in determining a matrix whose elements are of two different types, starting from its horizontal and vertical projections. It is known that the one color problem has a polynomial time reconstruction algorithm, while, with k ≥ 2, the k-color problem is NP-complete. Thus, the 2-color problem constitutes an interesting example of a problem just in the frontier between hard and easy problems.In this paper we define a linear time algorithm to solve a set of its instances, where some values of the horizontal and vertical projections are constant, while the others are upper bounded by a positive number proportional to the dimension of the problem. Our algorithm relies on classical studies for the solution of the one color problem.

The 1-Color Problem and the Brylawski Model

“…This result was also proved by induction in [1] with the purpose to furnish a property that allows the decomposition of a two color problem of a certain family in several one color problems that always admit a solution.…”

“…Finally we show that all the computed vectors still form a lattice that inherits many properties of that of Brylawski. This approach reveals its usefulness each time the vectors of projections are not given as an input, but they have to be computed through the decomposition of a more complex problem, as in the n-color case. For example, in [1] the authors decompose some instances of 2-color into four 1-color 's ones that allow a solution to exist.…”