Weighted Improper Colouring

In this paper, we study a colouring problem motivated by a practical frequency assignment problem and up to our best knowledge new. In wireless networks, a node interferes with the other nodes the level of interference depending on numerous parameters: distance between the nodes, geographical topography, obstacles, etc. We model this with a weighted graph

G

where the weights on the edges represent the noise (interference) between the two end-nodes. The total interference in a node is then the sum of all the noises of the nodes emitting on the same frequency. A weighted

t

-improper

k

-colouring of

G

is a

k

-colouring of the nodes of

G

(assignment of

k

frequencies) such that the interference at each node does not exceed some threshold

t

. The Weighted Improper Colouring problem, that we consider here consists in determining the weighted

t

-improper chromatic number defined as the minimum integer

k

such that

G

admits a weighted

t

-improper

k

-colouring. We also consider the dual problem, denoted the Threshold Improper Colouring problem, where given a number

k

of colours (frequencies) we want to determine the minimum real

t

such that

G

admits a weighted

t

-improper

k

-colouring. We show that both problems are NP-hard and first present general upper bounds; in particular we show a generalisation of Lovász’s Theorem for the weighted

t

-improper chromatic number. We then show how to transform an instance of the Threshold Improper Colouring problem into another equivalent one where the weights are either 1 or

M

, for a sufficient big value

M

. Motivated by the original application, we study a special interference model on various grids (square, triangular, hexagonal) where a node produces a noise of intensity 1 for its neighbours and a noise of intensity 1/2 for the nodes that are at distance 2. Consequently, the problem consists of determining the weighted

t

-improper chromatic number when

G

is the square of a grid and the weights of the edges are 1, if their end nodes are adjacent in the grid, and 1/2 otherwise. Finally, we model the problem using linear integer programming, propose and test heuristic and exact Branch-and-Bound algorithms on random cell-like graphs, namely the Poisson-Voronoi tessellations.