### G-2015-61

# Constructive algorithms for the partial directed weighted improper coloring problem

## Alain Hertz, Romain Montagné, and François Gagnon

Given a complete directed graph `\(G\)`

with weights on the vertices and on the arcs, a `\(\theta\)`

-improper `\(k\)`

-coloring is an assignment of at most `\(k\)`

different
colors to the vertices of `\(G\)`

such that the weight of every vertex `\(v\)`

is greater, by a given factor `\(\frac{1}{\theta}\)`

, than the sum of the weights
on the arcs `\((u,v)\)`

entering `\(v\)`

with the tail `\(u\)`

of the same color as `\(v\)`

.
For a given real number `\(\theta\)`

and an integer `\(k\)`

, the Partial Directed Weigthed Improper Coloring Problem (PDWICP) is to
determine the order of the largest induced subgraph `\(G'\)`

of `\(G\)`

such that `\(G'\)`

admits a `\(\theta\)`

-improper `\(k\)`

-coloring. This problem is motivated by a practical channel assignment application
where the objective is to maximize the number of simultaneously communicating mobiles in a wireless network.
We consider three constructive algorithms for the standard vertex coloring problem, and adapt them to the PDWICP. We show that they

perform better than today's phone operator systems based on decentralized channel assignment strategies such as fractional frequency reuse.

Published **June 2015**
,
24 pages