On the span in channel assignment problems: bounds, computing and counting

• Authors: Colin McDiarmid
• Venue: Discrete Mathematics - Special issue: The 18th British combinatorial conference
• Year: 2003

The channel assignment problem involves assigning radio channels to transmitters, using a small span of channels but without causing excessive interference. We consider a standard model for channel assignment, the constraint matrix model, which extends ideas of graph colouring. Given a graph G = (V,E) and a length l(uv) for each edge uv of G, we call an assignment φ : V → {1,...,t} feasible if |φ(u)-φ(v)| ≥ l(uv) for each edge uv. The least t for which there is a feasible assignment is the span of the problem. We first derive two bounds on the span, an upper bound (from a sequential assignment method) and a lower bound. We then see that an extension of the Gallai-Roy theorem on chromatic number and orientations shows that the span can be calculated in O(n!) steps for a graph with n nodes, neglecting a polynomial factor. We prove that, if the edge-lengths are bounded, then we may calculate the span in exponential time, that is, in time O(cn) for a constant c. Finally we consider counting feasible assignments and related quantities.