Résumé en anglais

The HAPPYNET problem is defined as follows : Given an undirected simple graph G with integer weights wvu on its edges vu ∈ E(G), find a function s : V(G) → {-1, 1} such that ∀v ∈ V (G), v is happy in G, i.e. such that ∑u∈Γs(v)s(u) wvu ≥ 0. It is easy to see [3] that HAPPYNET has always a solution, no matter what the input is. However, no polynomial algorithm is known for this problem, which is complete for the class PLS (see [4] for a definition). Parberry et al. have shown in [7] that in the case of cubic graphs (i.e. of maximum degree 3) is as difficult as for arbitrary graphs. A ρ-approximate solution to a HAPPYNET instance of size n can be defined for 0 ≤ ρ ≤ 1 as a natural extension of the solution function, with at least ρn happy vertices. In this paper, we present a polynomial-time algorithm that finds a ρ-approximate solution for the HAPPYNET problem on cubic graphs, with ρ ≥ ¾.