Approximate solutions of HAPPYNET on cubic graphs
نویسندگان
چکیده
The HAPPYNET problem is defined as follows : Given a 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∈Γ(v) s(v)s(u)wuv ≥ 0. It is easy to see 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. Parberry et al. have shown that in the case of cubic graphs (i.e. of maximum degree 3) HAPPYNET is as difficult as for arbitrary graphs. A ρapproximate 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 ≥ 34 . ∗LAMSADE, Université Paris-Dauphine, 75775 Paris cedex 16, France. {giannako, olivier.pottie}@lamsade.dauphine.fr Cahiers du LAMSADE 1
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