Given a graph G = (V, E), a satisfying bisection of G is a partition of the vertex set V into two sets V1, V2, such that |V1| = |V2|, and such that every vertex v [set membership, variant] V has at least as many neighbors in its own set as in the other set. The problem of deciding whether a graph G admits such a partition is -complete. In Bazgan et al. (2008) [C. Bazgan, Z. Tuza, D. Vanderpooten, Approximation of satisfactory bisection problems, Journal of Computer and System Sciences 75 (5) (2008) 875-883], the authors present a polynomial-time 3-approximation for maximizing the number of satisfied vertices in a bisection. It remained an open problem whether one could find a (3 - c)-approximation, for c > 0 (see Bazgan et al. (2010) [C. Bazgan, Z. Tuza, D. Vanderpooten, Satisfactory graph partition, variants, and generalizations, European Journal of Operational Research 206 (2) (2010) 271-280]). In this paper, we solve this problem by presenting a polynomial-time 2-approximation algorithm for the maximum number of satisfied vertices in a satisfying bisection.
