We consider the design of semidefinite programming （SDP） based approximation algorithm for the problem Max Hypergraph Cut with Limited Unbalance （MHC-LU）： Find a partition of the vertices of a weighted hypergraph H = （V, E） into two subsets V1, V2 with ｜｜V2｜ - ｜1/1 ｜｜ ≤ u for some given u and maximizing the total weight of the edges meeting both V1 and V2. The problem MHC-LU generalizes several other combinatorial optimization problems including Max Cut, Max Cut with Limited Unbalance （MC-LU）, Max Set Splitting, Max Ek-Set Splitting and Max Hypergraph Bisection. By generalizing several earlier ideas, we present an SDP randomized approximation algorithm for MHC-LU with guaranteed worst-case performance ratios for various unbalance parameters τ = u/｜V｜. We also give the worst-case performance ratio of the SDP-algorithm for approximating MHC-LU regardless of the value of τ. Our strengthened SDP relaxation and rounding method improve a result of Ageev and Sviridenko （2000） on Max Hypergraph Bisection （MHC-LU with u = 0）, and results of Andersson and Engebretsen （1999）, Gaur and Krishnamurti （2001） and Zhang et al. （2004） on Max Set Splitting （MHC-LU with u = ｜V｜）. Furthermore, our new formula for the performance ratio by a tighter analysis compared with that in Galbiati and Maffioli （2007） is responsible for the improvement of a result of Galbiati and Maffioli （2007） on MC-LU for some range of τ.
