Graph partitioning is a key fundamental problem in the area of big graph computation. In this paper, motivated by optimizing the big data computing applications, two novel variants of the classical balanced graph partitioning are studied. The first one is to optimize the performance of specific workloads by graph partitioning, and the second one is to optimize the computation of motifs by graph partitioning. The formal definitions of the above two problems are introduced first, and the semidefinite programming representations are designed based on analyzing their properties. For the motif based partitioning problem, it is proved to be \(\textsf {NP}\) -complete even for the special case of \(k=2\) and the motif is a triangle, and its inapproximability is also shown by proving that there are no efficient algorithms with finite approximation ratio. Finally, by solving the semidefinite programs and utilizing sophisticated rounding techniques, the bi-criteria \(O(\sqrt{\log n\log k})\) -approximation algorithms with polynomial time cost are designed and analyzed for them.