  We study the optimal order (or sequence) of contracting a tensor network with
a minimal computational cost. We conclude 2 different versions of this optimal
sequence: that minimize the operation number (OMS) and that minimize the time
complexity (CMS). Existing results only shows that OMS is NP-hard, but no
conclusion on CMS problem. In this work, we firstly reduce CMS to CMS-0, which
is a sub-problem of CMS with no free indices. Then we prove that CMS is easier
than OMS, both in general and in tree cases. Last but not least, we prove that
CMS is still NP-hard. Based on our results, we have built up relationships of
hardness of different tensor network contraction problems.
