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.

翻译：我们研究以最小计算成本收缩张量网络的最优顺序（或序列）。我们归纳出该最优顺序的两种不同版本：最小化运算次数（OMS）和最小化时间复杂度（CMS）。现有结果仅表明OMS是NP难的，但关于CMS问题尚无结论。在本工作中，我们首先将CMS归约为CMS-0（CMS中无自由指标的子问题）。随后证明无论是在一般情况还是树形情况下，CMS均比OMS更易处理。最后但同样重要的是，我们证明了CMS仍然是NP难的。基于上述结果，我们构建了不同张量网络收缩问题难度之间的关系体系。