关于群与色同构问题
On the Group and Color Isomorphism Problems
摘要：
本文证明了群的复杂性与色同构问题之间的关系。色同构问题的难度与所涉及的置换群的组成因子密切相关。以前的工作主要关注的是将颜色同构应用于有界度图同构，因此集中在交替的连通因子，因为这些是图同构的情况下的瓶颈。我们考虑的颜色同构问题的组成因素限制到那些以外的交替组，表明组同构减少在n^（O（log log n））时间到这个问题，相反，这种颜色同构问题的一个特殊情况减少到轻微的推广组同构。然后，我们锐化我们的结果，通过确定的投影特殊线性群的主要障碍，快速算法组同构，并证明上述reduction从组同构到颜色同构实际上只产生循环和投影特殊线性因子。我们的研究结果表明，就像交替群是30年来图同构快速算法的障碍一样，特殊线性群也是群同构快速算法的障碍。
英文摘要：
In this paper, we prove results on the relationship between the complexity of the group and color isomorphism problems. The difficulty of color isomorphism problems is known to be closely linked to the the composition factors of the permutation group involved. Previous works are primarily concerned with applying color isomorphism to bou nded degree graph isomorphism, and have therefore focused on the alternating composit ion factors, since those are the bottleneck in the case of graph isomorphism. We consider the color isomorphism problem with composition factors restricted to those other than the alternating group, show that group isomorphism reduces in n^(O(log log n)) time to this problem, and, conversely, that a special case of this color isomorphism problem reduces to a slight generalization of group isomorphism. We then sharpen our results by identifying the projective special linear group as the main obstacle to faster algorithms for group isomorphism and prove that the aforementioned reduc tion from group isomorphism to color isomorphism in fact produces only cyclic and projective special linear factors. Our results demonstrate that, just as the alternatin g group was a barrier to faster algorithms for graph isomorphism for three decades, the projective special linear group is an obstacle to faster algorithms for group isomorphism.
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