We give a new FPT algorithm testing isomorphism of n -vertex graphs of tree-width k in time 2 kpolylog(k) n 3 , improving the FPT algorithm due to Lokshtanov, Pilipczuk, Pilipczuk, and Saurabh (FOCS 2014), which runs in time 2 O(k5 log k) n 5 . Based on an improved version of the isomorphism-invariant graph decomposition technique introduced by Lokshtanov et al., we prove restrictions on the structure of the automorphism groups of graphs of tree-width k . Our algorithm then makes heavy use of the group theoretic techniques introduced by Luks (JCSS 1982) in his isomorphism test for bounded degree graphs and Babai (STOC 2016) in his quasipolynomial isomorphism test. In fact, we even use Babai’s algorithm as a black box in one place. We also give a second algorithm that, at the price of a slightly worse running time 2 O(k2 log k) n 3 , avoids the use of Babai’s algorithm and, more importantly, has the additional benefit that it can also be used as a canonization algorithm.

中文翻译：

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有界树宽图的改进同构检验

我们给出了一种新的 FPT 算法测试同构性n- 树宽的顶点图ķ及时2 kpolylog(k) n3 ，由于 Lokshtanov、Pilipczuk、Pilipczuk 和 Saurabh (FOCS 2014) 改进了 FPT 算法，该算法在时间 2 中运行O(k5 日志 k) n 5. 基于Lokshtanov等人提出的同构不变图分解技术的改进版本，我们证明了对树宽图的自同构群结构的限制ķ. 然后，我们的算法大量使用 Luks (JCSS 1982) 在他的有界度图的同构测试和 Babai (STOC 2016) 在他的拟多项式同构测试中引入的群论技术。事实上，我们甚至在一个地方使用了 Babai 的算法作为一个黑匣子。我们还给出了第二种算法，其代价是运行时间稍差 2O(k2 log k) n 3，避免了使用 Babai 的算法，更重要的是，它还有一个额外的好处，即它也可以用作规范化算法。