寻找具有附加约束的小权同构是固定参数易处理的
Finding Small Weight Isomorphisms with Additional Constraints is Fixed-Parameter Tractable
摘要：
Lubiw证明了图同构的几个变体是NP完全的，其中的解需要满足某些附加约束[SICOMP 10，1981]。其中之一，所谓的同构与限制，是决定两个给定的图$X_1=(V,E_1)$和$X_2=(V,E_2)$和一个子集$R\subseteq V\times V$的禁止对是否有一个同构$π$从$X_1$到$X_2$，使$π(i)\neq j$的所有$(i,j)\in R$。我们证明，这个问题和它的几个推广实际上是在FPT：-决定是否有两个图之间的同构，移动k个顶点，并满足Lubiw式的约束的问题是在FPT，与k和大小的$R$作为参数。如果允许这样的约束的CNF，问题仍然存在于FPT中。因此，问题，以决定是否有一个同构，移动正是k个顶点是在FPT。这解决了我们在关于精确权重自同构的文章[STACS 2017]中留下的一个问题。- 当权重和复杂度不受限制时，找到满足Lubiw风格约束的CNF的同构可以在FPT中通过访问GI oracle来解决。- 检查复杂度为t的两个图之间是否存在同构$π$也是在以t为参数的FPT中，其中置换的复杂度是定义为最小数量t的Cayley测度，使得$π$可以表示为t个置换的乘积。- 我们考虑一个更一般的问题，其中图X的顶点集被划分为红色和蓝色，我们感兴趣的是一个自同构，稳定红色和蓝色和移动正好k个顶点在蓝色，其中k是参数。这个问题是由[Downey and Fellows 1999]引入的，我们证明了[STACS 2017]，即使红色内部的颜色类大小为4，它也是W[1]-hard的。现在，对于红色内部大小最多为3的颜色类，我们证明问题在FPT中。
英文摘要：
Lubiw showed that several variants of Graph Isomorphism are NP-complete, where the solutions are required to satisfy certain additional constraints [SICOMP 10, 1981]. One of these, called Isomorphism With Restrictions, is to decide for two given graphs $X_1=(V,E_1)$ and $X_2=(V,E_2)$ and a subset $R\subseteq V\times V$ of forbidden pairs whether there is an isomorphism $π$ from $X_1$ to $X_2$ such that $π(i)\neq j$ for all $(i,j)\in R$. We prove that this problem and several of its generalizations are in fact in FPT: - The problem of deciding whether there is an isomorphism between two graphs that moves k vertices and satisfies Lubiw-style constraints is in FPT, with k and the size of $R$ as parameters. The problem remains in FPT if a CNF of such constraints is allowed. It follows that the problem to decide whether there is an isomorphism that moves exactly k vertices is in FPT. This solves a question left open in our article on exact weight automorphisms [STACS 2017]. - When the weight and complexity are unrestricted, finding isomorphisms that satisfy a CNF of Lubiw-style constraints can be solved in FPT with access to a GI oracle. - Checking if there is an isomorphism $π$ between two graphs with complexity t is also in FPT with t as parameter, where the complexity of a permutation is the Cayley measure defined as the minimum number t such that $π$ can be expressed as a product of t transpositions. - We consider a more general problem in which the vertex set of a graph X is partitioned into Red and Blue, and we are interested in an automorphism that stabilizes Red and Blue and moves exactly k vertices in Blue, where k is the parameter. This problem was introduced by [Downey and Fellows 1999], and we showed [STACS 2017] that it is W[1]-hard even with color classes of size 4 inside Red. Now, for color classes of size at most 3 inside Red, we show the problem is in FPT.
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