局部约束同态的数学框架
An Algorithmic Framework for Locally Constrained Homomorphisms
摘要：
从客图$G$到主图$H$的同态$f$是局部双射、内射或满射的，如果对每个$u\in V(G)$，$f$到$u$的邻域的限制分别是双射、内射或满射的.相应的决策问题，LBHOM，LIHOM和LSHOM，在一般的图和特殊的图类，得到了很好的研究。这三个问题的参数化复杂性除了用客图的树宽和最大度参数化后得到的复杂性结果外，还缺乏深入的研究。本文填补了这一空白：通过考虑客体图$G$的参数层次，我们证明了一些新的FPT，W[1]-hard和para-NP-complete结果.对于我们的FPT结果，我们通过开发一个新的算法框架来实现这一点，该框架涉及到一个通用的ILP模型。为了说明新框架的适用性，我们还用它来证明FPT结果的角色分配问题，这源于社会网络理论，是密切相关的局部满同态。
英文摘要：
A homomorphism $f$ from a guest graph $G$ to a host graph $H$ is locally bijective, injective or surjective if for every $u\in V(G)$, the restriction of $f$ to the neighbourhood of $u$ is bijective, injective or surjective, respectively. The corresponding decision problems, LBHOM, LIHOM and LSHOM, are well studied both on general graphs and on special graph classes. Apart from complexity results when the problems are parameterized by the treewidth and maximum degree of the guest graph, the three problems still lack a thorough study of their parameterized complexity. This paper fills this gap: we prove a number of new FPT, W[1]-hard and para-NP-complete results by considering a hierarchy of parameters of the guest graph $G$. For our FPT results, we do this through the development of a new algorithmic framework that involves a general ILP model. To illustrate the applicability of the new framework, we also use it to prove FPT results for the Role Assignment problem, which originates from social network theory and is closely related to locally surjective homomorphisms.
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