工作流可满足性问题的多项式核与用户约简
Polynomial Kernels and User Reductions for the Workflow Satisfiability Problem
摘要：
工作流可满足性问题（WSP）是一个具有实际意义的问题，每当任务需要由授权用户执行时，都会受到业务规则定义的约束。我们需要决定是否存在一个计划--将任务分配给授权的用户--这样所有的约束都得到了满足。实际上，WSP是保守的约束满足问题（即，对于每个变量，这里称为任务，我们有一个一元授权约束），因此是NP完全的。Wang和Li（2010）观察到任务的数量k通常很小，因此可以用作参数，并且随后的几项工作研究了WSP关于参数k的参数化复杂度。我们更详细地研究了当Γ表示有限或无限的允许约束集时WSP（Γ）的核化复杂度。我们的主要结果是在所有约束都是正则的情况下的二分法：（1）我们能够将用户数n减少到n '<= k。这需要核化以确定有限Γ的poly（k）的大小，并且在温和的技术条件下，确定无限Γ的poly（k + m）的大小，其中m表示约束的数量。(2)对于某些R \in Γ，WSP（R）已经不允许k + m中的多项式核化，除非多项式层次崩溃。
英文摘要：
The Workflow Satisfiability Problem (WSP) is a problem of practical interest that arises whenever tasks need to be performed by authorized users, subject to constraints defined by business rules. We are required to decide whether there exists a plan -- an assignment of tasks to authorized users -- such that all constraints are satisfied. The WSP is, in fact, the conservative Constraint Satisfaction Problem (i.e., for each variable, here called task, we have a unary authorization constraint) and is, thus, NP-complete. It was observed by Wang and Li (2010) that the number k of tasks is often quite small and so can be used as a parameter, and several subsequent works have studied the parameterized complexity of WSP regarding parameter k. We take a more detailed look at the kernelization complexity of WSP(Γ) when Γ denotes a finite or infinite set of allowed constraints. Our main result is a dichotomy for the case that all constraints in Γ are regular: (1) We are able to reduce the number n of users to n' <= k. This entails a kernelization to size poly(k) for finite Γ, and, under mild technical conditions, to size poly(k+m) for infinite Γ, where m denotes the number of constraints. (2) Already WSP(R) for some R \in Γ allows no polynomial kernelization in k+m unless the polynomial hierarchy collapses.
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