Polynomial Kernels and User Reductions for the Workflow Satisfiability Problem
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(\(\varGamma \)) when \(\varGamma \) denotes a finite or infinite set of allowed constraints. Our main result is a dichotomy for the case that all constraints in \(\varGamma \) are regular: (1) We are able to reduce the number \(n\) of users to \(n'\le k\). This entails a kernelization to size poly\((k)\) for finite \(\varGamma \), and, under mild technical conditions, to size poly\((k+m)\) for infinite \(\varGamma \), where \(m\) denotes the number of constraints. (2) Already WSP(\(R\)) for some \(R\in \varGamma \) allows no polynomial kernelization in \(k+m\) unless the polynomial hierarchy collapses.
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