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Polynomial Kernels and User Reductions for the Workflow Satisfiability Problem

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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, \(\mathsf {NP}\)-complete. It was observed by Wang and Li (ACM Trans Inf Syst Secur 13(4):40, 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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  1. The term partial kernel was first used in [4].

  2. Such reductions are of interest by themselves as some practical wsp algorithms iterate over users in search for a valid plan [5].


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Our research was partially supported by EPSRC Grant EP/K005162/1. We are grateful to the referees for their useful comments and suggestions.

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Correspondence to Stefan Kratsch.

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Gutin, G., Kratsch, S. & Wahlström, M. Polynomial Kernels and User Reductions for the Workflow Satisfiability Problem. Algorithmica 75, 383–402 (2016).

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