Polynomial Kernels and User Reductions for the Workflow Satisfiability Problem

TL;DR

This paper investigates the kernelization complexity of the Workflow Satisfiability Problem (WSP), providing a dichotomy for regular constraints that includes polynomial kernelization results and hardness proofs.

Contribution

It establishes a clear dichotomy for the kernelization of WSP with regular constraints, showing when polynomial kernels are possible and when they are unlikely.

Findings

Polynomial kernelization reduces user count to at most k for regular constraints.

Finite constraint sets allow polynomial kernels of size poly(k).

Some constraints prevent polynomial kernelization unless the polynomial hierarchy collapses.

Abstract

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(\Gamma) when \Gamma\ denotes a finite or infinite set of allowed constraints. Our main result is a dichotomy for the case that all constraints in \Gamma\ 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 \Gamma, and, under mild technical conditions, to size poly(k+m) for infinite \Gamma, where m denotes the number of constraints. (2) Already WSP(R) for some R \in \Gamma\ allows no polynomial kernelization in k+m unless the polynomial hierarchy collapses.