The \emph{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 \emph{plan} -- an assignment of tasks to authorized users -- such that all constraints are satisfied. Several bespoke algorithms have been constructed for solving the WSP, optimised to deal with constraints (business rules) of particular types. It is natural to see the WSP as a subclass of the {\em Constraint Satisfaction Problem (CSP)} in which the variables are tasks and the domain is the set of users. What makes the WSP distinctive as a CSP is that we can assume that the number of tasks is very small compared to the number of users. This is in sharp contrast with traditional CSP models where the domain is small and the number of variables is very large. As such, it is appropriate to ask for which constraint languages the WSP is fixed-parameter tractable (FPT), parameterized by the number of tasks. We have identified a new FPT constraint language, user-independent constraint, that includes many of the constraints of interest in business processing systems. We are also able to prove that the union of FPT languages remains FPT if they satisfy a simple compatibility condition. In this paper we present our generic algorithm, in which plans are grouped into equivalence classes, each class being associated with a \emph{pattern}. We demonstrate that our generic algorithm has running time $O^*(2^{k\log k})$, where $k$ is the number of tasks, for the language of user-independent constraints. We also show that there is no algorithm of running time $O^*(2^{o(k\log k)})$ for user-independent constraints unless the Exponential Time Hypothesis fails

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The workflow satisfiability problem (WSP) asks whether there exists an assignment of authorized users to the steps in a workflow specification that satisfies the constraints in the specification. The problem is NP-hard in general, but several subclasses of the problem are known to be fixed-parameter tractable (FPT) when parameterized by the number of steps in the specification. In this paper, we consider the WSP with user-independent counting constraints, a large class of constraints for which the WSP is known to be FPT. We describe an efficient implementation of an FPT algorithm for solving this subclass of the WSP and an experimental evaluation of this algorithm. The algorithm iteratively generates all equivalence classes of possible partial solutions until, whenever possible, it finds a complete solution to the problem. We also provide a reduction from a WSP instance to a pseudo-Boolean SAT instance. We apply this reduction to the instances used in our experiments and solve the resulting PB SAT problems using SAT4J, a PB SAT solver. We compare the performance of our algorithm with that of SAT4J and discuss which of the two approaches would be more effective in practice.

The fixed parameter tractable (FPT) approach is a powerful tool in tackling computationally hard problems. In this paper, we link FPT results to classic artificial intelligence (AI) techniques to show how they complement each other. Specifically, we consider the workflow satisfiability problem (WSP) which asks whether there exists an assignment of authorised users to the steps in a workflow specification, subject to certain constraints on the assignment. It was shown by Cohen et al. (JAIR 2014) that WSP restricted to the class of user-independent constraints (UI), covering many practical cases, admits FPT algorithms, i.e. can be solved in time exponential only in the number of steps k and polynomial in the number of users n. Since usually k ≪ n in WSP, such FPT algorithms are of great practical interest.We present a new interpretation of the FPT nature of the WSP with UI constraints giving a decomposition of the problem into two levels. Exploiting this two-level split, we develop a new FPT algorithm that is by many orders of magnitude faster than the previous state-of-the-art WSP algorithm and also has only polynomial-space complexity. We also introduce new pseudo-Boolean (PB) and Constraint Satisfaction (CSP) formulations of the WSP with UI constraints which efficiently exploit this new decomposition of the problem and raise the novel issue of how to use general-purpose solvers to tackle FPT problems in a fashion that meets FPT efficiency expectations. In our computational study, we investigate, for the first time, the phase transition (PT) properties of the WSP, under a model for generation of random instances. We show how PT studies can be extended, in a novel fashion, to support empirical evaluation of scaling of FPT algorithms. 1 or cross-checking of work, etc. [4, 16, 47, 50]. Furthermore, different users have different capabilities and security permissions, and will generally not be authorised to process all of the steps. In the Workflow Satisfiability Problem (WSP), the aim is to assign authorised users to the steps in a workflow specification, subject to constraints arising from business rules and practices. (Note the term "workflow" originally arose from the flow of the steps between users, however, in this context, the time ordering is not relevant -the challenge is to make a feasible assignment for all the steps.) The WSP has important applications and has been extensively studied in the security research community [4,5,6,14,15,17,18,25,50].The WSP is NP-complete, and it has been difficult to solve, even for some moderately-sized instances [12,47]. Work in WSP has attempted to render solving of the WSP practical by finding a subclass of problems that admit fixed parameter tractable (FPT) algorithms; informally speaking, this means that there is a small parameter k such that the problem is exponential in k but polynomial in the size of the problem. In the case of the WSP, the parameter k is naturally the number of steps -in real-life instances this number is usually much smaller than the number n of u...