Pattern-Based Approach to the Workflow Satisfiability Problem with User-Independent Constraints

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 fixed parameter tractable (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 as they significantly extend the size of the problem that can be routinely solved. We give a new view of the FPT nature of the WSP with UI constraints, showing that it decomposes into two levels. Exploiting this two-level split, we develop a new FPT algorithm that is by many orders of magnitude faster then the previous state-of-the-art WSP algorithm, and it also has only polynomial space complexity whereas the old algorithm takes memory exponential in k, which limits its application. The WSP with UI constraints we consider can also be viewed as an extension of the hypergraph list colouring problem, and this influenced our research. In particular, inspired by a classic graph colouring method called Zykov’s Contraction, we designed a new pseudo-boolean (PB) formulation of WSP with UI constraints that also exploits the two-level split of the problem. Our experiments showed that, in many cases, this formulation being solved with a general purpose PB solver demonstrated performance comparable to that of our bespoke FPT algorithm. This raises the potential of using general purpose solvers to tackle FPT problems efficiently. We also study the practical, average-case, performance of various algorithms to complement the overly-pessimistic worst-case analysis that is usually done in FPT studies. To support this we extend studies of phase transition phenomena in the understanding of the average computational effort needed to solve decision problems. We investigate, for the first time, the phase transition properties of the WSP, under a model for generation of random instances, and note that the methods of the phase transition study need to be adjusted to FPT problems. We also discuss questions specific to experimental methodology when dealing with FPT problems.