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 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 that WSP restricted to the class of user-independent constraints (UI), covering many practical cases, admits FPT algorithms.
We show that the FPT nature of WSP with UI constraints decomposes the problem into two levels, and exploit this in a new FPT algorithm that is by many orders of magnitude faster then the previous state-of-the-art WSP algorithm.
The WSP with UI constraints can also be viewed as an extension of the hypergraph list colouring problem. 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 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.
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