Problems of satisfiability and resiliency in workflows have been widely
studied in the last decade. Recent work has shown that many such problems may
be viewed as special cases of the authorization policy existence problem
(APEP), which returns an authorization policy if one exists and ‘No’ otherwise.
A solution may not exist because of the restrictions imposed by the base
authorization relation and constraints that form part of the input to APEP.
However, in many practical settings it would be more useful to obtain a ‘least
bad’ policy than just a ‘No’, where ‘least bad’ is characterized by some
numerical value associated with the policy indicating the extent to which the
policy violates the base authorization relation and constraints. Accordingly,
we introduce the Valued APEP, which returns an authorization policy of minimum
weight, where the (non-negative) weight is determined by the constraints
violated by the returned solution (and is 0 if all constraints are satisfied).
We then establish a number of results concerning the parameterized complexity
of Valued APEP. We prove that the problem is fixed-parameter tractable if the
set of constraints satisfies two restrictions, but is intractable if only one
of these restrictions holds. (Most constraints known to be of practical use
satisfy these restrictions.) We introduce the notion of a user profile for a
weighted constraint, which enables us to prove a powerful result, a corollary
of which improves on known complexity results for APEP. Finally, we consider
Valued APEP when restricted to particular sub-classes of constraints and show
that instances of such problems can be reduced to the Valued WSP, enabling us
to exploit known algorithms to solve these particular instances.

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