M. S. Ramanujan

Patrick O'Hara, M. S. Ramanujan, Theodoros Damoulas

We introduce the Constrained Least-cost Tour (CLT) problem: given an undirected graph with weight and cost functions on the edges, minimise the total cost of a tour rooted at a start vertex such that the total weight lies within a given range. CLT is related to the family of Travelling Salesman Problems with Profits, but differs by defining the weight function on edges instead of vertices, and by requiring the total weight to be within a range instead of being at least some quota. We prove CLT is $\mathcal{NP}$-hard, even in the simple case when the input graph is a path. We derive an informative lower bound by relaxing the integrality of edges and propose a heuristic motivated by this relaxation. For the case that requires the tour to be a simple cycle, we develop two heuristics which exploit Suurballe's algorithm to find low-cost, weight-feasible cycles. We demonstrate our algorithms by addressing a real-world problem that affects urban populations: finding routes that minimise air pollution exposure for walking, running and cycling in the city of London.

J. Crampton, R. Crowston, G. Gutin et al.

The workflow satisfiability problem is concerned with determining whether it is possible to find an allocation of authorized users to the steps in a workflow in such a way that all constraints are satisfied. The problem is NP-hard in general, but is known to be fixed-parameter tractable for certain classes of constraints. The known results on fixed-parameter tractability rely on the symmetry (in some sense) of the constraints. In this paper, we provide the first results that establish fixed-parameter tractability of the satisfiability problem when the constraints are asymmetric. In particular, we introduce the notion of seniority constraints, in which the execution of steps is determined, in part, by the relative seniority of the users that perform them. Our results require new techniques, which make use of tree decompositions of the graph of the binary relation defining the constraint. Finally, we establish a lower bound for the hardness of the workflow satisfiability problem.