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We investigate issues of complexity related to welfare maximization in congestion games. In particular, we provide a full classification of complexity results for the problem of finding a minimum cost solution to a congestion game, under the model of Rosenthal. We consider both network and general congestion games, and we examine several variants of the problem concerning the structure of the game and the properties of its associated cost functions. Many of these problem variants turn out to be NP-hard, and some are hard to approximate to within any finite factor, unless P = NP. We also identify several versions of the problem that are solvable in polynomial time.

The integer equal flow problem is an NP-hard network flow problem, in which all arcs in given sets R1, . . . , R must carry equal flow. We show this problem is effectively inapproximable, even if the cardinality of each set R k is two. When is fixed, it is solvable in polynomial time.

Abstract. We describe the use of very large-scale neighborhood search (VLSN) techniques in examination timetabling problems. We detail three applications of VLSN algorithms that illustrate the versatility and potential of such algorithms in timetabling. The first of these uses cyclic exchange neighborhoods, in which an ordered subset of exams in disjoint time slots are swapped cyclically such that each exam moves to the time slot of the exam following it in the order. The neighborhood of all such cyclic exchanges may be searched effectively for an improving set of moves, making this technique computationally reasonable in practice. We next describe the idea of optimized crossover in genetic algorithms, where the parent solutions used in the genetic algorithm perform an optimization routine to produce the 'most fit' of their children under the crossover operation. This technique can be viewed as a form of multivariate large-scale neighborhood search, and it has been applied successfully in several areas outside timetabling. The final topic we discuss is functional annealing, which gives a method of incorporating neighborhood search techniques into simulated annealing algorithms. Under this technique, the objective function is perturbed slightly to avoid stopping at local optima. We conclude by encouraging the timetabling community to further examine the promising potential of these techniques in practice.

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