Moritz Mühlenthaler

Felix Hommelsheim, Pia Jehmlich, Moritz Mühlenthaler

Given an edge-colored graph, the Maximum Rainbow Matching problem asks for a maximum-cardinality matching of the graph that contains at most one edge from each color. We provide the following complexity dichotomy for this problem based on the structure of the color classes: Maximum Rainbow Matching admits a polynomial-time algorithm if almost every color class is a complete multipartite graph and it is NP-hard otherwise. To prove the NP-hardness-part of the dichotomy, we first show that the problem remains NP-hard even if every color class is a subgraph on four vertices that is either a matching of size two, a path on four vertices or a paw. We then leverage this result to all color classes that are not complete multipartite graphs. For this purpose, we introduce color-closed graph classes, which seem to be an appropriate notion for obtaining complexity classifications for rainbow problems and may be of independent interest. To prove the positive part of the dichotomy, we show that the problem essentially reduces to computing a maximum $(l, u)$-matching, where we heavily exploit that almost all color classes are complete multipartite graphs. In the case where all color classes are complete multipartite, we provide a polynomial-time algorithm that computes a maximum matching containing at most $m_i$ edges from each color class $i$.

Moritz Mühlenthaler, Rolf Wanka

We propose a decomposition of the max-min fair curriculum-based course timetabling (MMF-CB-CTT) problem. The decomposition models the room assignment subproblem as a generalized lexicographic bottleneck optimization problem (LBOP). We show that the generalized LBOP can be solved efficiently if the corresponding sum optimization problem can be solved efficiently. As a consequence, the room assignment subproblem of the MMF-CB-CTT problem can be solved efficiently. We use this insight to improve a previously proposed heuristic algorithm for the MMF-CB-CTT problem. Our experimental results indicate that using the new decomposition improves the performance of the algorithm on most of the 21 ITC2007 test instances with respect to the quality of the best solution found. Furthermore, we introduce a measure of the quality of a solution to a max-min fair optimization problem. This measure helps to overcome some limitations imposed by the qualitative nature of max-min fairness and aids the statistical evaluation of the performance of randomized algorithms for such problems. We use this measure to show that using the new decomposition the algorithm outperforms the original one on most instances with respect to the average solution quality.

Moritz Mühlenthaler, Rolf Wanka

We consider the problem of creating fair course timetables in the setting of a university. Our motivation is to improve the overall satisfaction of individuals concerned (students, teachers, etc.) by providing a fair timetable to them. The central idea is that undesirable arrangements in the course timetable, i.e., violations of soft constraints, should be distributed in a fair way among the individuals. We propose two formulations for the fair course timetabling problem that are based on max-min fairness and Jain's fairness index, respectively. Furthermore, we present and experimentally evaluate an optimization algorithm based on simulated annealing for solving max-min fair course timetabling problems. The new contribution is concerned with measuring the energy difference between two timetables, i.e., how much worse a timetable is compared to another timetable with respect to max-min fairness. We introduce three different energy difference measures and evaluate their impact on the overall algorithm performance. The second proposed problem formulation focuses on the tradeoff between fairness and the total amount of soft constraint violations. Our experimental evaluation shows that the known best solutions to the ITC2007 curriculum-based course timetabling instances are quite fair with respect to Jain's fairness index. However, the experiments also show that the fairness can be improved further for only a rather small increase in the total amount of soft constraint violations.