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Technical Program TECHNICAL PROGRAM Wednesday, 9:00-10:30 Wednesday, 10:50-12:20 WA-01 WB-02 Wednesday, 9:00-10:30 - Fo1 Wednesday, 10:50-12:20 - Fo2 Opening Ceremony Mixed Integer Linear Programming Stream: Invited Presentations and Ceremonies Stream: Discrete and Combinatorial Optimization, Plenary session Graphs and Networks Chair: Marco Lübbecke Invited session Chair: Alexander Martin 1 - Sports Scheduling meets Business Analytics Michael Trick 1 - Error bounds for mixed integer linear optimization Faster computers and algorithms have transformed how sports sched- problems ules have been created in practice in a wide range of sports. Techniques Oliver Stein such as Combinatorial Benders Decomposition, Large Scale Neighbor- hood Search, and Brand-and-Price have greatly increased the range of We introduce computable a-priori and a-posteriori error bounds for op- sports leagues that can use operations research methods to create their timality and feasibility of a point generated as the rounding of an op- schedules. With this increase in computational and algorithmic power timal point of the LP relaxation of a mixed integer linear optimization comes the opportunity to create not just playable schedules but more problem. Treating the mesh size of integer vectors as a parameter al- profitable schedules. Using data mining and other predictive analytics lows us to study the effect of different ‘granularities’ in the discrete techniques, it is possible to model attendance and other revenue effects variables on the error bounds. Our analysis mainly bases on the con- of the schedule. Combining these models with advanced schedule cre- struction of a so-called grid relaxation retract. Relations to proximity ation approaches leads to schedules that can generate more revenue for results and the integer rounding property are highlighted. teams and leagues. These concepts are illustrated with experiences in professional and college sports leagues. 2 - Separation of Generic Cutting Planes in Branch-and- Price Using a Basis Jonas Timon Witt, Marco Lübbecke When reformulating a given mixed integer program by the use of clas- sical Dantzig-Wolfe decomposition, a subset of the constraints is par- tially convexified, which corresponds to implicitly adding all valid in- equalities for the associated integer hull. Since these inequalities are not known, a solution of the original linear programming (LP) relax- ation which is obtained by transferring an optimal basic solution of the reformulated LP relaxation is in general not basic. In order to obtain an optimal basic solution we would have to explicitly add valid in- equalities for the integer hull associated with the partially convexified constraints such that the considered solution becomes basic. Hence, cutting planes which are separated using a basis like Gomory mixed integer cuts or strong Chvatal-Gomory cuts are usually not directly ap- plicable when separating such a solution in the original problem. Nev- ertheless, we can use some crossover method in order to obtain a basic solution which is nearby the considered non-basic solution and sepa- rate this auxiliary solution by applying all separators including those using a basis. The generated cutting planes might not only cut off the auxiliary solution, but also the solution we originally wanted to sepa- rate. So far, this problem was only considered extensively by Range, who proposed the previously described approach including a particular crossover method to find such a nearby basic solution. We present a modified crossover method and extend this procedure by considering additional valid inequalities strengthening the original LP relaxation. Furthermore, we provide the first full implementation of a separator like this and tested it on instances of several problem classes. 3 - A Lagrangian Relaxation Algorithm for Modularity Maximization Problem Kotohumi Inaba, Yoichi Izunaga, Yoshitsugu Yamamoto The modularity proposed by Newman and Girvan is one of the most common measures when the nodes of a graph are grouped into commu- nities consisting of tightly connected nodes. Due to the NP-hardness of the problem, few exact algorithms have been proposed. Aloise et al. formulated the problem as a set partitioning problem, which has to take into account all, exponentially many, nonempty subsets of the node set, and makes it difficult to secure the computational resource when the number of nodes is large. Their algorithm is based on the linear programming relaxation, LP relaxation for short, and uses the column generation technique. Although it provides a tight upper bound of the optimal value, it can suffer a high degeneracy due to the set partitioning constraints. In this study, we propose an algorithm based on the La- grangian relaxation. We relax the set partitioning constraints and add them to the objective function as a penalty with Lagrangian multipli- ers, and obtain the Lagrangian relaxation problem with only the binary variable constraints. For a given Lagrangian multiplier vector, an opti- mal solution of the Lagrangian relaxation problem can be obtained by 1 WB-03 OR 2014 - Aachen checking the sign of coefficients, but it is hard to compute all the co- Our result complements the results obtained in the companion paper efficients of variables. Then we propose to use the column generation of Krumke and Thielen, where a nondeterministic polynomial time al- technique in order to alleviate the computational burden. Namely, we gorithm for the more general problem of deciding of strong imple- start the algorithm with a small number of variables and gradually add mentability via indirect mechanisms is given. This more general prob- variables as the computation goes on. We also propose some methods lem is expected to be NP-complete. to accelerate the convergence. WB-04 WB-03 Wednesday, 10:50-12:20 - Fo4 Wednesday, 10:50-12:20 - Fo3 Robust knapsack problems Computational Social Choice Stream: Robust and Stochastic Optimization Stream: Algorithmic Game Theory Invited session Invited session Chair: Marc Goerigk Chair: Stephan Westphal 1 - Packing a Knapsack of Unknown Capacity 1 - On the Discriminative Power of Tournament Solu- Yann Disser, Max Klimm, Nicole Megow, Sebastian Stiller tions We study the problem of packing a knapsack without knowing its ca- Hans Georg Seedig, Felix Brandt pacity. Whenever we attempt to pack an item that does not fit, the item is discarded; if the item fits, we have to include it in the packing. We Tournament solutions constitute an important class of social choice show that there is always a policy that packs a value within factor 2 of functions that only depend on the pairwise majority comparisons be- the optimum packing, irrespective of the actual capacity. If all items tween alternatives. Recent analytical results have shown that several have unit density, we achieve a factor equal to the golden ratio 1.618. concepts with appealing axiomatic properties such as the Banks set or Both factors are shown to be best possible. In fact, we obtain the above the minimal covering set tend to not discriminate at all when the tour- factors using packing policies that are universal in the sense that they naments are chosen from the uniform distribution. This is in sharp fix a particular order of the items and try to pack the items in this order, contrast to empirical studies which have found that real-world prefer- independent of the observations made while packing. We give efficient ence profiles often exhibit Condorcet winners, i.e. alternatives that all algorithms computing these policies. On the other hand, we show that, tournament solutions select as the unique winner. In this work, we aim for any alpha > 1, the problem of deciding whether a given universal to fill the gap between these extremes by examining the distribution of policy achieves a factor of alpha is coNP-complete. If alphe is part of the number of alternatives returned by common tournament solutions the input, the same problem is shown to be coNP-complete for items for empirical data as well as data generated according to stochastic with unit densities. Finally, we show that it is coNP-hard to decide, preference models such as impartial culture, impartial anonymous cul- for given alpha, whether a set of items admits a universal policy with ture, Mallows mixtures, spatial models, and Polya-Eggenberger urn factor alpha, even if all items have unit densities. models. 2 - Algorithms for the Recoverable Robust Knapsack 2 - Complexity of Strong Implementation of Social Problem Choice Functions in Dominant Strategies Christina Büsing, Sebastian Goderbauer, Arie Koster, Manuel Sven Krumke, Clemens Thielen Kutschka We consider the classical mechanism design problem of strongly im- In this talk we present a recoverable robust knapsack problem, where plementing social choice functions in a setting where monetary trans- the uncertainty of the item weights follows the approach of Bertsimas fers are allowed. In contrast to weak implementation, where only one and Sim. In contrast to the classical robust setting, a limited recovery equilibrium of a mechanism needs to yield the desired outcomes given action is allowed, i.e., up to k items may be removed when the actual by the social choice function, strong implementation (also known as weights are known. We will introduce several algorithms based on a full implementation) means that a mechanism is sought in which all compact integer linear programming formulation, different robustness equilibria yield the desired outcomes. For strong implementation, one cuts and robust extended cover inequalities and compare their run-time cannot restrict attention to incentive compatible direct revelation mech- w.r.t. the recovery action and the scenario set. anisms via the Revelation Principle, so the question whether a given social choice function is strongly implementable cannot be answered 3 - The Robust Knapsack Problem with Queries as easily as for weak implementation. When considering Bayes Nash Marc Goerigk, Manoj Gupta, Jonas Ide, Anita Schöbel, equilibria, the Augmented Revelation Principle states that it suffices Sandeep Sen to consider mechanisms in which the set of types of each agent is a subset of the set of her possible bids.
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