We consider a variant of the generalized assignment problem (GAP) where the amount of space used in each bin is restricted to be either zero (if the bin is not opened) or above a given lower bound (a minimum quantity). We provide several complexity results for different versions of the problem and give polynomial time exact algorithms and approximation algorithms for restricted cases. For the most general version of the problem, we show that it does not admit a polynomial time approximation algorithm (unless P = NP), even for the case of a single bin. This motivates to study dual approximation algorithms that compute solutions violating the bin capacities and minimum quantities by a constant factor. When the number of bins is fixed and the minimum quantity of each bin is at least a factor δ > 1 larger than the largest size of an item in the bin, we show how to obtain a polynomial time dual approximation algorithm that computes a solution violating the minimum quantities and bin capacities by at most a factor 1 − 1 δ and 1 + 1 δ , respectively, and whose profit is at least as large as the profit of the best solution that satisfies the minimum quantities and bin capacities strictly. In particular, for δ = 2, we obtain a polynomial time (1, 2)-approximation algorithm.
We consider the traveling tournament problem, which is a well-known benchmark problem in tournament timetabling. It consists of designing a schedule for a sports league of n teams such that the total traveling costs of the teams are minimized. The most important variant of the traveling tournament problem imposes restrictions on the number of consecutive home games or away games a team may have. We consider the case where at most two consecutive home games or away games are allowed. We show that the well-known independent lower bound for this case cannot be reached and present two approximation algorithms for the problem. The first algorithm has an approximation ratio of 3/2 + 6 n−4 in the case that n/2 is odd, and of 3/2 + 5 n−1 in the case that n/2 is even. Furthermore, we show that this algorithm is applicable to real world problems as it yields close to optimal tournaments for many standard benchmark instances. The second algorithm we propose is only suitable for the case that n/2 is even and n ≥ 12, and achieves an approximation ratio of 1 + 16/n in this case, which makes it the first 1 + O(1/n)-approximation for the problem.Parts of this work appeared as an extended abstract in:
We determine the power of the weighted sum scalarization with respect to the computation of approximations for general multiobjective minimization and maximization problems. Additionally, we introduce a new multi-factor notion of approximation that is specifically tailored to the multiobjective case and its inherent trade-offs between different objectives. For minimization problems, we provide an efficient algorithm that computes an approximation of a multiobjective problem by using an exact or approximate algorithm for its weighted sum scalarization. In case that an exact algorithm for the weighted sum scalarization is used, this algorithm comes arbitrarily close to the best approximation quality that is obtainable by supported solutions – both with respect to the common notion of approximation and with respect to the new multi-factor notion. Moreover, the algorithm yields the currently best approximation results for several well-known multiobjective minimization problems. For maximization problems, however, we show that a polynomial approximation guarantee can, in general, not be obtained in more than one of the objective functions simultaneously by supported solutions.
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