Clemens Thielen

Complexity of the traveling tournament problem

We consider the complexity of the traveling tournament problem, which is a well-known benchmark problem in tournament timetabling. The problem was supposed to be computationally hard ever since its proposal in 2001. Recently, the first NP-completeness proof has been given for the variant of the problem were no constraints on the number of consecutive home games or away games of a team are considered. The complexity of the original traveling tournament problem including these constraints, however, is still open. In this paper, we show that this variant of the problem is strongly NP-complete when the upper bound on the maximal number of consecutive away games is set to 3.

The generalized assignment problem with minimum quantities

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 1-1δ and 1+1δ, 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.

Interval scheduling on related machines

We consider the problem of scheduling n intervals (jobs with fixed starting times) on m machines with different speeds with the objective to maximize the number of accepted intervals. We prove that the offline version of the problem is strongly NP-hard to solve. For the online version, we show a lower bound of 5 3 on the competitive ratio of any deterministic online algorithm for the problem. Moreover, we present two simple greedy rules for online algorithms and show that any online algorithm using these rules is 2-competitive. One of these 2-competitive algorithms is shown to run in O ( n log m ) time. Additionally, we prove that our greedy rules impose no loss in the sense that every online algorithm for the problem can be modified to use the rules without reducing the number of accepted intervals on any instance.

Extensions to online delay management on a single train line: new bounds for delay minimization and profit maximization

We present extensions to the Online Delay Management Problem on a Single Train Line. While a train travels along the line, it learns at each station how many of the passengers wanting to board the train have a delay of δ. If the train does not wait for them, they get delayed even more since they have to wait for the next train. Otherwise, the train waits and those passengers who were on time are delayed by δ. The problem consists in deciding when to wait in order to minimize the total delay of all passengers on the train line. We provide an improved lower bound on the competitive ratio of any deterministic online algorithm solving the problem using game tree evaluation. For the extension of the original model to two possible passenger delays δ1 and δ2, we present a 3-competitive deterministic online algorithm. Moreover, we study an objective function modeling the refund system of the German national railway company, which pays passengers with a delay of at least Δ a part of their ticket price back. In this setting, the aim is to maximize the profit. We show that there cannot be a deterministic competitive online algorithm for this problem and present a 2-competitive randomized algorithm.

The online knapsack problem with incremental capacity

We consider an online knapsack problem with incremental capacity. In each time period, a set of items, each with a specific weight and value, is revealed and, without knowledge of future items, it has to be decided which of these items to accept. Additionally, the knapsack capacity is not fully available from the start but increases by a constant amount in each time period. The goal is to maximize the overall value of the accepted items. This setting extends the basic online knapsack problem by introducing a dynamic instead of a static knapsack capacity and is applicable to classic problems such as resource allocation or one-way trading. In contrast to the basic online knapsack problem, for which no competitive algorithms exist, the setting of incremental capacity facilitates the development of competitive algorithms for a bounded time horizon. We provide a competitive analysis of deterministic and randomized online algorithms for the online knapsack problem with incremental capacity and present lower bounds on the competitive ratio achievable by online algorithms for the problem. Most of these lower bounds match the competitive ratios achieved by our online algorithms exactly or differ only by a constant factor.

Budget-constrained minimum cost flows

We study an extension of the well-known minimum cost flow problem in which a second kind of costs (called usage fees) is associated with each edge. The goal is to minimize the first kind of costs as in traditional minimum cost flows while the total usage fee of a flow must additionally fulfill a budget constraint. We distinguish three variants of computing the usage fees. The continuous case, in which the usage fee incurred on an edge depends linearly on the flow on the edge, can be seen as the

On the complexity and approximability of budget-constrained minimum cost flows☆

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A Constant Factor Approximation for the Generalized Assignment Problem with Minimum Quantities and Unit Size Items

ε-constraint method applied to the bicriteria minimum cost flow problem. We present the first strongly polynomial-time algorithm for this problem. In the integral case, in which the fees are incurred in integral steps, we show weak