Hard multidimensional multiple choice knapsack problems, an empirical study

Abstract

Recent advances in algorithms for the multidimensional multiple choice knapsack problems have enabled us to solve rather large problem instances. However, these algorithms are evaluated with very limited benchmark instances. In this study, we propose new methods to systematically generate comprehensive benchmark instances. Some instances with special correlation properties between parameters are found to be several orders of magnitude harder than those currently used for benchmarking the algorithms. Experiments on an existing exact algorithm and two generic solvers show that instances whose weights are uncorrelated with the profits are easier compared with weakly or strongly correlated cases. Instances with classes containing similar set of profits for items and with weights strongly correlated to the profits are the hardest among all instance groups investigated. These hard instances deserve further study and understanding their properties may shed light to better algorithms.

A Trust Region Method for the Solution of the Surrogate Dual in Integer Programming

“…The results on the instances in Classes II and III which took longer than 100 s to solve in CPLEX are particularly noteworthy; the use of the surrogate dual bound appears to reduce the solution times by several orders of magnitude. While the heuristics presented in [13] perform better than CPLEX, in general, we note that the relative improvements from those heuristics are nowhere near what we obtain using the surrogate dual value in these experiments. more efficient to revisit previous values that are not yet resolved: at such values, some elements of the model function known to be supporting hyperplanes have already been found.…”

“…We only use problems for which the LP solution is not optimal, and refer to the remaining instances in each class as Class I (53 instances), Class II (50 instances), and Class III (47 instances), respectively. Set (B) consists of 13 MCMDK problems taken from [13]. Some of these problems remain unsolved.…”

“…Set (A) consists of 150 multiple choice multidimensional Knapsack problems (MCMDK) belonging to three of the hardest classes of problems presented in [13], specifically the G-L-D(SU), G-C(L)-S, and G-C(L)-D(S) instances (refer to Table 3, for example, in [13]). We only use problems for which the LP solution is not optimal, and refer to the remaining instances in each class as Class I (53 instances), Class II (50 instances), and Class III (47 instances), respectively.…”

Cross-layer QoE-driven admission control and resource allocation for adaptive multimedia services in LTE

“…In practise, however, only sessions that have increased their resource consumption since admission will be considered by the optimization problem. Since the utility and profit values, and thus the objective function, are calculated based on the configurations' bandwidth requirements, a correlation in the data set is present, thus making the resulting MMKP problem more difficult (Han et al, 2010). However, this is also expected to be the case in the real network.…”

A scalable GPU-based approach to accelerate the multiple-choice knapsack problem

“…In each set, we generated 5 instances and the running times were averaged over these 5 instances. The values and weights in the problem instances were randomly generated, with no correlation between the value and weight of an item [11], [5]. In all the instances, the number of choices in each class (n i ) was between 10 and 1024.…”