Reducing Mixed-Integer to Zero-One Programs

1. The 0-1 knapsack problem with a single continuous variable (KPC) is a natural extension of the binary knapsack problem (KP), where the capacity is not any longer fixed but can be extended which is expressed by a continuous variable. This variable might be unbounded or restricted by a lower or upper bound, respectively. This paper concerns techniques in order to reduce several variants of KPC to KP which enables us to employ approaches for KP. We propose both, an equivalent reformulation and a heuristic one bringing along less computational effort. We show that the heuristic reformulation can be customized in order to provide solutions having an objective value arbitrarily close to the one of the original problem. Obviously, this article has been developed together with Dirk Briskorn, where the workload has been split equally. First, we highlight special cases of the KPC, which result from different restrictive bounds on the continuous variable. Additionally, we show, that the KPC can always be optimized by solving at most three standard KPs. If therefore all coefficients are integers, each KP can be handled very efficient by the algorithm combo. This algorithm was developed by Martello et al. [8] and it is currently the state–of–the–art approach for standard KPs with integer coefficients, see Martello et al. [9]. But, the main importance of this paper for the whole thesis results from its fundamental behavior regarding the treatment of a MIP. Here, we present for the first time the replacement of a continuous variable by several binary ones. Since we limit the solution space by using binary instead of continuous variables, we only get an upper bound on the objective function value for a maximization problem. However, as already mentioned above, we are able to show, that this bound can be arbitrarily close to the optimal objective function value by adapting the IP-reformulation appropriate. Since the KPC is one of the basic MIPs, it is only built by the objective and one constraint, we used it as a starting point for transferring the developed idea of replacing a continuous variable to handle more complex MIPs. This is the content of our next articles. 2.2 Reducing the Elastic Generalized Assignment Problem to the Standard Generalized Assignment Problem Marcel Büther (2007): Reducing the Elastic Generalized Assignment Problem to the Standard Generalized Assignment Problem, Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel, No. 632 Abstract 2. The elastic generalized assignment problem (eGAP) is a natural extension of the generalized assignment problem (GAP) where the capacities are not fixed but