Solving elementary shortest-path problems as mixed-integer programs

Ibrahim et al. (Int Trans Oper Res 16:361–369, 2009) presented and analyzed two integer programming formulations for the elementary shortest-path problem (ESPP), which is known to be NP-hard if the underlying digraph contains negative cycles. In fact, the authors showed that a formulation based on multi-commodity flows possesses a significantly stronger LP relaxation than a formulation based on arc flow variables. Since the ESPP is essentially an integer problem, the contribution of our paper lies in extending this research by comparing the formulations with regard to the computation time and memory requirements required for their integer solution. Moreover, we assess the quality of the lower bounds provided by an integer relaxation of the multi-commodity flow formulation.