Hybridizing Carousel Greedy and Kernel Search: A new approach for the maximum flow problem with conflict constraints

This work addresses a variant of the maximum flow problem where specific pairs of arcs are not allowed to carry positive flow simultaneously. Such restrictions are known in the literature as negative disjunctive constraints or conflict constraints . The problem is known to be strongly NP-hard and several exact approaches have been proposed in the literature. In this paper, we present a heuristic algorithm for the problem, based on two different approaches: Carousel Greedy and Kernel Search. These two approaches are merged to obtain a fast and effective matheuristic, named Kernousel. In particular, the computational results reveal that exploiting the information gathered by the Carousel Greedy to build the set of most promising variables (the kernel set ), makes the Kernel Search more effective. To validate the performance of the new hybrid method, we compare it with the two components running individually. Results are also evaluated against the best-known solutions available in the literature for the problem. The new hybrid method provides 15 new best-known values on benchmark instances. • We address a variant of the maximum flow problem involving conflicting arc pairs. • We present Kernousel, a hybrid method based on Carousel Greedy and Kernel Search. • We show that the hybrid method outperforms the two components running individually. • We evaluate the performance of our algorithms against state-of-the-art solutions. • We conduct an instance space analysis to assess the effectiveness of our method.