The rise of new multicore CPUs introduced new challenges in the process of design of concurrent data structures: in addition to traditional requirements like correctness, linearizability and progress, the scalability is of paramount importance. It is a common opinion that these two demands are partially in conflict each others, so that in these computational environments it is necessary to relax the requirements on the traditional features of the data structures. In this paper we introduce a relaxed approach for the management of heap based priority queues on multicore CPUs, with the aim to realize a tradeoff between efficiency and sequential correctness. The approach is based on a sharing of information among only a small number of cores, so that to improve performance without completely losing the features of the data structure. The results obtained on a numerical algorithm show significant benefits in terms of parallel efficiency. © Springer International Publishing AG, part of Springer Nature 2018.