Lower bounds and a nearly fastest general parallel branch-and-bound algorithm

In this paper, it is supposed that the B&B algorithm finds the first optimal solution after h nodes have been expanded and m active nodes have been created in the state-space tree. Then the lower bound ω(m + h log h) of the running time for the general sequential B&B algorithm and the lower bound ω(m/p + h log p) for the general parallel best-first B&B algorithm in PRAM-CREW are proposed, where p is the number of processors available. Moreover, the lower bound ω(M/p+ H + (H/p) log (H/p) is presented for the parallel algorithms on distributed memory system, where M and H represent total number of the active nodes and that of the expanded nodes processed by p processors, respectively. In addition, a nearly fastest general parallel best-first B&B algorithm is put forward. The parallel algorithm is the fastest one as
$p = \max\{h^{\epsilon}, r\}$
. where
$\epsilon = 1/\sqrt{\log h}$
, and r is the largest branch number of the nodes in the state-space tree.