Branch and bound on the network model (Q5941065)

Karp and Zhang developed a general randomized parallel algorithm for solving branch and bound problems. They showed that with high probability their algorithm attained optimal speedup within a constant factor (for \(p< n/(\log n)^{c}\), where \(p\) is the number of processors, \(n\) is the ``size'' of the problem, and \(c\) is a constant). Ranade later simplified the analysis and obtained a better processor bound. Karp and Zhang's algorithm works on models of computation where communication cost is constant. The present paper considers the Branch and Bound problem on networks where the communication cost is high. Suppose sending a message in a \(p\) processor network takes \(G=O(\log p)\) time and node expansion (defined below) takes unit time (other operations being free). Then a simple randomized algorithm is presented which is, asymptotically, nearly optimal for \(p= O(2^{\log^{c}n})\), where \(c\) is any constant \(<1/3\) and \(n\) is the number of nodes in the input tree with cost no greater than the cost of the optimal leaf in the tree.