Combining a Parallel Branch-and-Bound Algorithm with a Strong Heuristic to Solve the Sequential Ordering Problem

In this paper, we describe how to combine a parallel branch-and-bound (B&B) algorithm and a strong heuristic to solve the Sequential Ordering Problem (SOP), which is an NP-hard optimization problem. A parallel B&B algorithm is run in parallel with the Lin-Kernighan-Helsgaun heuristic algorithm, which is known to be one of the strongest heuristic algorithms for solving the SOP. The best solutions found by each algorithm are shared with the other algorithm, and each algorithm benefits from the better solutions found by the other. With the better solutions found by B&B, LKH can find even better solutions. With the better solutions found by LKH, B&B will have a tighter upper bound that enables it to prune at shallower tree nodes and thus complete it search faster. The combined algorithm is evaluated experimentally on the SOPLIB and TSPLIB benchmarks. The results show that the combined algorithm gives significantly better performance than any of the B&B algorithm or the LKH heuristic individually. Significant improvements in both speed and solution quality are seen on both benchmark suites. For example, the proposed algorithm delivers a geometric-mean speedup of 10.17 relative to LKH on the medium-difficulty SOPLIB instances. On the hard SOPLIB instances, it improves the cost by up to 22% relative to B&B and up to 90% relative to LKH