An exact exponential branch-and-merge algorithm for the single machine total tardiness problem

This paper proposes an exact exponential algorithm for the single machine total tardiness problem. It exploits the structure of a basic branch-and-reduce framework based on the well known Lawler's decomposition property that solves the problem with worst-case complexity in time O⁎(3n) and polynomial space. The proposed algorithm, called branch-and-merge, is an improvement of the branch-and-reduce technique with the embedding of a node merging operation. Its time complexity converges to O⁎(2n) keeping the space complexity polynomial. This improves upon the best-known complexity result for this problem provided by dynamic programming across the subsets with O⁎(2n) worst-case time and space complexity. The branch-and-merge technique is likely to be generalized to other sequencing problems with similar decomposition properties.