Interval-Memoized Backtracking on ZDDs for Fast Enumeration of All Lower Cost Solutions

In many real-life situations, we are often faced with combinatorial problems under linear cost functions. In this paper, we propose a fast method for exactly enumerating a very large number of all lower cost solutions for various combinatorial problems. Our method is based on backtracking for a given decision diagram which represents the feasible solutions of a problem. The main idea is to memoize the intervals of cost bounds to avoid duplicate search in the backtracking process. Although existing state-of-the-art methods with dynamic programming requires a pseudo-polynomial time with the total cost values, it may take an exponential time when the cost values become large. In contrast, the computation time of the proposed method does not directly depend on the total cost values, but is bounded by the input and output size of the decision diagrams. Therefore, it can be much faster than the existing methods if the cost values are large but the output of the decision diagrams are well-compressed. Our experimental results show that, for some practical instances of the Hamiltonian path problem, we succeeded in exactly enumerating billions of all lower cost solutions in a few seconds, which is hundred or more times faster than existing methods. Our method can be regarded as a novel search algorithm which integrates the two classical techniques, branch-and-bound and dynamic programming. This method would have many applications in various fields, including operations research, data mining, statistical testing, hardware/software system design, etc.