A Fast Algorithm for Optimal k-Set Selection Problems on A Decision Diagram

Abstract

Decision diagrams are data structures that can represent discrete structures in a compressed form. They support numerous queries and transformations, such as set operations and counting and retrieval, in polynomial time to the size of the decision diagrams. Since decision diagrams can exponentially compress targets at their best, there are a number of algorithms that have achieved significant speedups by processing compressed data in the decision diagram format. However, there has been no efficient algorithm for selecting k sets from decision diagrams to maximize the objective function that considers the relationship between sets. Many problems can be viewed within the framework of k-set selection. We propose an exponentially fast algorithm to solve k-set selection problems. In this paper, we clarify the conditions under which the objective function can be efficiently computed on decision diagrams and propose a new optimization algorithm for solving k-set selection problem by dynamic programming on a given decision diagram. Our method can solve problems in computation time that depends on the size of the given decision diagram when the objective function is decomposable into individual elements of a set. We implemented the proposed method and conducted computational experiments on multiple types of decision diagrams to confirm its performance.