Finding Pareto-optimal solutions is a basic approach in multi-objective combinatorial optimization. In this paper, we focus on the 0-1 multi-objective knapsack problem, and present an algorithm to enumerate all its Pareto-optimal solutions, which improves upon the method proposed by Bazgan et al. Our algorithm is based on dynamic programming techniques using an efficient data structure called zero-suppressed binary decision diagram (ZDD), which handles a set of combinations compactly. In our algorithm, we utilize ZDDs for storing all the feasible solutions compactly, and pruning inessential partial solutions as quickly as possible. As an output of the algorithm, we can obtain a useful ZDD indexing all the Pareto-optimal solutions. The results of our experiments show that our algorithm is faster than the previous method for various types of three- and four-objective instances, which are difficult problems to solve.

Discrete structure manipulation is a fundamental technique for many problems solved by computers. BDDs/ZDDs have attracted a great deal of attention for twenty years, because those data structures are useful to efficiently manipulate basic discrete structures such as logic functions and sets of combinations. Recently, one of the most interesting research topics related to BDDs/ZDDs is Frontier-based search method, a very efficient algorithm for enumerating and indexing the subsets of a graph to satisfy a given constraint. This work is important because many kinds of practical problems can be efficiently solved by some variations of this algorithm. In this article, we present recent research activity related to BDD and ZDD. We first briefly explain the basic techniques for BDD/ZDD manipulation, and then we present several examples of the state-of-the-art algorithms to show the power of enumeration.

Discrete structures are foundational material for computer science and mathematics, which are related to set theory, symbolic logic, inductive proof, graph theory, combinatorics, probability theory, etc. Many problems solved by computers can be decomposed into discrete structures using simple primitive algebraic operations. It is very important to represent discrete structures compactly and to execute efficiently tasks such as equivalency/validity checking, analysis of models, and optimization. Recently, BDDs (Binary Decision Diagrams) and ZDDs (Zero-suppressed BDDs) have attracted a great deal of attention, because they efficiently represent and manipulate large-scale combinational logic data, which are the basic discrete structures in various fields of application. Although a quarter of a century has passed since Bryant's first idea, there are still a lot of interesting and exciting research topics related to BDD and ZDD. BDD/ZDD is based on in-memory data processing techniques, and it enjoys the advantage of using random access memory. Recent commodity PCs are equipped with gigabytes of main memory, and we can now solve large-scale problems which used to be impossible due to memory shortage. Thus, especially since 2000, the scope of BDD/ZDD methods has increased. This survey paper describes the history of, and recent research activity pertaining to, techniques related to BDD and ZDD.