On Threshold BDDs and the Optimal Variable Ordering Problem

Abstract

Many combinatorial optimization problems can be formulated as 0/1 integer programs (0/1 IPs). The investigation of the structure of these problems raises the following tasks: count or enumerate the feasible solutions and find an optimal solution according to a given linear objective function. All these tasks can be accomplished using binary decision diagrams (BDDs), a very popular and effective datastructure in computational logics and hardware verification.

We present a novel approach for these tasks which consists of an output-sensitive algorithm for building a BDD for a linear constraint (a so-called threshold BDD) and a parallel AND operation on threshold BDDs. In particular our algorithm is capable of solving knapsack problems, subset sum problems and multidimensional knapsack problems.

BDDs are represented as a directed acyclic graph. The size of a BDD is the number of nodes of its graph. It heavily depends on the chosen variable ordering. Finding the optimal variable ordering is an NP-hard problem. We derive a 0/1 IP for finding an optimal variable ordering of a threshold BDD. This 0/1 IP formulation provides the basis for the computation of the variable ordering spectrum of a threshold function.

We introduce our new tool azove 2.0 as an enhancement to azove 1.1 which is a tool for counting and enumerating 0/1 points. Computational results on benchmarks from the literature show the strength of our new method.