An Optimal Algorithm for Computing the Integer Closure of UTVPI Constraints

Abstract

In this paper, we study the problem of computing the lattice point closure of a conjunction of Unit Two Variable Per Inequality (UTVPI) constraints. We accomplish this by adapting Johnson’s all pairs shortest path algorithm to UTVPI constraints. This problem has been extremely well-studied in the literature, inasmuch as it arises in a number of applications, including but not limited to, program verification and operations research. The complexity of solving this problem has steadily improved over the past several decades with the fastest algorithm for this problem running in time \(O(n^3)\) on a UTVPI constraint system with n variables and m constraints. For the same input parameters, we detail an algorithm that runs in time \(O(m\cdot n +n^2 \cdot \log n)\). It is clear that our algorithm is superior to the state of the art when the constraint system is sparse (\(m \in O(n)\)), and no worse than the state of the art when the constraint system is dense (\(m \in \varTheta (n^2)\)). It is worth noting that our algorithm is time optimal in the following sense: The best known running time for computing the closure of a conjunction of difference constraints (m constraints, n variables) is \(O(m\cdot n +n^2 \cdot \log n)\), and UTVPI constraints subsume difference constraints.