Analyzing Lattice Point Feasibility in UTVPI Constraints

Abstract

This paper is concerned with the design and analysis of a time-optimal and space-optimal, certifying algorithm for checking the lattice point feasibility of a class of constraints called Unit Two Variable Per Inequality (UTVPI) constraints. A UTVPI constraint has at most two non-zero variables and the coefficients of the non-zero variables belong to the set $\{+1, -1\}$. These constraints occur in a number of application domains, including but not limited to program verification, abstract interpretation, and operations research. As per the literature, the fastest known model-generating algorithm for checking lattice point feasibility in UTVPI constraint systems runs in $O(m \cdot n+n^{2} \cdot \log n)$ time and $O(n^{2})$ space, where m represents the number of constraints and n represents the number of variables in the constraint system. In this paper, we design and analyze a new algorithm for checking the lattice point feasibility of UTVPI constraints. The presented algorithm runs in $O(m \cdot n)$ time and $O(m+n)$ space. Additionally, it is certifying in that it produces a satisfying assignment in the event that it is presented with feasible instances and a refutation in the event that it is presented with infeasible instance. Our approach for the lattice point feasibility problem in UTVPI constraint systems is fundamentally different from existing approaches for this problem.

K. Subramani—This work was supported by the Air Force Research Laboratory under US Air Force contract FA8750-16-3-6003. The views expressed are those of the authors and do not reflect the official policy or position of the Department of Defense or the U.S. Government.

P. Wojciechowski—This research was supported in part by the National Science Foundation through Award CCF-1305054.