A New Algorithm for Linear and Integer Feasibility in Horn Constraints

Abstract

In this paper, we detail a new algorithm for the problem of checking linear and integer feasibility of a system of Horn constraints. For certain special cases, the new algorithm is faster than the “Lifting Algorithm” described in [1]. Moreover, the new approach is based on different ideas and in fact exploits several properties of Horn constraint systems (HCS) which are not known to be part of the literature. In the case of constraints of bounded width (corresponding to “loosely coupled” systems), our algorithm can be modified to run in $O(n^3 + m \cdot n + \frac{m\cdot n^2}{\log (\max (m,n))})$ time. Our main result establishes that checking the feasibility of an HCS can be reduced to three subproblems: negative-cost cycle detection in networks (NCCD), matrix-vector multiplication (MV), and the conversion of an HCS to a non-redundant set of difference constraints (H2D). The MV and NCCD problems have been extremely well-studied, and specialized, fast algorithms exist for relevant special cases. We have identified a new problem, H2D, which warrants future research, since improved algorithms for H2D could be implemented in our algorithm to decrease the running time.

This research was supported in part by the National Science Foundation through Award CCF-0827397.