On a Generalization of Horn Constraint Systems

Abstract

In this paper, we study linear constraint systems in which each constraint is a fractional Horn constraint. A constraint is fractional horn, if it can be written in the form: $\sum _{i=1}^{n}a_{i}\cdot x_{i}\ge c$, where the $a_{i}$ and c are integral, and at most one of the $a_{i}>0$ and all negative $a_{i}$ are equal to $-1$. A conjunction of fractional Horn constraints is called a Fractional Horn Systems (FHS). FHSs generalize a number of specialized constraint systems such as Difference Constraint Systems and Horn Constraint Systems. We show that the problem of checking linear feasibility in these systems is in P, whereas the problem of checking lattice point feasibility is NP-complete. We then study a sub-class of fractional horn systems called Binary fractional horn systems (BFHS) in which each constraint has at most two non-zero coefficients with at most one being positive. In this case, we show that the problem of lattice point feasibility is in P.

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

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.