Efficient algorithms for the conjunction of UTVPI and congruence constraints

Determine whether efficient (low-degree polynomial-time) algorithms exist for constraint systems formed by conjuncting Unit Two Variable Per Inequality (UTVPI) constraints with modular congruence constraints, i.e., for domains that simultaneously enforce UTVPI inequalities and congruence equalities on integer variables.

Background

The paper reviews numerical abstract domains used in analysis and optimization, highlighting Difference Bound Matrices (DBM) and the octagon (UTVPI) domain for relational inequalities, alongside congruence constraints that capture modular arithmetic properties. Combining these two families of constraints is common in compiler analyses and transformations, especially in machine learning compilers where strides, tiling, and vector sizes induce modular structure while loop bounds and dependencies are captured by UTVPI.

Historically, while each domain admits efficient algorithms on its own, obtaining efficient algorithms when both types of constraints are present has been regarded as open. The paper introduces Strided DBMs (SDBM) as a specific combined domain, proves NP-hardness of satisfiability in general, and provides pseudo-polynomial algorithms and efficient results for the harmonic subclass (HSDBM), but the broader question of efficient algorithms for the general conjunction remains explicitly posed.