Strongly-local reductions and the complexity/efficient approximability of algebra and optimization on abstract algebraic structures

Abstract

We demonstrate how the concepts of algebraic representability and strongly-local reductions developed here and in [20] can be used to characterize the computational complexity/efficient approximability of a number of basic problems and their variants, on various abstract algebraic structures F. These problems include the following: Algebra: Determine the solvability, unique solvability, number of solutions, etc., of a system of equations on F. Determine the equivalence of two formulas or straight-line programs on F.Optimization: Let ∈ > 0.Determine the maximum number of simultaneously satisfiable equations in a system of equations on F; or approximate this number within a multiplicative factor or n∈. Determine the maximum value of an objective function subject to satisfiable algebraically-expressed constraints on F; or approximate this maximum value within a multiplicative factor of n∈Given a formula or straight-line program, find a minimum size equivalent formula or straight-line program; or find an equivalent formula or straight-line program of size l f(minimum). Both finite and infinite algebraic structures are considered. These finite structures include all finite non-degenerate lattices and all finite rings or semi-rings with a nonzero element idempotent under multiplication (e.g. all non-degenerate finite unitary rings or semi-rings); and these infinite structures include the natural numbers, integers, real numbers, various algebras on these structures, all ordered rings, many cancellative semi-rings, and all infinite lattices with two elements a,b such that a is covered by b.