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

  • Authors:

  • Harry B. Hunt, III;Madhav V. Marathe;Richard E. Stearns

  • Affiliations:

  • Univ. at Albany-State Univ. of New York, Albany;Univ. at Albany-State Univ. of New York, Albany;Univ. at Albany-State Univ. of New York, Albany

  • Venue:
  • Proceedings of the 2001 international symposium on Symbolic and algebraic computation
  • Year:
  • 2001

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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 ⪇ 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.