Satisfiability of cross product terms is complete for real nondeterministic polytime Blum-Shub-Smale machines

Abstract

Nondeterministic polynomial-time Blum-Shub-Smale Machines over the reals give rise to a discrete complexity class between NP and PSPACE. Several problems, mostly from real algebraic geometry / polynomial systems, have been shown complete (under many-one reduction by polynomial-time Turing machines) for this class. We exhibit a new one based on questions about expressions built from cross products only. MotivationThe Millennium Question "P vs. NP" asks whether polynomial-time algorithms that may guess, and then verify, bits can be turned into deterministic ones. It arose from the Cook-Levin-Theorem asserting Boolean Satisfiability to be complete for NP; which initiated the identification of more and more other natural problems also complete [GaJo79].The Millennium Question is posed [Smal98] also for models able to guess objects more general than bits. More precisely a Blum-Shub-Smale (BSS) machine over a ring R may operate on elements from R within unit time. It induces the nondeterministic polynomial-time complexity class NP R ; for which the following problem FEAS R has been shown complete [BSS89, MAIN THEOREM]: Given a system of multivariate polynomials with 0s and ±1s as coefficients, does it admit a joint root from R ?BSS machines over R coincide with the real-RAM model from Computational Geometry [BKOS97] and underlie algorithms in Semialgebraic Geometry [Gius91, Lece00, BüSc09]. They give rise to a particularly rich structural complexity theory resembling the classical Turing Machine-based one -but often (unavoidably) with surprisingly different proofs [Bürg00,BaMe13]. It is known that NP ⊆ BP(NP 0 R ) ⊆ PSPACE holds [Grig88,Cann88,HRS90,Rene92]. FEAS R and FEAS 0 R are sometimes referred to as existential theory over the reals. However even in this highly important case R = R, and in striking contrast to NP, relatively few other natural problems have yet been identified as complete: * Supported in parts by the Marie Curie International Research Staff Exchange Scheme Fellowship 294962 within the 7th European Community Framework Programme † e.g. as lists of monomials and their coefficients or as algebraic expressions

Optimal Curve Straightening is $\exists\mathbb{R}$-Complete

“…Mnëv's universality theorem has since been used to prove the ∃ -hardness of several other geometric and topological problems. Examples include realizability of abstract 4-polytopes [54], cross-product systems [38], Delaunay triangulations [2,50], and planar linkages [40,58]; recognition of several types of graphs [12,16,17,42,45,46,58]; several problems in graph drawing [6,16,18,20,27,39,57]; several problems related to fixed points and Nash equilibria [7,8,29,59]; problems in social choice theory [51]; optimal polytope nesting [24,61]; nonnegative matrix factorization [22,24,61]; minimum rank of a matrix with a given sign pattern [3,5]; real tensor rank [60]; and the art gallery problem [1]. (Some of these papers claim only NP-hardness but describe reductions from pseudoline stretchability that imply ∃ -hardness; others make no claims about computational complexity but derive appropriate universality theorems that imply ∃ -hardness.…”