The complexity of quantifier elimination and cylindrical algebraic decomposition

Abstract

This paper has two parts. In the first part we give a simple and constructive proof that quantifier elimination in real algebra is doubly exponential, even when there is only one free variable and all polynomials in the quantified input are linear. The general result is not new, but we hope the simple and explicit nature of the proof makes it interesting. The second part of the paper uses the construction of the first part to prove some results on the effects of projection order on CAD construction -roughly that there are CAD construction problems for which one order produces a constant number of cells and another produces a doubly exponential number of cells, and that there are problems for which all orders produce a doubly exponential number of cells. The second of these results implies that there is a true singly vs. doubly exponential gap between the worst-case running times of several modern quantifier elimination algorithms and CAD-based quantifier elimination when the number of quantifier alternations is constant.

CAD and Topology of Semi-Algebraic Sets

“…Remark 3.6 (About complexity) CAD complexity is dominated by the number of cells, which is doubly exponential in the worst case [8]. Our modification of the algorithm does not change the cells to be computed.…”

Some lower bounds for the complexity of the linear programming feasibility problem over the reals

“…Davenport and Brown presented,in [7], a simplified proof of this doubly-exponential lower bound that works for both, dense and sparse codification of polynomials. Thus, in order of magnitude, upper and lower complexity bounds meet for classic data structures (i.e., when polynomials are represented in dense or sparse form).…”

$$\mathsf {SC}^\mathsf{2} $$ : Satisfiability Checking Meets Symbolic Computation

“…This has produced many new applications and surprising developments: in an area everyone believed was solved, non-linear solving over the reals (using cylindrical algebraic decomposition -CAD) has recently found a new algorithm for computing square roots [35]. CAD is another area where practice is (sometimes) well ahead of theory: the theory [18,29] states that the complexity is doubly exponential in the number of variables, but useful problems can still be solved in practice ( [3] points out that CAD is the most significant engine in the "Todai robot" project).…”