On Topological Lower Bounds for Algebraic Computation Trees

We prove that the height of any algebraic computation tree for deciding membership in a semialgebraic set Sigma subset of R-n is bounded from below by c(1) log(b(m) (Sigma))/m + 1 - c(2)n, where b(m)(Sigma) is the mth Betti number of Sigma with respect to "ordinary" (singular) homology and c(1), c(2) are some (absolute) positive constants. This result complements the well-known lower bound by Yao (J Comput Syst Sci 55:36-43, 1997) for locally closed semialgebraic sets in terms of the total Borel-Moore Betti number. We also prove that if rho : R-n -> Rn-r is the projection map, then the height of any tree deciding membership in Sigma is bounded from below by c(1) log(b(m)(rho(Sigma)))/(m + 1)(2) - c(2)n/m + 1 for some positive constants c(1), c(2). We illustrate these general results by examples of lower complexity bounds for some specific computational problems.