Computing the Homology of Semialgebraic Sets I: Lax Formulas

TL;DR

This paper introduces an algorithm to compute the homology of semialgebraic sets efficiently, achieving exponential speedup over previous methods by operating in weak exponential time for most inputs.

Contribution

It presents the first algorithm with exponential acceleration for computing homology of semialgebraic sets, reducing complexity from doubly exponential to single exponential for most data.

Findings

Algorithm computes Betti numbers and torsion coefficients.

Operates in weak exponential time, faster than previous methods.

Achieves exponential speedup for almost all input data.

Abstract

We describe and analyze an algorithm for computing the homology (Betti numbers and torsion coefficients) of closed semialgebraic sets given by Boolean formulas without negations over lax polynomial inequalities. The algorithm works in weak exponential time. This means that outside a subset of data having exponentially small measure, the cost of the algorithm is single exponential in the size of the data. All previous algorithms solving this problem have doubly exponential complexity (and this is so for almost all input data). Our algorithm thus represents an exponential acceleration over state-of-the-art algorithms for all input data outside a set that vanishes exponentially fast.