Generation Problems

Abstract

Given a fixed computable binary operation f, we study the complexity of the following generation problem: The input consists of strings a ₁,...,a _n ,b. The question is whether b is in the closure of {a ₁, ..., a _n } under operation f.

For several subclasses of operations we prove tight upper and lower bounds for the generation problems. For example, we prove exponential-time upper and lower bounds for generation problems of length-monotonic polynomial-time computable operations. Other bounds involve classes like NP and PSPACE.

Here the class of bivariate polynomials with positive coefficients turns out to be the most interesting class of operations. We show that many of the corresponding generation problems belong to NP. However, we do not know this for all of them, e.g., for x ²+2y this is an open question. We prove NP-completeness for polynomials x ^a y ^b c where a,b,c≥ 1. Also, we show NP-hardness for polynomials like x ²+2y. As a by-product we obtain NP-completeness of the extended sum-of-subset problem SOS _c ={(w ₁,...,w _n ,z): $\exists I\subseteq\{1,\ldots,n\}({\sum_{i\in I}w_i^{c}=z})\}$ where c≥ 1.