On Minimizing Regular Expressions Without Kleene Star

Abstract

Finite languages lie at the heart of literally every regular expression. Therefore, we investigate the approximation complexity of minimizing regular expressions without Kleene star, or, equivalently, regular expressions describing finite languages. On the side of approximation hardness, given such an expression of size s, we prove that it is impossible to approximate the minimum size required by an equivalent regular expression within a factor of $O\left( \frac{s}{(\log s)^{\delta }}\right)$ if the running time is bounded by a quasipolynomial function depending on $\delta$, for every $\delta >1$, unless the exponential time hypothesis (ETH) fails. For approximation ratio $O(s^{1-\delta })$, we prove an exponential-time lower bound depending on $\delta$, assuming ETH. The lower bounds apply to alphabets of constant size. On the algorithmic side, we show that the problem can be approximated in polynomial time within $O(\frac{s\log \log s}{\log s})$, with s being the size of the given regular expression. For constant alphabet size, the bound improves to $O(\frac{s}{\log s})$. Finally, we devise a family of superpolynomial approximation algorithms with approximation ratios matching the lower bounds, while the running times are just above the lower bounds excluded by the exponential time hypothesis.