Digraph Complexity Measures and Applications in Formal Language Theory

TL;DR

This paper studies the complexity of digraph measures like cycle rank and their relation to star height in formal language theory, providing new algorithms and complexity results for these problems.

Contribution

It establishes the NP-completeness of cycle rank computation, offers approximation and exact algorithms, and analyzes the complexity of star height in bideterministic languages.

Findings

Cycle rank computation is NP-complete for sparse digraphs.

Provides a polynomial-time approximation algorithm with O((log n)^(3/2)) ratio.

Develops an exponential-time exact algorithm with O*(1.9129^n) complexity.

Abstract

We investigate structural complexity measures on digraphs, in particular the cycle rank. This concept is intimately related to a classical topic in formal language theory, namely the star height of regular languages. We explore this connection, and obtain several new algorithmic insights regarding both cycle rank and star height. Among other results, we show that computing the cycle rank is NP-complete, even for sparse digraphs of maximum outdegree 2. Notwithstanding, we provide both a polynomial-time approximation algorithm and an exponential-time exact algorithm for this problem. The former algorithm yields an O((log n)^(3/2))- approximation in polynomial time, whereas the latter yields the optimum solution, and runs in time and space O*(1.9129^n) on digraphs of maximum outdegree at most two. Regarding the star height problem, we identify a subclass of the regular languages for which we can precisely determine the computational complexity of the star height problem. Namely, the star height problem for bideterministic languages is NP-complete, and this holds already for binary alphabets. Then we translate the algorithmic results concerning cycle rank to the bideterministic star height problem, thus giving a polynomial-time approximation as well as a reasonably fast exact exponential algorithm for bideterministic star height.