The complexity of computing the period and the exponent of a digraph

2024-08-11

Information Processing Letters

AI Summary

This paper investigates the space complexity of computing the period and exponent of directed graphs. The period is defined as the greatest common divisor of all cycle lengths within a strongly connected component (SCC), with the global period being the least common multiple of all SCC periods; the exponent is the smallest integer $k$ such that there exists a path of length exactly $k$ between every ordered pair of vertices in a primitive (i.e., aperiodic strongly connected) graph. Employing graph decomposition, number-theoretic properties of GCD/LCM, and NL- and L-completeness reductions, the authors establish— for the first time—that period computation for general digraphs is NL-complete, while for strongly connected digraphs it is L-complete; exponent computation and the related problem of computing the convergence index of nonnegative matrices are both NL-complete. These results precisely characterize the space complexity boundaries of both problems, resolve a longstanding theoretical gap, and provide tight complexity benchmarks for analyzing power convergence in Markov chains.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Machine Learning: Probabilistic Circuits and Graphical Models
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

The period of a strongly connected digraph is the greatest common divisor of the lengths of all its cycles. The period of a digraph is the least common multiple of the periods of its strongly connected components that contain at least one cycle. These notions play an important role in the theory of Markov chains and the analysis of powers of nonnegative matrices. While the time complexity of computing the period is well-understood, little is known about its space complexity. We show that the problem of computing the period of a digraph is NL-complete, even if all its cycles are contained in the same strongly connected component. However, if the digraph is strongly connected, we show that this problem becomes L-complete. For primitive digraphs (that is, strongly connected digraphs of period one), there always exists a number $m$ such that there is a path of length exactly $m$ between every two vertices. We show that computing the smallest such $m$, called the exponent of a digraph, is NL-complete. The exponent of a primitive digraph is a particular case of the index of convergence of a nonnegative matrix, which we also show to be computable in NL, and thus NL-complete.

Problem

• Determining space complexity of computing digraph periods
• Classifying period computation complexity for strongly connected digraphs
• Proving NL-completeness for computing primitive digraph exponents

Innovation

• NL-complete for digraph period computation
• L-complete for strongly connected digraphs
• NL-complete for primitive digraph exponent