The Computational Complexity of Factored Graphs

TL;DR

This paper explores the computational complexity of graph problems on factored graphs, which are represented as combinations of smaller graphs, revealing diverse complexity results and implications for algorithm efficiency.

Contribution

It introduces a parameterized framework for analyzing graph problems on factored graphs and characterizes their complexity, including FPT and XP classifications.

Findings

Lexicographically first maximal independent set is XP-complete.

Clique counting is fixed-parameter tractable (FPT).

Reachability is XNL-complete.

Abstract

While graphs and abstract data structures can be large and complex, practical instances are often regular or highly structured. If the instance has sufficient structure, we might hope to compress the object into a more succinct representation. An efficient algorithm (with respect to the compressed input size) could then lead to more efficient computations than algorithms taking the explicit, uncompressed object as input. This leads to a natural question: when does knowing the input instance has a more succinct representation make computation easier? We initiate the study of the computational complexity of problems on factored graphs: graphs that are given as a formula of products and unions on smaller graphs. For any graph problem, we define a parameterized version that takes factored graphs as input, parameterized by the number of (smaller) ordinary graphs used to construct the factored graph. In this setting, we characterize the parameterized complexity of several natural graph problems, exhibiting a variety of complexities. We show that a decision version of lexicographically first maximal independent set is $\mathbf{XP}$-complete, and therefore unconditionally not fixed-parameter tractable ($\mathbf{FPT}$). On the other hand, we show that clique counting is $\mathbf{FPT}$. Finally, we show that reachability is $\mathbf{XNL}$-complete. Moreover, $\mathbf{XNL}$ is contained in $\mathbf{FPT}$ if and only if $\mathbf{NL}$ is contained in some fixed polynomial time.