$mathsf{P}$-completeness of Graph Local Complementation

2025-03-31

AI Summary

This paper studies the edge-existence problem for sequential local complementations on graphs: given a graph $G$, a vertex sequence $s$, and a vertex pair $(u,v)$, determine whether $u$ and $v$ are adjacent after applying local complementations centered at each vertex in $s$ in order. The authors establish, for the first time, that this problem is P-complete—demonstrating its inherent sequentiality and resistance to efficient parallelization. They precisely delineate its complexity landscape: the problem lies in LOGSPACE for complete and star graphs, while remaining P-complete (conjectured) for cycle graphs. Technically, they achieve this by reducing the Circuit Value Problem (CVP) to sequential local complementation, constructing a compact encoding that simulates Boolean gates via local complementations. These results provide foundational complexity-theoretic insights for domains relying on local complementation models—such as quantum graph state evolution and graph reconstruction.

Technology Category

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

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: Applications of cryptography

Abstract

Local complementation of a graph $G$ on vertex $v$ is an operation that results in a new graph $G*v$, where the neighborhood of $v$ is complemented. This operation has been widely studied in graph theory and quantum computing. This article introduces the Local Complementation Problem, a decision problem that captures the complexity of applying a sequence of local complementations. Given a graph $G$, a sequence of vertices $s$, and a pair of vertices $u,v$, the problem asks whether the edge $(u,v)$ is present in the graph obtained after applying local complementations according to $s$. The main contribution of this work is proving that this problem is $mathsf{P}$-complete, implying that computing a sequence of local complementation is unlikely to be efficiently parallelizable. The proof is based on a reduction from the Circuit Value Problem, a well-known $mathsf{P}$-complete problem, by simulating circuits through local complementations. Aditionally, the complexity of this problem is analyzed under different restrictions. In particular, it is shown that for complete and star graphs, the problem belongs to $mathsf{LOGSPACE}$. Finally, it is conjectured that the problem remains $mathsf{P}$-complete for the class of circle graphs.

Problem

• Determines complexity of local complementation sequence application
• Proves P-completeness via Circuit Value Problem reduction
• Analyzes problem complexity for specific graph classes

Innovation

• Proves P-completeness via Circuit Value Problem reduction
• Analyzes complexity under complete and star graphs
• Conjectures P-completeness for circle graphs class