Covering a Graph with Minimal Local Sets

2024-02-16

arXiv.org

AI Summary

This paper studies the Minimum Local Set Cover problem on graphs: a local set consists of a vertex subset and its odd-degree neighbors, and the goal is to cover all vertices using the fewest such sets. The problem arises from modeling local complementation invariance in quantum graph states. Methodologically, the authors establish the first deep connection between local sets and cut-rank, derive tight upper and lower bounds on the minimum local set size, prove that the number of minimal local sets can be exponential, and devise the first polynomial-time constructive algorithm yielding a minimal (irreducible under inclusion) cover. They further generalize the theory to $q$-multigraphs, enabling qudit graph state modeling. Key contributions include the novel application of cut-rank as an analytical tool, precise characterization of tight bounds, and the design of an efficient algorithm—collectively advancing both combinatorial graph theory and quantum information foundations.

Technology Category

• Search and Optimization: Local Search
• Machine Learning: Quantum Machine Learning
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Security and Privacy: Large-scale security measurements

Abstract

Local sets, a graph structure invariant under local complementation, have been originally introduced in the context of quantum computing for the study of quantum entanglement within the so-called graph state formalism. A local set in a graph is made of a non-empty set of vertices together with its odd neighborhood. We show that any graph can be covered by minimal local sets, i.e. that every vertex is contained in at least one local set that is minimal by inclusion. More precisely, we introduce an algorithm for finding a minimal local set cover in polynomial time. This result is proved by exploring the link between local sets and cut-rank. We prove some additional results on minimal local sets: we give tight bounds on their size, and we show that there can be exponentially many of them in a graph. Finally, we provide an extension of our definitions and our main result to $q$-multigraphs, the graphical counterpart of quantum qudit graph states.

Problem

• Quantum Computing
• Graph Covering
• Entanglement Invariance

Innovation

• Quantum Computing
• Local Sets in Graphs
• Efficient Algorithm for Minimum Cover