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Related Experiment Video

Updated: Jul 10, 2026

Setting Limits on Supersymmetry Using Simplified Models

Setting Limits on Supersymmetry Using Simplified Models

Published on: November 15, 2013

Statistical mechanics of the hitting set problem.

Marc Mézard¹, Marco Tarzia

¹CNRS, Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris-Sud, UMR 8626, Orsay Cedex 91405, France.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|November 13, 2007

Summary

This study analyzes the hitting set problem in random hypergraphs using statistical mechanics. Researchers determined the minimal particle density and phase space structure, offering efficient algorithms for solving complex instances.

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Area of Science:

Statistical mechanics
Theoretical computer science
Graph theory

Background:

The hitting set (HS) problem generalizes vertex cover to hypergraphs.
It involves finding a minimal density configuration of particles to cover all hyperedges.
Applications include group testing for defective item detection.

Purpose of the Study:

To conduct a detailed study of the hitting set problem on random regular hypergraphs.
To investigate the phase diagram and critical properties of the HS problem.
To develop efficient algorithms for solving large instances of the HS problem.

Main Methods:

Statistical mechanics approach utilizing the cavity method.
Analysis of the phase diagram, including replica symmetric and replica symmetry breaking phases.
Application of decimation procedures: belief propagation and survey propagation algorithms.

Main Results:

Explicit results on minimal particle density and phase space structure were derived for different degrees.
The HS problem exhibits replica symmetric or one-step replica symmetry breaking phases.
Belief propagation and survey propagation algorithms demonstrate high efficiency for large HS problem instances.

Conclusions:

The hitting set problem on random regular hypergraphs is more tractable than the vertex cover problem.
The study provides a comprehensive understanding of the HS problem's phase diagram and solution strategies.
Efficient algorithms are presented for practical applications of the hitting set problem.