Table of Contents
International Journal of Statistical Mechanics
Volume 2014 (2014), Article ID 136829, 13 pages
http://dx.doi.org/10.1155/2014/136829
Research Article

The Statistical Mechanics of Random Set Packing and a Generalization of the Karp-Sipser Algorithm

1Dipartimento di Fisica, Università “La Sapienza”, Piazzale Aldo Moro 2, 00185 Rome, Italy
2Dipartimento di Fisica, INFN-Sezione di Roma1, CNR-IPCF UOS Roma Kerberos, Università “La Sapienza”, Piazzale Aldo Moro 2, 00185 Rome, Italy

Received 19 November 2013; Accepted 8 January 2014; Published 10 March 2014

Academic Editor: Hyunggyu Park

Copyright © 2014 C. Lucibello and F. Ricci-Tersenghi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We analyse the asymptotic behaviour of random instances of the maximum set packing (MSP) optimization problem, also known as maximum matching or maximum strong independent set on hypergraphs. We give an analytic prediction of the MSPs size using the 1RSB cavity method from statistical mechanics of disordered systems. We also propose a heuristic algorithm, a generalization of the celebrated Karp-Sipser one, which allows us to rigorously prove that the replica symmetric cavity method prediction is exact for certain problem ensembles and breaks down when a core survives the leaf removal process. The -phenomena threshold discovered by Karp and Sipser, marking the onset of core emergence and of replica symmetry breaking, is elegantly generalized to for one of the ensembles considered, where is the size of the sets.