Large Independent Sets on Random d-Regular Graphs with Fixed Degree d

Author:

Marino Raffaele1ORCID,Kirkpatrick Scott2

Affiliation:

1. Dipartimento di Fisica e Astronomia, Università degli Studi di Firenze, Via Giovanni Sansone 1, Sesto Fiorentino, 50019 Firenze, Italy

2. School of Computer Science and Engineering, The Hebrew University of Jerusalem, Edmond Safra Campus, Givat Ram, Jerusalem 91904, Israel

Abstract

The maximum independent set problem is a classic and fundamental combinatorial challenge, where the objective is to find the largest subset of vertices in a graph such that no two vertices are adjacent. In this paper, we introduce a novel linear prioritized local algorithm tailored to address this problem on random d-regular graphs with a small and fixed degree d. Through exhaustive numerical simulations, we empirically investigated the independence ratio, i.e., the ratio between the cardinality of the independent set found and the order of the graph, which was achieved by our algorithm across random d-regular graphs with degree d ranging from 5 to 100. Remarkably, for every d within this range, our results surpassed the existing lower bounds determined by theoretical methods. Consequently, our findings suggest new conjectured lower bounds for the MIS problem on such graph structures. This finding has been obtained using a prioritized local algorithm. This algorithm is termed ‘prioritized’ because it strategically assigns priority in vertex selection, thereby iteratively adding them to the independent set.

Funder

Federman Cyber Security Center at the Hebrew University of Jerusalem

#NEXTGENERATIONEU

Ministry of University and Research

National Recovery and Resilience Plan

MNESYS

Publisher

MDPI AG

Subject

Applied Mathematics,Modeling and Simulation,General Computer Science,Theoretical Computer Science

Reference29 articles.

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4. Mezard, M., and Montanari, A. (2009). Information, Physics, and Computation, Oxford University Press.

5. Mohseni, M., Eppens, D., Strumpfer, J., Marino, R., Denchev, V., Ho, A.K., Isakov, S.V., Boixo, S., Ricci-Tersenghi, F., and Neven, H. (2021). Nonequilibrium Monte Carlo for unfreezing variables in hard combinatorial optimization. arXiv.

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