Algorithmic obstructions in the random number partitioning problem
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314fa3bb8a1bb4aa416421ae62900fcd
Author(s)
Gamarnik, David
Kızıldağ, Eren C
Date Issued
December 2023
Journal
The Annals of Applied Probability
Publisher
Institute of Mathematical Statistics
Citation
David Gamarnik. Eren C. Kızıldağ. "Algorithmic obstructions in the random number partitioning problem." Ann. Appl. Probab. 33 (6B) 5497 - 5563, December 2023.
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Final published version
Abstract
We consider the algorithmic problem of finding a near-optimal solution for the number partitioning problem (NPPNPPpossesses a so-called statistical-to-computational gap: when its input X has distribution N(0,I_n)NPPis Θ(√n 2^(-n))w.h.p., whereas the best-known polynomial-time algorithm achieves an objective value of only 2^(-Θ(〖log〗^2 n))w.h.p.
In this paper we initiate the study of the nature of this gap. Inspired by insights from statistical physics, we study the landscape of the NPPand establish the presence of the overlap gap property (OGP), an intricate geometrical property which is known to be a rigorous evidence of an algorithmic hardness for large classes of algorithms. By leveraging the OGP, we establish that: (a) any sufficiently stable algorithm, appropriately defined, fails to find a near-optimal solution with energy below 2^(-ω(n〖log〗^(-1/5) n)) 2^(-Θ(〖log〗^2 n))is indeed stable, but formally verifying this is left as an open problem.
OGP regards the overlap structure of m-tuples of solutions achieving a certain objective value. When m is constant, we prove the presence of OGP for the objective values of order 2^(-Θ(n))and the absence of it in the regime 2^(-o(n))m, we prove the presence of the OGP up to the level 2^(-ω(√nlogn)) 2^(-ω(n〖log〗^(-1/5) n))employs methods from Ramsey theory from the extremal combinatorics and is of independent interest.
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DOI of Published Version
https://doi.org/10.1214/23-aap1953
