Wishart planted ensemble: A tunably rugged pairwise Ising model with a firstorder phase transition
Abstract
We propose the Wishart planted ensemble, a class of zerofield Ising models with tunable algorithmic hardness and specifiable (or planted) ground state. The problem class arises from a simple procedure for generating a family of random integer programming problems with specific statistical symmetry properties but turns out to have intimate connections to a signinverted variant of the Hopfield model. The Hamiltonian contains only 2spin interactions, with the coupler matrix following a type of Wishart distribution. The class exhibits a classical firstorder phase transition in temperature. For some parameter settings the model has a locally stable paramagnetic state, a feature which correlates strongly with difficulty in finding the ground state and suggests an extremely rugged energy landscape. We analytically probe the ensemble thermodynamic properties by deriving the ThoulessAndersonPalmer equations and free energy and corroborate the results with a replica and annealed approximation analysis; extensive Monte Carlo simulations confirm our predictions of the firstorder transition temperature. The class exhibits a wide variation in algorithmic hardness as a generation parameter is varied, with a pronounced easyhardeasy profile and peak in solution time towering many orders of magnitude over that of the easy regimes. By deriving the ensembleaveraged energy distribution and taking into account finiteprecision representation, we propose an analytical expression for the location of the hardness peak and show that at fixed precision, the number of constraints in the integer program must increase with system size to yield truly hard problems. The Wishart planted ensemble is interesting for its peculiar physical properties and provides a useful and analytically transparent set of problems for benchmarking optimization algorithms.
 Publication:

Physical Review E
 Pub Date:
 May 2020
 DOI:
 10.1103/PhysRevE.101.052102
 arXiv:
 arXiv:1906.00275
 Bibcode:
 2020PhRvE.101e2102H
 Keywords:

 Condensed Matter  Disordered Systems and Neural Networks;
 Quantum Physics
 EPrint:
 40 pages, 19 figures, 1 lonely table