Replica symmetry breaking for Ulam's problem

TL;DR

This paper introduces an efficient algorithm to sample increasing subsequences in permutation sequences at any temperature, revealing complex replica symmetry breaking behavior in Ulam's problem, a classic combinatorial optimization challenge.

Contribution

The authors develop a polynomial-time algorithm for exact sampling of increasing subsequences at any temperature, enabling large-scale analysis of the problem's complex landscape.

Findings

Distribution of overlaps remains broad at low temperatures

Configuration landscape exhibits replica symmetry breaking

Algorithm confirms analytical predictions for moments of IS count

Abstract

We study increasing subsequences (IS) for an ensemble of sequences given by permutation of numbers {1,2,...,n}. We consider a Boltzmann ensemble at temperature T. Thus each IS appears with the corresponding Boltzmann probability where the energy is the negative length -l of the IS. For T -> 0, only ground states, i.e. longest IS (LIS) contribute, also called Ulam's problem. We introduce an algorithm which allows us to directly sample IS in perfect equilibrium in polynomial time, for any given sequence and any temperature. Thus, we can study very large sizes. We obtain averages for the first and second moments of number of IS as function of $n$ and confirm analytical predictions. Furthermore, we analyze for low temperature $T$ the sampled ISs by computing the distribution of overlaps and performing hierarchical cluster analyses. In the thermodynamic limit the distribution of overlaps stays broad and the configuration landscape remains complex. Thus, Ulam's problem exhibits replica symmetry breaking. This means it constitutes a model with complex behavior which can be studied numerically exactly in a highly efficient way, in contrast to other RSB-showing models, like spin glasses or NP-hard optimization problems, where no fast exact algorithms are known.