TL;DR
This paper establishes bounds on the threshold for the q-overlap-k-Exact Cover problem, inspired by statistical physics, aiming to rigorously analyze the overlap distribution's behavior.
Contribution
It provides the first rigorous bounds for the q-overlap threshold in the k-Exact Cover problem, connecting statistical physics insights with combinatorial thresholds.
Findings
Derived upper and lower bounds for the q-overlap threshold
Connected replica symmetry breaking concepts to combinatorial thresholds
Paved the way for rigorous analysis of overlap distributions
Abstract
We prove upper and lower bounds for the threshold of the q-overlap-k-Exact cover problem. These results are motivated by the one-step replica symmetry breaking approach of Statistical Physics, and the hope of using an approach based on that of Mezard et al. (2005) to rigorously prove that for some values of the order parameter the overlap distribution of k-Exact Cover has discontinuous support.
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