Strong Refutation of Random Ordering CSPs

In this work, we initiate the study of strongly refuting the satisfiability of random ordering constraint satisfaction problems. We show that there is a polynomial-time $\epsilon$-refutation algorithm for random ordering CSP with predicate $P$ when the number of clauses is above the threshold $\tilde{\mathrm{\Omega }}\left(n^{d\mathrm{/}2}\mathrm{/}{\epsilon }^2\right)$, where $d$ is the coordinate degree of the predicate $P$. We further give a smooth three-way tradeoff between the running time, the clause density, and the refutation strength $\epsilon$ using the Kikuchi method. Finally, we complement our algorithmic results with a computational lower bound based on the class of low coordinate degree algorithms, providing evidence that the established three-way tradeoff is near optimal.