Strong Refutation of Ordering, Phylogenetic, and Ordinary CSPs, and New Satisfiability and Refutation Thresholds for Triplet and Quartet Reconstruction

2026-07-15

AI Summary

This work investigates constraint satisfaction problems (CSPs) arising in hierarchical clustering and ranking, with a focus on the satisfiability and strong refutation of triplet and quartet reconstruction. By analyzing the relationship between constraint density and the fraction of constraints satisfied by an optimal solution, and leveraging probabilistic methods, random CSP theory, and combinatorial optimization, the study establishes the sharp satisfiability threshold for triplet reconstruction at approximately 1.2277—the first exact determination of this phase transition. It further introduces a general strong refutation framework that operates without variable negation. For constraint counts $m = \Omega(n)$ (triplets) and $m = \Omega(n^{3/2})$ (quartets), the framework achieves strong refutation with $\text{val}(T^*) \leq 5/9 + \varepsilon$, and at higher densities, it proves that no solution significantly outperforms random guessing, yielding $\text{val} \leq 1/3 + \varepsilon$.

Abstract

We study phase transitions and algorithms for refuting CSPs arising in hierarchical clustering (as well as ranking, and ordinary CSPs). Here, $n$ variables are assigned to leaves of a tree, so as to satisfy $m$ constraints, specifying evolutionary relationships. Two canonical $NP$-hard optimization problems are Triplet and Quartet Reconstruction, where the input consists of triplets $xy|z$ or quartets $xy|zw$, and the goal is to find a tree $T^*$ maximizing agreement with constraints. Our main results are (as density $λ=m/n$ increases): 1. We show the existence and precisely locate the sharp threshold $λ^*\approx1.2277$ for Triplets (via closed-form solution). To the best of our knowledge, this is the first sharp threshold for the broad family of Phylogenetic CSPs. Moreover, we give a lower and upper bound for Quartets. 2. We provide strong refutation algorithms that certify that $val(T^*)\le5/9 + ε$, where $val(T^*)$ is the fraction of constraints satisfied by the (unknown) optimal tree. For triplets, our algorithm succeeds w.h.p if $m =Ω(n)$, and for quartets if $m = Ω(n^{3/2})$. 3. We obtain strongest possible refutations at slightly larger densities (for triplets $m=O(n^{3/2}\log ^3n)$, for quartets $m=O(n^2)$): we certify that $T^*$ is no better than a random assignment, i.e., $val(T^*)\le 1/3+ε$. In fact, we obtain strongest possible refutations for finite-alphabet CSPs with or without negations. Our refutations above are instantiations of our general theorem that applies more broadly to Phylogenetic and Ordering CSPs (and all CSPs failing to support $t$-wise independence), and generalizes the current algorithmic frontier on refuting random CSPs~\citep{allen2015refute}. A crucial difference here, unlike Boolean CSPs, is that there are no negated variables, so prior works relying on negations -- a source of randomness -- do not apply.

Problem

• Phylogenetic CSPs
• Triplet Reconstruction
• Quartet Reconstruction
• Strong Refutation
• Phase Transitions

Innovation

• sharp threshold
• strong refutation
• phylogenetic CSP
• triplet reconstruction
• constraint satisfaction problem