Testing Full Quartet Consistency: Adaptive Reconstruction, Random Verification, and Constant-Query Testability

Abstract

We study dense property testing for full systems of resolved quartet topologies on $n$ taxa: determining whether a system is induced by a phylogenetic tree or is $\varepsilon$-far from every tree-induced system. Our main result is an explicit polynomial-time adaptive one-sided-error tester. It reconstructs a candidate tree through anchored quartet queries and verifies the candidate using uniformly random quartet queries. With error probability $\delta$, it uses $O\!\left(n\log n+\varepsilon^{-1}\log(1/\delta)\right)$ queries. We also give a non-adaptive cached-anchor variant using $\binom{n-1}{3}+O\!\left(\varepsilon^{-1}\log(1/\delta)\right)$ queries. Both improve the previous explicit $O(n^3/\varepsilon)$ query bound. Since the input contains $\binom{n}{4}=\Theta(n^4)$ quartet entries, both testers use $o\!\left(\binom{n}{4}\right)$ queries for fixed~$\varepsilon$ and~$\delta$. We additionally encode full quartet systems, equivariantly under relabeling, as directed, three-colored $4$-ary structures. Hereditary directed-hypergraph testing then yields an $n$-independent one-sided-error tester, although its dependence on $\varepsilon$ is quantitatively impractical. Finally, we prove lower bounds. In ordinary property testing, every adaptive randomized tester, even with two-sided error, requires asymptotically at least $\ln((1-\delta)/\delta)/\ln(1/(1-\varepsilon))$ queries as $n\to\infty$. Every one-sided-error tester requires $\ln(1/\delta)/\ln(1/(1-\varepsilon))$ queries, matching the random-verification term up to rounding. For the stronger reconstruct-or-reject task, our upper bounds are optimal up to constant factors: the adaptive and non-adaptive complexities are $\Theta(n\log n+\varepsilon^{-1}\log(1/\delta))$ and $\Theta(n^3+\varepsilon^{-1}\log(1/\delta))$, respectively.