对排序、系统发育和普通约束满足问题的有力反驳，以及三元组和四元组重建的新可满足性和反驳阈值
Strong Refutation of Ordering, Phylogenetic, and Ordinary CSPs, and New Satisfiability and Refutation Thresholds for Triplet and Quartet Reconstruction
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中文总结 AI 辅助
研究层次聚类等中CSPs的相变与反驳算法，确定三元组尖锐阈值\(\lambda^*\approx1.2277\)并给出四元组上下界，提供有力反驳算法，在特定密度下得到最强反驳，推广了反驳随机CSPs的算法前沿。
中文摘要 AI 辅助
我们研究了在层次聚类（以及排序和普通约束满足问题）中出现的约束满足问题（CSPs）的相变和反驳算法。这里，\(n\)个变量被分配到一棵树的叶子上，以满足\(m\)个指定进化关系的约束。两个典型的NP难优化问题是三元组和四元组重建，输入由三元组\(xy|z\)或四元组\(xy|zw\)组成，目标是找到与约束最大程度一致的树\(T^*\)。我们的主要结果如下（随着密度\(\lambda = m/n\)增加）：1. 我们证明了三元组存在尖锐阈值\(\lambda^*\approx1.2277\)并精确确定其位置（通过闭式解）。据我们所知，这是系统发育CSPs广泛家族中的第一个尖锐阈值。此外，我们给出了四元组的上下界。2. 我们提供了有力的反驳算法，可证明\(val(T^*)\leq5/9+\epsilon\)，其中\(val(T^*)\)是（未知）最优树满足的约束比例。对于三元组，如果\(m=\Omega(n)\)，我们的算法以高概率成功；对于四元组，如果\(m=\Omega(n^{3/2})\)。3. 在稍大密度下（对于三元组\(m = O(n^{3/2}\log^3n)\)，对于四元组\(m = O(n^2)\)），我们得到了最强可能的反驳：我们证明\(T^*\)不比随机赋值好，即\(val(T^*)\leq1/3+\epsilon\)。实际上，我们对有或无否定的有限字母CSPs都得到了最强可能的反驳。我们上述的反驳是我们一般定理的实例，该定理更广泛地适用于系统发育和排序CSPs（以及所有不支持\(t\) - 明智独立性的CSPs），并推广了当前反驳随机CSPs的算法前沿。这里一个关键区别是，与布尔CSPs不同，没有否定变量，所以依赖否定（一种随机性来源）的先前工作不适用。
英文摘要
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.
