Motivated by recent works on streaming algorithms for constraint satisfaction problems (CSPs), we define and analyze oblivious algorithms for the Max-$k$AND problem. This generalizes the definition by Feige and Jozeph (Algorithmica '15) of oblivious algorithms for Max-DICUT, a special case of Max-$2$AND. Oblivious algorithms round each variable with probability depending only on a quantity called the variable's bias. For each oblivious algorithm, we design a so-called "factor-revealing linear program" (LP) which captures its worst-case instance, generalizing one of Feige and Jozeph for Max-DICUT. Then, departing from their work, we perform a fully explicit analysis of these (infinitely many!) LPs. In particular, we show that for all $k$, oblivious algorithms for Max-$k$AND provably outperform a special subclass of algorithms we call "superoblivious" algorithms. Our result has implications for streaming algorithms: Generalizing the result for Max-DICUT of Saxena, Singer, Sudan, and Velusamy (SODA'23), we prove that certain separation results hold between streaming models for infinitely many CSPs: for every $k$, $O(\log n)$-space sketching algorithms for Max-$k$AND known to be optimal in $o(\sqrt n)$-space can be beaten in (a) $O(\log n)$-space under a random-ordering assumption, and (b) $O(n^{1-1/k} D^{1/k})$ space under a maximum-degree-$D$ assumption. Even in the previously-known case of Max-DICUT, our analytic proof gives a fuller, computer-free picture of these separation results.

翻译：受近期关于约束满足问题（CSP）流算法的研究启发，我们定义并分析了Max-$k$AND问题的隐匿算法。这推广了Feige与Jozeph（Algorithmica '15）针对Max-DICUT（Max-$2$AND的特例）提出的隐匿算法定义。隐匿算法中每个变量的舍入概率仅取决于该变量的"偏差"量。针对每种隐匿算法，我们设计了一个称为"因子揭示线性规划"（LP）的模型来刻画其最坏情况实例，这推广了Feige与Jozeph针对Max-DICUT提出的方法。不同于他们的工作，我们对这些（无穷多个！）线性规划进行了完全显式的分析。特别地，我们证明：对所有$k$，Max-$k$AND的隐匿算法严格优于一类称为"超隐匿"算法的子类。我们的结论对流算法具有重要启示：推广Saxena、Singer、Sudan与Velusamy（SODA'23）关于Max-DICUT的结果，我们证明对于无穷多个CSP，流模型之间存在特定的分离结果——对每个$k$，已知在$o(\sqrt n)$空间下最优的Max-$k$AND的$O(\log n)$空间草图算法，可在以下条件下被超越：(a) 随机序假设下的$O(\log n)$空间；以及(b) 最大度$D$假设下的$O(n^{1-1/k} D^{1/k})$空间。即使在先前已知的Max-DICUT情形中，我们的分析性证明也为这些分离结果提供了更完整、无需计算机辅助的图景。