We consider the problem of query-efficient global max-cut on a weighted undirected graph in the value oracle model examined by [RSW18]. This model arises as a natural special case of submodular function maximization: on query $S \subseteq V$, the oracle returns the total weight of the cut between $S$ and $V \backslash S$. For most constants $c \in (0,1]$, we nail down the query complexity of achieving a $c$-approximation, for both deterministic and randomized algorithms (up to logarithmic factors). Analogously to general submodular function maximization in the same model, we observe a phase transition at $c = 1/2$: we design a deterministic algorithm for global $c$-approximate max-cut in $O(\log n)$ queries for any $c < 1/2$, and show that any randomized algorithm requires $\tilde{\Omega}(n)$ queries to find a $c$-approximate max-cut for any $c > 1/2$. Additionally, we show that any deterministic algorithm requires $\Omega(n^2)$ queries to find an exact max-cut (enough to learn the entire graph), and develop a $\tilde{O}(n)$-query randomized $c$-approximation for any $c < 1$. Our approach provides two technical contributions that may be of independent interest. One is a query-efficient sparsifier for undirected weighted graphs (prior work of [RSW18] holds only for unweighted graphs). Another is an extension of the cut dimension to rule out approximation (prior work of [GPRW20] introducing the cut dimension only rules out exact solutions).

翻译：我们研究[RSW18]中所考察的估值预言机模型中加权无向图上的全局最大割查询高效问题。该模型是子模函数最大化的自然特例：查询$S \subseteq V$时，预言机返回$S$与$V \backslash S$之间割的总权重。对于大多数常数$c \in (0,1]$，我们确定了实现$c$近似时确定性与随机化算法的查询复杂度（至多对数因子）。与同一模型中一般子模函数最大化类似，我们观察到在$c = 1/2$处存在相变：针对任意$c < 1/2$，我们设计出$O(\log n)$次查询的确定性全局$c$近似最大割算法；并证明对于任意$c > 1/2$，任何随机化算法需要$\tilde{\Omega}(n)$次查询才能找到$c$近似最大割。此外，我们证明任何确定性算法需要$\Omega(n^2)$次查询才能求得精确最大割（足以学习整个图），并开发出针对任意$c < 1$的$\tilde{O}(n)$次查询随机化$c$近似算法。我们的方法包含两项可能独立引起兴趣的技术贡献：其一是面向加权无向图的查询高效稀疏化器（[RSW18]的先前工作仅适用于无权图）；其二是将割维数扩展以排除近似解（[GPRW20]引入的割维数仅能排除精确解）。