论文标题
关于近似最大切割的切割复杂性
On the cut-query complexity of approximating max-cut
论文作者
论文摘要
我们考虑了[RSW18]检查的值Oracle模型中的加权甲骨文模型中查询有效的全局最大切割的问题。最近已经研究了此切割查询模型和其他查询模型中的图形算法，以解决其他各种问题，例如最低，连通性，两人两角检测。剪切查询模型中的最大切割也可以看作是一种自然的特殊情况，即最大化的自然特殊情况：在查询$ s \ subseteq v $上，Oracle返回了$ s $和$ s $和$ v \ backslash s $的总重量。 我们的第一个主要技术结果是较低的说明，确定性算法对于任何$ c>> 1/2 $都需要$ω（n）$ QUERIES实现$ c $ approximation。这使用剪切维度的扩展来排除近似（[GPRW20]的先前工作仅引入切割维度，仅排除确切的解决方案）。其次，我们提供了一种随机算法，其中$ \ tilde {o}（n）$ Queries可以找到任何$ C <1 $的$ C $ -AppRoximation。我们使用无向加权图的查询效率稀疏器实现了这一点（[RSW18]的先前工作仅适用于未加权图）。 To complement these results, 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 $ - $ o（\ log n）$查询任何$ c <1/2 $的查询，并表明任何随机算法都需要$ω（n/\ log n）$查询以找到$ c $ c $ c $ apppro-apptroximate max-cut，以显示任何$ c> 1/2 $ $ quiery $ quor $。确切的最大切割（足以学习整个图）。
We consider the problem of query-efficient global max-cut on a weighted undirected graph in the value oracle model examined by [RSW18]. Graph algorithms in this cut query model and other query models have recently been studied for various other problems such as min-cut, connectivity, bipartiteness, and triangle detection. Max-cut in the cut query model can also be viewed 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 $V \backslash S$. Our first main technical result is a lower bound stating that a deterministic algorithm achieving a $c$-approximation for any $c > 1/2$ requires $Ω(n)$ queries. This uses an extension of the cut dimension to rule out approximation (prior work of [GPRW20] introducing the cut dimension only rules out exact solutions). Secondly, we provide a randomized algorithm with $\tilde{O}(n)$ queries that finds a $c$-approximation for any $c < 1$. We achieve this using a query-efficient sparsifier for undirected weighted graphs (prior work of [RSW18] holds only for unweighted graphs). To complement these results, 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 $Ω(n/\log n)$ queries to find a $c$-approximate max-cut for any $c > 1/2$. Additionally, we show that any deterministic algorithm requires $Ω(n^2)$ queries to find an exact max-cut (enough to learn the entire graph).
