This paper provides a cut-strategy that produces constant-hop expanders in the well-known cut-matching game framework. Constant-hop expanders strengthen expanders with constant conductance by guaranteeing that any demand can be (obliviously) routed along constant-hop paths - in contrast to the $\Omega(\log n)$-hop routes in expanders. Cut-matching games for expanders are key tools for obtaining close-to-linear-time approximation algorithms for many hard problems, including finding (balanced or approximately-largest) sparse cuts, certifying the expansion of a graph by embedding an (explicit) expander, as well as computing expander decompositions, hierarchical cut decompositions, oblivious routings, multi-cuts, and multicommodity flows. The cut-matching game provided in this paper is crucial in extending this versatile and powerful machinery to constant-hop expanders. It is also a key ingredient towards close-to-linear time algorithms for computing a constant approximation of multicommodity-flows and multi-cuts - the approximation factor being a constant relies on the expanders being constant-hop.
翻译:本文提供了一种切换策略, 在众所周知的切开游戏框架中产生恒定点扩张器。 常住点扩张器通过保证任何需求都可以( 明显地) 沿恒点路径( 与扩张器中的$\ omega( log n) $- hop 路径相反) 沿常点路径( 与 $\ omega (\ log n) / $- hop 路径) 路径( 与 $\ omega (\ log n) / $- hop ) 路径( ) 。 扩展器的切开匹配游戏是为许多棘手问题获取近距离至线时间近距离近距离近距离近距离近距离的算法的关键工具, 包括找到( 平衡或 约大一点 ) 的稀释削减, 通过嵌入 ( Exclid) 扩展器来验证图形的扩张器扩张, 以及计算扩张器变形器变形器变形器、 、 等级切割器变形器、 、 不断依赖常态的近离子要素 。 本文提供的剪动游戏对于将这一变换成常依赖一个恒的组合要素至关重要。