We consider the aggregation problem in radio networks: find a spanning tree in a given graph and a conflict-free schedule of the edges so as to minimize the latency of the computation. While a large body of literature exists on this and related problems, we give the first approximation results in graphs that are not induced by unit ranges in the plane. We give a polynomial-time $\tilde{\mathrm{O}}(\sqrt{dn})$-approximation algorithm, where d is the average degree and n the number of vertices in the graph, and show that the problem is $\mathrm{\Omega }(n^{1-ϵ})$-hard (and $\mathrm{\Omega }({(dn)}^{1/2-ϵ})$-hard) to approximate even on bipartite graphs, for any $ϵ>0$, rendering our algorithm essentially optimal. We also obtain a $O(\log n)$-approximation in interval graphs.

我们考虑无线电网络中的聚合问题：在给定图中找到生成树，并找到边缘的无冲突调度，以最大程度地减少计算的延迟。尽管存在大量关于此问题和相关问题的文献，但我们在图形中给出的第一近似结果不是由平面中的单位范围引起的。我们给出一个多项式时间$\stackrel{〜}{\mathrm{Ø}}（\sqrt{dñ}）$-近似算法，其中d是平均度，n是图形中的顶点数，表明问题是$\mathrm{\Omega }（{ñ}^{\mathrm{1个}-ϵ}）$-硬（和 $\mathrm{\Omega }（{（dñ）}^{\mathrm{1个}/2-ϵ}）$-hard）甚至在二部图上也近似 $ϵ>0$，从而使我们的算法本质上处于最佳状态。我们还获得了$Ø（\mathrm{日志}ñ）$-间隔图中的近似值。