具有边冲突的生成树和无线连通性
Spanning Trees With Edge Conflicts and Wireless Connectivity
摘要：
我们引入的问题，找到一个生成树，以及一个分区的树边缘成最少数量的可行集，在边缘上的约束定义的可行性。动机来自无线网络，我们试图模拟在实际无线环境中看到的不规则性。并非所有节点对都能够通信，即使在地理上接近--因此，可用的节点对使用链路图$\mathcal{L}=(V,E)$建模。此外，信号衰减不需要遵循精确的几何公式-因此，干扰由链路上的冲突（超）图$\mathcal{C}=(E,F)$建模。我们的目标是最大限度地提高通信的效率，或者等价地最小化着色形式的树边的调度的长度。我们发现，尽管这一切的一般性，问题可以近似线性方面的一个多功能的参数，感应独立的干扰图。具体地说，我们给出了一个简单的算法，达到了$O(ρ\log n)$ -近似，其中$n$是节点的数量和$ρ$是归纳的独立性，并表明近线性依赖$ρ$也是必要的。我们还处理扩展到斯坦纳树，建模多播，并获得了可比的结果。我们的研究结果表明，几个典型的假设的几何形状，规律性和“尼斯”在无线设置有时可以放松算法性能没有显着的打击。
英文摘要：
We introduce the problem of finding a spanning tree along with a partition of the tree edges into fewest number of feasible sets, where constraints on the edges define feasibility. The motivation comes from wireless networking, where we seek to model the irregularities seen in actual wireless environments. Not all node pairs may be able to communicate, even if geographically close --- thus, the available pairs are modeled with a link graph $\mathcal{L}=(V,E)$. Also, signal attenuation need not follow a nice geometric formulas --- hence, interference is modeled by a conflict (hyper)graph $\mathcal{C}=(E,F)$ on the links. The objective is to maximize the efficiency of the communication, or equivalently minimizing the length of a schedule of the tree edges in the form of a coloring. We find that in spite of all this generality, the problem can be approximated linearly in terms of a versatile parameter, the inductive independence of the interference graph. Specifically, we give a simple algorithm that attains a $O(ρ\log n)$-approximation, where $n$ is the number of nodes and $ρ$ is the inductive independence, and show that near-linear dependence on $ρ$ is also necessary. We also treat an extension to Steiner trees, modeling multicasting, and obtain a comparable result. Our results suggest that several canonical assumptions of geometry, regularity and "niceness" in wireless settings can sometimes be relaxed without a significant hit in algorithm performance.
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