保距离图压缩
Distance-preserving graph contractions
摘要：
压缩和稀疏化算法经常在分析或优化大型网络/图之前的预处理步骤中应用。在本文中，我们提出并研究了一个新的框架收缩边的图（合并顶点到超顶点）的目标是保持成对距离尽可能准确。形式上，给定边加权图，收缩应该保证对于距离为$d$的任何两个顶点，对应的超顶点在收缩图中保持至少$φ(d)$的距离，其中$φ$是限制允许的距离失真的容差函数。我们提出了一个全面的图片的仿射公差函数$φ(x)=x/α-β$，其中$α\geq 1$和$β\geq 0$是任意实值参数的收缩问题的算法复杂性。具体来说，我们提出了多项式时间算法的树木以及硬度和不逼近的结果，不同的图形类，精确地分离容易和困难的情况下。此外，我们分析了收缩的渐近行为，并找到有效的算法来计算（非最佳）收缩，尽管我们的硬度结果。
英文摘要：
Compression and sparsification algorithms are frequently applied in a preprocessing step before analyzing or optimizing large networks/graphs. In this paper we propose and study a new framework contracting edges of a graph (merging vertices into super-vertices) with the goal of preserving pairwise distances as accurately as possible. Formally, given an edge-weighted graph, the contraction should guarantee that for any two vertices at distance $d$, the corresponding super-vertices remain at distance at least $φ(d)$ in the contracted graph, where $φ$ is a tolerance function bounding the permitted distance distortion. We present a comprehensive picture of the algorithmic complexity of the contraction problem for affine tolerance functions $φ(x)=x/α-β$, where $α\geq 1$ and $β\geq 0$ are arbitrary real-valued parameters. Specifically, we present polynomial-time algorithms for trees as well as hardness and inapproximability results for different graph classes, precisely separating easy and hard cases. Further we analyze the asymptotic behavior of contractions, and find efficient algorithms to compute (non-optimal) contractions despite our hardness results.
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