Abstract
We present a new bound relating edge connectivity in a simple, unweighted graph with effective resistance in the corresponding electrical network. The bound is tight. While we believe the bound is of independent interest, our work is motivated by the problem of constructing combinatorial and spectral sparsifiers of a graph, i.e., sparse, weighted sub-graphs that preserve cut information (in the case of combinatorial sparsifiers) and additional spectral information (in the case of spectral sparsifiers). Recent results by Fung et al. (STOC 2011) and Spielman and Srivastava (SICOMP 2011) show that sampling edges with probability based on edge-connectivity gives rise to a combinatorial sparsifier whereas sampling edges with probability based on effective resistance gives rise to a spectral sparsifier. Our result implies that by simply increasing the sampling probability by a O(n 2/3) factor in the combinatorial sparsifier construction, we also preserve the spectral properties of the graph. Combining this with the algorithms of Ahn et al. (SODA 2012, PODS 2012) gives rise to the first data stream algorithm for the construction of spectral sparsifiers in the dynamic setting where edges can be added or removed from the stream. This was posed as an open question by Kelner and Levin (STACS 2011).
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Ahn, K.J., Guha, S., McGregor, A. (2013). Spectral Sparsification in Dynamic Graph Streams. In: Raghavendra, P., Raskhodnikova, S., Jansen, K., Rolim, J.D.P. (eds) Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques. APPROX RANDOM 2013 2013. Lecture Notes in Computer Science, vol 8096. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-40328-6_1
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