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Approximating Small Balanced Vertex Separators in Almost Linear Time

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Abstract

For a graph G with n vertices and m edges, we give a randomized Las Vegas algorithm that approximates a small balanced vertex separator of G in almost linear time. More precisely, we show the following, for any \(\frac{2}{3}\le \alpha <1\) and any \(0<\varepsilon <1-\alpha \): If G contains an \(\alpha \)-separator of size K, then our algorithm finds an \((\alpha +\varepsilon )\)-separator of size \({\mathcal {O}}(\varepsilon ^{-1}K^2\log ^{1+o(1)} n)\) in time \({\mathcal {O}}(\varepsilon ^{-1}K^3m\log ^{2+o(1)} n)\) w.h.p. In particular, if \(K\in {\mathcal {O}}(\hbox {polylog } n)\), then we obtain an \((\alpha +\varepsilon )\)-separator of size \({\mathcal {O}}(\varepsilon ^{-1}\hbox { polylog } n)\) in time \({\mathcal {O}}(\varepsilon ^{-1}m\hbox { polylog } n)\) w.h.p. The presented algorithm does not require knowledge of K.

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Notes

  1. Naturally, other factors like the nature of the considered problem or the exact model of parallelization play a role here.

  2. A vertex separator is said to be \(\alpha \)-balanced if the two vertex sets separated by the vertex separator contain at most \(\alpha n\) vertices each.

  3. Note that the main result holds also for graphs that are not simple as we study vertex separators which, by their nature, do not care if there are multiple edges between two vertices.

  4. For ease of presentation, we refrain from calling it an s–t-cut which would be the technically precise term.

  5. Such a v must exist since for \(v=t\) an i as described above exists.

  6. This applies not only for the remainder of the proof, but also for the comments in Algorithm 1.

  7. Note that \(d_s\) and \(c_t\) are not adjacent in \(\overline{G}\) since s and t are not adjacent in G.

  8. Note that after the above digression, we now consider again undirected graphs.

  9. Note that we have \(|\overline{V(H)}|\le |\overline{V}|\), \(|\overline{E(H)}|\le |\overline{E}|\) and \(k\le K\) at every time during the algorithm.

  10. Note that in this last iteration the found number of pairwise vertex-disjoint s–t-paths must be \(K'\) since there cannot be more such paths than \(K'\) (because \(S^*\) separates s and t and consists of \(K'\) vertices) and if the found number would be strictly smaller than \(K'\), then there would be at least one further iteration (since \(S^*\) separates \(s_0\) and \(t_0\), the last if condition must remain true throughout the whole Algorithm 4 if \(S^*\subseteq V_{t_0}(S)\cap V_{s_0}(T)\)).

  11. From this point on, deviating from the earlier convention, let s and t be as they are in the respective considered iteration (and not as they are at the end of Algorithm 4).

  12. We can obtain such X, Y by successively adding the vertices of \(V(H)\cap S^*\) to the smaller one of \(V(H)\cap A^*\) and \(V(H)\cap B^*\).

  13. The reason we switch to coins instead of iterations is simply that we do not stop throwing coins when we have reached a certain number of heads whereas Algorithm  5 terminates after at most K successful iterations which complicates the analysis.

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  1. Eidgenössische Technische Hochschule Zürich, Universitätstrasse 6, 8092, Zurich, Switzerland

    Sebastian Brandt

  2. Eidgenössische Technische Hochschule Zürich, Gloriastrasse 35, 8092, Zurich, Switzerland

    Roger Wattenhofer

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  1. Sebastian Brandt
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Brandt, S., Wattenhofer, R. Approximating Small Balanced Vertex Separators in Almost Linear Time. Algorithmica 81, 4070–4097 (2019). https://doi.org/10.1007/s00453-018-0490-x

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