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Vertex downgrading to minimize connectivity

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Abstract

We consider the problem of interdicting a directed graph by deleting nodes with the goal of minimizing the local edge connectivity of the remaining graph from a given source to a sink. We introduce and study a general downgrading variant of the interdiction problem where the capacity of an arc is a function of the subset of its endpoints that are downgraded, and the goal is to minimize the downgraded capacity of a minimum source-sink cut subject to a node downgrading budget. This models the case when both ends of an arc must be downgraded to remove it, for example. For this generalization, we provide a bicriteria (4, 2)-approximation that downgrades nodes with total weight at most 4 times the budget and provides a solution where the downgraded connectivity from the source to the sink is at most 2 times that in an optimal solution. We accomplish this with an LP relaxation and rounding using a ball-growing algorithm based on the LP values. Furthermore, we show that other bicriteria approximations exist where one can worsen the approximation factor for one of the costs in order to improve the other. We further generalize the downgrading problem to one where each vertex can be downgraded to one of k levels, and the arc capacities are functions of the pairs of levels to which its ends are downgraded. We generalize our LP rounding to get a (4k, 4k)-approximation for this case. Trade-offs between the two approximation ratios similar to the two-level case also exist for the generalized problem. By transferring node values to edge values, we also derive new bicriteria approximation results for the vertex interdiction versions of the multiway cut problem in digraphs and multicut problems in undirected graphs.

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Notes

  1. This is the major difference compared to NVDP where we no longer need to add dummy arcs nor rescale the weights \(x^*\). The relevant arcs in \(H'\) for NVDP forms a subdivision of a star. After deleting all the backward arcs, it no longer remains as a star but could have disconnected arcs. Then, the range of these arcs is no longer bounded around (vv) and thus needed other modifications.

  2. We can simply try all possible L values using binary search to find the smallest value for which the condition holds. To make this search polynomial time, we can simply search over multiplicative powers of \((1 + \epsilon ')\) for some small \(\epsilon ' > 0\) starting from 1 and up to the maximum possible cost of a cut using \(\log _{1 + \epsilon '} {mc_{max}}\) trials where m is the number of arcs and \(c_{max}\) is the largest arc cost.

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Acknowledgements

We thank the two anonymous reviewers whose comments/suggestions helped improve and clarify this manuscript. We especially thank an anonymous reviewer for pointing out the reduction given in Sect. 3.2.

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Authors and Affiliations

  1. Paris Dauphine University, Paris, France

    Hassene Aissi

  2. Carnegie Mellon University, Pittsburgh, USA

    Da Qi Chen

  3. Tepper School of Business, Carnegie Mellon University, Pittsburgh, USA

    R. Ravi

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  1. Hassene Aissi
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  2. Da Qi Chen
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  3. R. Ravi
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Correspondence to Da Qi Chen.

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An extended abstract of this paper appeared in the 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020)

This material is based upon work supported in part by the U. S. Office of Naval Research under Award Number N00014-21-1-2243 (N00014-18-1-2099) and the Air Force Office of Scientific Research under Award Number FA9550-20-1-0080.

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Aissi, H., Chen, D.Q. & Ravi, R. Vertex downgrading to minimize connectivity. Math. Program. 199, 215–249 (2023). https://doi.org/10.1007/s10107-022-01824-5

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