Improved Complexity for Power Edge Set Problem

We study the complexity of Power Edge Set (PES), a problem dedicated to the monitoring of an electric network. In such context we propose some new complexity results. We show that PES remains \(\mathcal {NP}\)-hard in planar graphs with degree at most five. This result is extended to bipartite planar graphs with degree at most six. We also show that PES is hard to approximate within a factor lower than Open image in new window in the bipartite case (resp. \(17/15-\epsilon \)), unless \(\mathcal {P}=\mathcal {NP}\), (resp. under \(\mathcal {UGC}\)). We also show that, assuming \(\mathcal {ETH}\), there is no \(2^{o(\sqrt{n})}\)-time algorithm and no \(2^{o(k)}n^{O(1)}\)-time parameterized algorithm, where n is the number of vertices and k the number of PMUs placed. These results improve the current best known bounds.

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