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On the Fixed-Parameter Tractability of the Maximum Connectivity Improvement Problem

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Abstract

In the Maximum Connectivity Improvement (MCI) problem, we are given a directed graph G = (V,E) and an integer B and we are asked to find B new edges to be added to G in order to maximize the number of connected pairs of vertices in the resulting graph. The MCI problem has been studied from the approximation point of view. In this paper, we approach it from the parameterized complexity perspective in the case of directed acyclic graphs. We show several hardness and algorithmic results with respect to different natural parameters. Our main result is that the problem is W[2]-hard for parameter B and it is FPT for parameters |V |− B and ν, the matching number of G. We further characterize the MCI problem with respect to other complementary parameters.

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Notes

  1. One can assume that |V | is an upper bound for B as otherwise the graph can be made strongly connected (see Theorem 1).

  2. We can equivalently define MCI as a decision problem without affecting (up to a poly-logarithmic factor) the complexity of the algorithms given in this paper.

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Correspondence to Federico Corò.

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This work has been partially supported by the Italian MIUR PRIN 2017 Project ALGADIMAR “Algorithms, Games, and Digital Markets.”

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Corò, F., D’Angelo, G. & Mkrtchyan, V. On the Fixed-Parameter Tractability of the Maximum Connectivity Improvement Problem. Theory Comput Syst 64, 1094–1109 (2020). https://doi.org/10.1007/s00224-020-09977-6

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