Parameterized Complexity of Min-Power Asymmetric Connectivity

TL;DR

This paper explores the parameterized complexity of Min-Power Asymmetric Connectivity, an NP-hard problem in wireless networks, providing efficient algorithms for specific cases and proving hardness results for others.

Contribution

It introduces linear-time algorithms for certain parameters and establishes W[2]-hardness, advancing understanding of the problem's computational complexity.

Findings

Linear-time algorithms for fixed parameters

W[2]-hardness with respect to solution cost

Complexity results for restricted graph classes

Abstract

We investigate parameterized algorithms for the NP-hard problem Min-Power Asymmetric Connectivity (MinPAC) that has applications in wireless sensor networks. Given a directed arc-weighted graph, MinPAC asks for a strongly connected spanning subgraph minimizing the summed vertex costs. Here, the cost of each vertex is the weight of its heaviest outgoing arc in the chosen subgraph. We present linear-time algorithms for the cases where the number of strongly connected components in a so-called obligatory subgraph or the feedback edge number in the underlying undirected graph is constant. Complementing these results, we prove that the problem is W[2]-hard with respect to the solution cost, even on restricted graphs with one feedback arc and binary arc weights.