On the Complexity of Sub-Tree Scheduling for Wireless Sensor Networks with Partial Coverage

TL;DR

This paper proves that the sub-tree scheduling problem for wireless sensor networks with partial coverage is NP-hard, even when the number of time slots is fixed, and provides polynomial algorithms for special cases.

Contribution

It establishes NP-hardness of the problem in general and fixed cases, and offers polynomial solutions for structured input instances.

Findings

NP-hardness proven for variable and fixed number of time slots

Polynomial algorithms developed for structured input cases

Reductions from the cardinality Steiner tree problem

Abstract

Given an undirected graph $G$ whose edge weights change over $s$ time slots, the sub-tree scheduling for wireless sensor networks with partial coverage asks to partition the vertices of $G$ in $s$ non-empty trees such that the total weight of the trees is minimized. In this note we show that the problem is NP-hard in both the cases where $s$ $(i)$ is part of the input and $(ii)$ is a fixed instance parameter. In both our proofs we reduce from the cardinality Steiner tree problem. We additionally give polynomial-time algorithms for structured inputs of the problem.