On the Minimum Number of Transmissions in Single-Hop Wireless Coding Networks

TL;DR

This paper investigates the minimum number of transmissions needed in wireless network coding, proving NP-completeness over GF(2), and proposes a heuristic solution with empirical validation.

Contribution

It establishes the NP-completeness of the optimal encoding problem in wireless network coding and introduces a graph coloring based heuristic approach.

Findings

NP-complete over GF(2)

Proposed a graph coloring heuristic

Empirical results on random settings

Abstract

The advent of network coding presents promising opportunities in many areas of communication and networking. It has been recently shown that network coding technique can significantly increase the overall throughput of wireless networks by taking advantage of their broadcast nature. In wireless networks, each transmitted packet is broadcasted within a certain area and can be overheard by the neighboring nodes. When a node needs to transmit packets, it employs the opportunistic coding approach that uses the knowledge of what the node's neighbors have heard in order to reduce the number of transmissions. With this approach, each transmitted packet is a linear combination of the original packets over a certain finite field. In this paper, we focus on the fundamental problem of finding the optimal encoding for the broadcasted packets that minimizes the overall number of transmissions. We show that this problem is NP-complete over GF(2) and establish several fundamental properties of the optimal solution. We also propose a simple heuristic solution for the problem based on graph coloring and present some empirical results for random settings.