On the Solvability of 3s/nt Sum-Network---A Region Decomposition and Weak Decentralized Code Method

TL;DR

This paper establishes a polynomial-time solvability criterion for 3s/nt sum-networks, introduces a faster algorithm for 3s/3t sum-networks, and generalizes region decomposition and decentralized coding methods for network coding problems.

Contribution

It provides a necessary and sufficient condition for the solvability of terminal-separable sum-networks and develops efficient algorithms and methods for network coding.

Findings

Polynomial-time decision criterion for terminal-separable sum-networks.

Faster algorithm (O(|E|)) for 3s/3t sum-network solvability.

Generalization of region decomposition and decentralized coding methods.

Abstract

We study the network coding problem of sum-networks with 3 sources and n terminals (3s/nt sum-network), for an arbitrary positive integer n, and derive a sufficient and necessary condition for the solvability of a family of so-called terminal-separable sum-network. Both the condition of terminal-separable and the solvability of a terminal-separable sum-network can be decided in polynomial time. Consequently, we give another necessary and sufficient condition, which yields a faster (O(|E|) time) algorithm than that of Shenvi and Dey ([18], (O(|E|^3) time), to determine the solvability of the 3s/3t sum-network. To obtain the results, we further develop the region decomposition method in [22], [23] and generalize the decentralized coding method in [21]. Our methods provide new efficient tools for multiple source multiple sink network coding problems.