[Paper Review] Minimum Cost Multicast with Decentralized Sources

Nebojsa Milosavljevic, Sameer Pawar+3 more|arXiv (Cornell University)|Mar 11, 2012

Cooperative Communication and Network Coding14 references4 citations

TL;DR

This paper proposes a polynomial-time algorithm for minimum-cost multicast in directed acyclic networks with decentralized sources, where nodes hold partial, correlated side information about a file. It uses submodular flow optimization to determine optimal link rates and constructs explicit transmission schemes via algebraic network coding, achieving optimal cost under linear objectives for both general and linearly correlated side information models.

ABSTRACT

In this paper we study the multisource multicast problem where every sink in a given directed acyclic graph is a client and is interested in a common file. We consider the case where each node can have partial knowledge about the file as a side information. Assuming that nodes can communicate over the capacity constrained links of the graph, the goal is for each client to gain access to the file, while minimizing some linear cost function of number of bits transmitted in the network. We consider three types of side-information settings:(ii) side information in the form of linearly correlated packets; and (iii) the general setting where the side information at the nodes have an arbitrary (i.i.d.) correlation structure. In this work we 1) provide a polynomial time feasibility test, i.e., whether or not all the clients can recover the file, and 2) we provide a polynomial-time algorithm that finds the optimal rate allocation among the links of the graph, and then determines an explicit transmission scheme for cases (i) and (ii).

Motivation & Objective

• To determine feasibility of multicast delivery when nodes hold partial, correlated side information about a file.
• To minimize total communication cost in a directed acyclic network with capacity-constrained links.
• To design an explicit transmission scheme that achieves optimal rate allocation under linear cost functions.
• To extend existing network coding results to multisource multicast with arbitrary side information correlation structures.
• To provide a scalable, polynomial-time solution for both general and linearly correlated side information models.

Proposed method

• Formulates the problem as a linear program with exponentially many rate-flow constraints.
• Introduces 'capacity flow' conditions to verify feasibility in polynomial time using the common base algorithm.
• Applies minimum cost submodular flow algorithms to solve the rate allocation problem for single-client cases.
• Extends single-client solutions to multi-client scenarios using a decomposition approach.
• Constructs explicit transmission schemes via algebraic network coding by introducing a super-source node and solving for transfer matrices.
• Uses simultaneous transfer matrix completion to design a network code in polynomial time, assuming field size > number of clients.

Experimental results

Research questions

• RQ1 Under what conditions can all clients in a directed acyclic network recover the full file when nodes hold partial, correlated side information?
• RQ2 Can the feasibility of multicast delivery be tested in polynomial time for arbitrary side information correlation structures?
• RQ3 Is there a polynomial-time algorithm to compute the optimal rate allocation that minimizes a linear cost function over network links?
• RQ4 How can an explicit transmission scheme be constructed for the case of linearly correlated side information?
• RQ5 Can the solution scale from single-client to multi-client multicast scenarios while preserving optimality and polynomial complexity?

Key findings

• The paper provides a polynomial-time feasibility test for multisource multicast with arbitrary i.i.d. side information correlation structures.
• It presents a polynomial-time algorithm to compute the optimal rate vector minimizing any linear cost function over network links.
• For the linearly correlated side information model, the paper constructs an explicit transmission scheme using algebraic network coding in polynomial time.
• The optimal rate vector is derived as the maximum of individual client requirements across all links, ensuring minimal total transmission cost.
• The solution leverages submodular flow and transfer matrix completion to scale from single-client to multi-client multicast with guaranteed optimality.
• The algorithm's complexity is bounded by O(k·(|E|+N)^3 log(|E|+N)) for the network code construction, assuming |F_q| > k.