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Representations of the Multicast Network Problem

Sarah E. Anderson, Wael Halbawi, Nathan Kaplan, Hiram H. López, Felice Manganiello, Emina Soljanin, Judy Walker

TL;DR

This work investigates when multicast network capacity can be achieved by linear network coding over finite fields ${\mathbb F}_q$ and develops multiple representations to study the problem. It introduces coding points and an ILP-based method to minimize their number, then constructs code graphs that capture the essential coding structure via labeled vertices and edge-disjoint paths. The authors connect the labeling problem to algebraic geometry by formulating vector labelings as ${\mathbb F}_q$-points on determinantal varieties and Grassmannians, and discuss how field size $q$ influences feasibility, including nonmonotonic and universality phenomena. The results provide a framework linking network coding with algebraic geometry, enabling rigorous analysis of field-size requirements and highlighting open questions such as which unions of strata arise from actual networks and how obstructions like the Fano plane constrain realizability.

Abstract

We approach the problem of linear network coding for multicast networks from different perspectives. We introduce the notion of the coding points of a network, which are edges of the network where messages combine and coding occurs. We give an integer linear program that leads to choices of paths through the network that minimize the number of coding points. We introduce the code graph of a network, a simplified directed graph that maintains the information essential to understanding the coding properties of the network. One of the main problems in network coding is to understand when the capacity of a multicast network is achieved with linear network coding over a finite field of size q. We explain how this problem can be interpreted in terms of rational points on certain algebraic varieties.

Representations of the Multicast Network Problem

TL;DR

This work investigates when multicast network capacity can be achieved by linear network coding over finite fields and develops multiple representations to study the problem. It introduces coding points and an ILP-based method to minimize their number, then constructs code graphs that capture the essential coding structure via labeled vertices and edge-disjoint paths. The authors connect the labeling problem to algebraic geometry by formulating vector labelings as -points on determinantal varieties and Grassmannians, and discuss how field size influences feasibility, including nonmonotonic and universality phenomena. The results provide a framework linking network coding with algebraic geometry, enabling rigorous analysis of field-size requirements and highlighting open questions such as which unions of strata arise from actual networks and how obstructions like the Fano plane constrain realizability.

Abstract

We approach the problem of linear network coding for multicast networks from different perspectives. We introduce the notion of the coding points of a network, which are edges of the network where messages combine and coding occurs. We give an integer linear program that leads to choices of paths through the network that minimize the number of coding points. We introduce the code graph of a network, a simplified directed graph that maintains the information essential to understanding the coding properties of the network. One of the main problems in network coding is to understand when the capacity of a multicast network is achieved with linear network coding over a finite field of size q. We explain how this problem can be interpreted in terms of rational points on certain algebraic varieties.

Paper Structure

This paper contains 8 sections, 8 theorems, 26 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Achieving multicast network requirements
  3. Coding Points and Reduced Multicast Networks
  4. Code Graphs
  5. ${\mathbb F}_q$-vector labelings and matrices
  6. ${\mathbb F}_q$-vector labelings and rational points
  7. ${\mathbb F}_q$-vector labelings and Grassmannians
  8. An open question on ${\mathbb F}_q$-vector labelings of code graphs

Key Result

Theorem 2.3

Let $G = (\mathcal{V}, \mathcal{E},\mathcal{S},\mathcal{R})$ be the underlying graph of a multicast network where $\mathcal{V}=\{V_i \mid i=1,\dots, t+L\}$, $\mathcal{S}=\{V_i\mid i=1,\dots,N\}$ and $\mathcal{R}=\{V_{t+i}\mid i=1,\dots,L\}$. Consider the following integer optimization problem: where the $x$'s and the $z$'s are variables. A solution $x^*_{k\ell ij}, z^*_{ij}\in \{0,1\}$ of this op

Figures (6)

  • Figure 1: The butterfly network.
  • Figure 2: A multicast network with four sources and three receivers.
  • Figure 3: Modified Butterfly Network
  • Figure 4: Networks with their code graph
  • Figure 5: Combination network and its code graph.
  • ...and 1 more figures

Theorems & Definitions (29)

  • Definition 1.1
  • Definition 2.1
  • Example 2.2
  • Theorem 2.3
  • proof
  • Example 2.4
  • Definition 2.5
  • Definition 3.1
  • Lemma 3.2
  • proof
  • ...and 19 more