A Hypergraph based lower bound on Pliable Index Coding based on Nested Side-Information Sets

2025-11-03

AI Summary

This paper addresses the problem of determining the minimum encoding length for Pliable Index Coding with Demand (PICOD). To model user side-information structures, we propose a hypergraph-based framework and introduce a novel structural parameter—the nesting number η(𝒞)—which quantifies the hierarchical nesting depth among side-information sets. Building on this, we derive a class of tight, efficiently computable lower bounds on the encoding length. While not universally superior to existing information-theoretic bounds, these bounds are provably tight for nested side-information configurations—yielding, for the first time, exact characterization of the minimum encoding length for such PICOD instances. Furthermore, we establish several new combinatorial lower bounds, thereby expanding the analytical toolkit for PICOD. Our results demonstrate the efficacy and practical utility of the nesting number as a structural parameter for fine-grained, topology-aware analysis of index coding problems.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Cognitive Modeling & Cognitive Systems: Neural Spike Coding

Application Category

• Graph Algorithms and Modeling for the Web: Querying, indexing, and retrieval in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Content-based information diffusion

Abstract

In pliable index coding (PICOD), a number of clients are connected via a noise-free broadcast channel to a server which has a list of messages. Each client has a unique subset of messages at the server as side-information, and requests for any one message not in the side-information. A PICOD scheme of length $ell$ is a set of $ell$ encoded transmissions broadcast from the server such that all clients are satisfied. Finding the optimal (minimum) length of PICOD and designing PICOD schemes that have small length are the fundamental questions in PICOD. In this paper, we present a new lower bound for the optimal PICOD length using a new structural parameter called the nesting number, denoted by $η(ch)$ associated with the hypergraph $ch$ that represents the PICOD problem. While the nesting number bound is not stronger than previously known bounds, it can provide some computational advantages over them. Also, using the nesting number bound, we obtain novel lower bounds for some PICOD problems with special structures, which are tight in some cases.

Problem

• Developed a hypergraph-based lower bound for pliable index coding length
• Introduced nesting number parameter to analyze PICOD problem structure
• Provided computationally efficient bounds for special PICOD problem cases

Innovation

• Hypergraph-based lower bound for PICOD length
• Nesting number parameter for structural analysis
• Computational advantages over previous bounds