Pliable Index Coding via Conflict-Free Colorings of Hypergraphs

In the pliable index coding (PICOD) problem, a server is to serve multiple clients, each of which possesses a unique subset of the complete message set as side information and requests a new message which it does not have. The goal of the server is to do this using as few transmissions as possible. This work presents a hypergraph coloring approach to the PICOD problem. A \textit{conflict-free coloring} of a hypergraph is known from literature as an assignment of colors to its vertices so that each edge of the graph contains one uniquely colored vertex. For a given PICOD problem represented by a hypergraph consisting of messages as vertices and request-sets as edges, we present achievable PICOD schemes using conflict-free colorings of the PICOD hypergraph. Various graph theoretic parameters arising out of such colorings (and some new variants) then give a number of upper bounds on the optimal PICOD length, which we study in this work. Our achievable schemes based on hypergraph coloring include scalar as well as vector linear PICOD schemes. For the scalar case, using the correspondence with conflict-free coloring, we show the existence of an achievable scheme which has length $O(\log^2\Gamma),$ where $\Gamma$ refers to a parameter of the hypergraph that captures the maximum `incidence' number of other edges on any edge. This result improves upon known achievability results in PICOD literature, in some parameter regimes.Comment: 21 page

Bounding the Optimal Length of Pliable Index Coding via a Hypergraph-based Approach

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 use a hypergraph-based approach to derive new achievability and converse results for PICOD. We present an algorithm which gives an achievable scheme for PICOD with length at most $\Delta(\mathcal{H})$, where $\Delta(\mathcal{H})$ is the maximum degree of any vertex in a hypergraph that represents the PICOD problem. We also give a lower bound for the optimal PICOD length using a new structural parameter associated with the PICOD hypergraph called the nesting number. We extend some of our results to the PICOD problem where each client demands $t$ messages, rather than just one. Finally, we identify a class of problems for which our converse is tight, and also characterize the optimal PICOD lengths of problems with $\Delta(\mathcal{H})\in\{1,2,3\}$.Comment: Accepted at the IEEE Information Theory Workshop, 202

Making recommendations bandwidth aware

This paper asks how much we can gain in terms of bandwidth and user satisfaction, if recommender systems became bandwidth aware and took into account not only the user preferences, but also the fact that they may need to serve these users under bandwidth constraints, as is the case over wireless networks. We formulate this as a new problem in the context of index coding: we relax the index coding requirements to capture scenarios where each client has preferences associated with messages. The client is satisfied to receive any message she does not already have, with a satisfaction proportional to her preference for that message. We consistently find, over a number of scenarios we sample, that although the optimization problems are in general NP-hard, significant bandwidth savings are possible even when restricted to polynomial time algorithms

Optimal-Rate Characterisation for Pliable Index Coding using Absent Receivers

We characterise the optimal broadcast rate for a few classes of pliable-index-coding problems. This is achieved by devising new lower bounds that utilise the set of absent receivers to construct decoding chains with skipped messages. This work complements existing works by considering problems that are not complete-S, i.e., problems considered in this work do not require that all receivers with a certain side-information cardinality to be either present or absent from the problem. We show that for a certain class, the set of receivers is critical in the sense that adding any receiver strictly increases the broadcast rate.Comment: Authors' cop

Content-Type Coding

Traditionally, we use networks to securely and efficiently convey specific information messages to one or more receivers. However, communication networks today are increasingly used to serve a fundamentally different traffic, that delivers types of content rather than specific messages. For instance, when we want to find photos of an event, we may not care which specific photos we receive - we only care about the type of content, namely, that they are photos of the correct event. Content-type traffic pervades a host of applications today, e.g., search engines, recommender systems, and advertising networks. Research on content-type networks is very popular. Most of the existing work looks at how to classify content into types; what to replicate, where and how to store, and from where to retrieve specific data. In contrast, we investigate a totally different question: are there benefits in designing information transmission schemes specifically tailored to content-type traffic?Our research indicates that in some cases, these benefits can be significant. We design a polynomial-time algorithm for pliable index coding that requires at most $O(\log^2(n))$ broadcast transmissions to serve $n$ clients, as compared to $O(n)$ broadcast transmissions for conventional index coding. This indicates that the exponential benefits of pliable index coding can be effectively realized. Moreover, we explore two applications: recommender systems and distributed computing. In recommender systems, we ask: how much we can gain in terms of bandwidth and user satisfaction, if recommender systems took into account not only the user preferences, but also the fact that they may need to serve these users under bandwidth constraints, as is the case over wireless networks. In other words, what if the recommender systems became bandwidth aware? In this setup, the user is satisfied to receive any message she does not already have, with a satisfaction proportional to her preference for that message. We show, through a number of scenaria, that although the optimization problems are in general NP-hard, polynomial time algorithms with constant approximation ratio can be designed to achieve more than 80\% of the satisfaction and to save 90\% of bandwidth. In distributed computing, to improve the communication efficiency in the data shuffling phase, we examine the pliable index coding problem under data shuffling constraints, where each of the $m$ messages can satisfy at most $c$ out of $n$ clients. We show that the constrained pliable index coding can achieve up to $O(n)$ (best case) benefits over index coding. We prove that the problem is NP-hard and the optimal broadcast transmissions for random instances is almost surely upper bounded by $O(\min\{\frac{n}{c\log(n)},\frac{n}{\log(m)}\})$ for $c=o(\frac{n^{1/7}}{\log^2(n)})$ and $O(\min\{\frac{n}{c}+\log(c),\frac{n}{\log(m)}\})$ for $c=\Omega(\frac{n^{1/7}}{\log^2(n)})$. Building upon constrained pliable index coding, we design a hierarchical data shuffling scheme for distributed computing. By leveraging the many possible shuffling choices, our proposed shuffling scheme is able to reduce the communication cost and achieve benefits up to $O(ns/m)$ over the index coding method, where $ns/m$ is the average number of workers caching a message, and $m$, $n$, and $s$ are the numbers of messages, workers, and cache size, respectively. In addition, we study the beneficial cases of content-type coding over large scale networks and over erasure channels. Compared with message-specific coding, we show that the benefits can be up to the number of messages in the content type for the former case and up to $19.5\%$ for the latter case with symmetric setting