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

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

Private information retrieval and function computation for noncolluding coded databases

The rapid development of information and communication technologies has motivated many data-centric paradigms such as big data and cloud computing. The resulting paradigmatic shift to cloud/network-centric applications and the accessibility of information over public networking platforms has brought information privacy to the focal point of current research challenges. Motivated by the emerging privacy concerns, the problem of private information retrieval (PIR), a standard problem of information privacy that originated in theoretical computer science, has recently attracted much attention in the information theory and coding communities. The goal of PIR is to allow a user to download a message from a dataset stored on multiple (public) databases without revealing the identity of the message to the databases and with the minimum communication cost. Thus, the primary performance metric for a PIR scheme is the PIR rate, which is defined as the ratio between the size of the desired message and the total amount of downloaded information. The first part of this dissertation focuses on a generalization of the PIR problem known as private computation (PC) from distributed storage system (DSS). In PC, a user wishes to compute a function of f variables (or messages) stored in n noncolluding coded databases, i.e., databases storing data encoded with an [n, k] linear storage code, while revealing no information about the desired function to the databases. Here, colluding databases refers to databases that communicate with each other in order to deduce the identity of the computed function. First, the problem of private linear computation (PLC) for linearly encoded DSS is considered. In PLC, a user wishes to privately compute a linear combination over the f messages. For the PLC problem, the PLC capacity, i.e., the maximum achievable PLC rate, is characterized. Next, the problem of private polynomial computation (PPC) for linearly encoded DSS is considered. In PPC, a user wishes to privately compute a multivariate polynomial of degree at most g over f messages. For the PPC problem an outer bound on the PPC rate is derived, and two novel PPC schemes are constructed. The first scheme considers Reed-Solomon coded databases with Lagrange encoding and leverages ideas from recently proposed star-product PIR and Lagrange coded computation. The second scheme considers databases coded with systematic Lagrange encoding. Both schemes yield improved rates compared to known PPC schemes. Finally, the general problem of PC for arbitrary nonlinear functions from a replicated DSS is considered. For this problem, upper and lower bounds on the achievable PC rate are derived and compared. In the second part of this dissertation, a new variant of the PIR problem, denoted as pliable private information retrieval (PPIR) is formulated. In PPIR, the user is pliable, i.e., interested in any message from a desired subset of the available dataset. In the considered setup, f messages are replicated in n noncolluding databases and classified into F classes. The user wishes to retrieve any one or more messages from multiple desired classes, while revealing no information about the identity of the desired classes to the databases. This problem is termed as multi-message PPIR (M-PPIR), and the single-message PPIR (PPIR) problem is introduced as an elementary special case of M-PPIR. In PPIR, the user wishes to retrieve any one message from one desired class. For the two considered scenarios, outer bounds on the M-PPIR rate are derived for arbitrary number of databases. Next, achievable schemes are designed for n replicated databases and arbitrary n. Interestingly, the capacity of PPIR, i.e., the maximum achievable PPIR rate, is shown to match the capacity of PIR from n replicated databases storing F messages. A similar insight is shown to hold for the general case of M-PPIR

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