Research Projects

For many communication models, the capacity characterization problem has been shown to be related to the entropy region. Initially, bounds in terms of the entropy region for distributed source coding and for network coding were given. Even in classical multi-user information theory, the capacity of the broadcast channel has a direct relation to a constrained inequality for five random variables. Due to the structure of communication models, constraints are imposed on the random variables involved, e.g., functional dependence, independence, and Markov chain. Hence, it is of fundamental importance to study constrained entropy regions. An implicit characterization of multi-source multi-sink network coding capacity is known, but the explicit capacity is unknown even for some small network coding instances. For undirected networks, even an implicit characterization is not known yet. However, it was conjectured in 2004 that there is no rate advantage of network coding over routing in undirected multiple-unicast networks. This statement is referred to as the undirected unicast network coding conjecture. Numerous efforts have been made to resolve the conjecture and its relaxed variant. These efforts resulted in the affirmation of the conjecture for special classes of networks. The conjecture is unresolved yet and has far-reaching implications in seemingly unrelated areas, e.g., theoretical computer science. This project focuses on the following four major problems concerning constrained entropy vectors and undirected unicast network information capacity. (1) The four-atom conjecture originated from the study of properties of random variable and their entropy vectors in 1996 and was refuted explicitly in 2016. Despite many attempts, an explicit construction of entropy vectors violating the conjecture remains open. A solution to this problem will provide important insights into entropy vectors and will also pave the way to construct practical network codes. (2) Quasi-uniform entropy vectors are significant due to their one-to-one correspondence with a large class of codes called quasi-uniform codes, which subsumes classes of practical codes such as linear, affine, and group-induced codes. Due to its practical and theoretical significance, we aim to construct constrained quasi-uniform and generic entropy vectors. (3) In our recent work, we derived a first information-theoretic upper bound on the capacity of undirected unicast networks. This bound is related to the well-known independent set problem in graph theory. We aim to derive a stronger bound by exploring the relation between fractional coloring and network flow problems. (4) Resolving the undirected unicast network coding conjecture is one of the most important open problems in network coding theory. We aim to tackle the problem with our recently proposed asymmetric demand network model. In summary, the focus of this project is on obtaining fundamental insights into some of the well-known problems in information theory and thus contributing to the research discipline. The outcomes of the project will bring significant advancements in the well-known open problems in the field and a new deeper understanding and implications for real-world communication system design. This proposed project aims to continue the work done on two projects sanctioned by SERB, DST, Government of India, under (i) Extra Mural Research Funding Scheme (No. SB/S3/EECE/265/2016, Duration: 19/01/2017 - 18/07/2020), and (ii) Core Research Grant (No. CRG/2020/003331, Duration: 05/04/2021 - 04/04/2024). As a part of the outcomes of these two projects, we have so far produced high-quality research publications, including those in IEEE Trans. Inf. Theory, IEEE Trans. Commun., IEEE Trans. Green Commun. Netw., IEEE Commun. Lett., IEEE ISIT, IEEE ITW and ISITA among publications and more works are under preparation.