A Study of Communication Networks through the Lens of Reduction

A central goal of information theory is to characterize the capacity regions of communication networks. Due to the difficulty of the general problem, research is primarily focused on families of problems defined by various classifiers. These classifiers include the channel transition function (i.e., noisy, deterministic, network coding), demand type (i.e., single-source, 2-unicast), network topology (i.e. acyclic network coding, index coding). To date, the families of networks that are fully solved remain limited. Moreover, results derived for one specific family often do not extend easily to other families of problems. Our work shifts from the traditional focus on solving example networks to one that builds connections between problem solutions so that we can say where and when solving a problem in one domain would also solve a corresponding problem in another domain. Central to our approach is a technique called "reduction", in which we connect the solutions and results of communication problems. We say that problem A reduces to problem B when A can be solved by first transforming it to B and then applying a solution for B. We focus on two notions of reduction: reduction in code design and reduction in capacity region. Our central results demonstrate reductions with respect to a variety of classifiers. We show that finding multiple multicast network capacity regions reduces to finding multiple unicast network capacity regions both when capacity is defined as the maximal rate over all possible codes and when capacity is defined as the optimal rate over linear codes. As a corollary to this result, we show that the same capacity reduction holds for when network types are limited to either network coding networks or index coding networks. In several instances, we show that a reduction in code design extends to a reduction in capacity region if and only if the edge removal conjecture holds. Here, the edge removal conjecture states that removing an edge of negligible capacity from a network does not change its capacity region. One of the key challenges in network coding research is how to handle networks containing cycles. As a result, many papers on network coding restrict attention to acyclic networks and some results derived for acyclic networks do not extend to networks containing cycles. We consider a streaming model for network communication where information is streamed to its destination under a constraint on maximal delay at the decoder. Restricting our attention to this scenario enables us to prove a code reduction from network coding to index coding in both acyclic and cyclic networks. Since index coding networks are acyclic, a consequence of this reduction is that under the streaming model, there is no fundamental difference between acyclic and cyclic networks.</p

A Geometric Framework for Investigating the Multiple Unicast Network Coding Conjecture

The multiple unicast network coding conjecture states that for multiple unicast sessions in an undirected network, network coding is equivalent to routing. Simple and intuitive as it appears, the conjecture has remained open since its proposal in 2004 [1], [2], and is now a well-known unsolved problem in the field of network coding. Based on a recently proposed tool of space information flow [3]-[5], we present a geometric framework for analyzing the multiple unicast conjecture. The framework consists of four major steps, in which the conjecture is transformed from its throughput version to cost version, from the graph domain to the space domain, and then from high dimension to 1-D, where it is to be eventually proved. We apply the geometric framework to derive unified proofs to known results of the conjecture, as well as new results previously unknown. A possible proof to the conjecture based on this framework is outlined.published_or_final_versio

Precoding-Based Network Alignment For Three Unicast Sessions

We consider the problem of network coding across three unicast sessions over a directed acyclic graph, where each sender and the receiver is connected to the network via a single edge of unit capacity. We consider a network model in which the middle of the network only performs random linear network coding, and restrict our approaches to precoding-based linear schemes, where the senders use precoding matrices to encode source symbols. We adapt a precoding-based interference alignment technique, originally developed for the wireless interference channel, to construct a precoding-based linear scheme, which we refer to as as a {\em precoding-based network alignment scheme (PBNA)}. A primary difference between this setting and the wireless interference channel is that the network topology can introduce dependencies between elements of the transfer matrix, which we refer to as coupling relations, and can potentially affect the achievable rate of PBNA. We identify all possible such coupling relations, and interpret these coupling relations in terms of network topology and present polynomial-time algorithms to check the presence of these coupling relations. Finally, we show that, depending on the coupling relations present in the network, the optimal symmetric rate achieved by precoding-based linear scheme can take only three possible values, all of which can be achieved by PBNA.Comment: arXiv admin note: text overlap with arXiv:1202.340

Algorithmic Aspects of Energy-Delay Tradeoff in Multihop Cooperative Wireless Networks

We consider the problem of energy-efficient transmission in delay constrained cooperative multihop wireless networks. The combinatorial nature of cooperative multihop schemes makes it difficult to design efficient polynomial-time algorithms for deciding which nodes should take part in cooperation, and when and with what power they should transmit. In this work, we tackle this problem in memoryless networks with or without delay constraints, i.e., quality of service guarantee. We analyze a wide class of setups, including unicast, multicast, and broadcast, and two main cooperative approaches, namely: energy accumulation (EA) and mutual information accumulation (MIA). We provide a generalized algorithmic formulation of the problem that encompasses all those cases. We investigate the similarities and differences of EA and MIA in our generalized formulation. We prove that the broadcast and multicast problems are, in general, not only NP hard but also o(log(n)) inapproximable. We break these problems into three parts: ordering, scheduling and power control, and propose a novel algorithm that, given an ordering, can optimally solve the joint power allocation and scheduling problems simultaneously in polynomial time. We further show empirically that this algorithm used in conjunction with an ordering derived heuristically using the Dijkstra's shortest path algorithm yields near-optimal performance in typical settings. For the unicast case, we prove that although the problem remains NP hard with MIA, it can be solved optimally and in polynomial time when EA is used. We further use our algorithm to study numerically the trade-off between delay and power-efficiency in cooperative broadcast and compare the performance of EA vs MIA as well as the performance of our cooperative algorithm with a smart noncooperative algorithm in a broadcast setting.Comment: 12 pages, 9 figure