This tutorial is devoted to the notion of max-min fairness (MMF), associated optimization problems, and their applications to multi-commodity flow networks.We first introduce a theoretical background for a generic MMF optimization problem and discuss its relation to lexicographic optimization. We next present resolution algorithms for convex MMF optimization, and analyze their properties. In the second half of the tutorial we discuss its applications to communication networks, in particular to routing and load-balancing. We state several properties with respect to each of the studied problems and analyze the behavior of the algorithms.

Optimization models related to designing and operating complex systems are mainly focused on some efficiency metrics such as response time, queue length, throughput, and cost. However, in systems which serve many entities there is also a need for respecting fairness: each system entity ought to be provided with an adequate share of the system’s services. Still, due to system operations-dependant constraints, fair treatment of the entities does not directly imply that each of them is assigned equal amount of the services. That leads to concepts of fair optimization expressed by the equitable models that represent inequality averse optimization rather than strict inequality minimization; a particular widely applied example of that concept is the so-called lexicographic maximin optimization (max-min fairness). The fair optimization methodology delivers a variety of techniques to generate fair and efficient solutions. This paper reviews fair optimization models and methods applied to systems that are based on some kind of network of connections and dependencies, especially, fair optimization methods for the location problems and for the resource allocation problems in communication networks.

Established approaches to data aggregation in wireless sensor networks (WSNs) do not cover the variety of new use cases developing with the advent of the Internet of Things. In particular, the current push towards fog computing, in which control, computation, and storage are moved to nodes close to the network edge, induces a need to collect data at multiple sinks, rather than the single sink typically considered in WSN aggregation algorithms. Moreover, for machine-to-machine communication scenarios, actuators subscribing to sensor measurements may also be present, in which case data should be not only aggregated and processed in-network, but also disseminated to actuator nodes. In this paper, we present mixed-integer programming formulations and algorithms for the problem of energy-optimal routing and multiple-sink aggregation, as well as joint aggregation and dissemination, of sensor measurement data in IoT edge networks. We consider optimisation of the network for both minimal total energy usage, and min-max per-node energy usage. We also provide a formulation and algorithm for throughput-optimal scheduling of transmissions under the physical interference model in the pure aggregation case. We have conducted a numerical study to compare the energy required for the two use cases, as well as the time to solve them, in generated network scenarios with varying topologies and between 10 and 40 nodes. Although aggregation only accounts for less than 15% of total energy usage in all cases tested, it provides substantial energy savings. Our results show more than 13 times greater energy usage for 40-node networks using direct, shortest-path flows from sensors to actuators, compared with our aggregation and dissemination solutions.

With the densification of nodes in cellular networks, free space optic (FSO) connections are becoming an appealing low cost and high rate alternative to copper and fiber backhaul solutions for wireless communication systems. To ensure a reliable cellular backhaul, provisions for redundant disjoint paths between the nodes must be made in the design phase. This paper aims at finding a cost-effective solution to upgrade the cellular backhaul with pre-deployed optical fibers using FSO links and mirror components. Since the quality of the FSO links depends on several factors, such as transmission distance, power, and weather conditions, we adopt an elaborate formulation to calculate link reliability. We present a novel integer linear programming model to approach optimal FSO backhaul design, guaranteeing $K$-disjoint paths connecting each node pair. Next, we derive a column generation method to a path-oriented mathematical formulation. Applying the method in a sequential manner enables high computational scalability. We use realistic scenarios to demonstrate that our approaches efficiently provide optimal or near-optimal solutions, and thereby allow for accurately dealing with the trade-off between cost and reliability.