Preallocation-Based Combinatorial Auction for Efficient Fair Channel Assignments in Multi-Connectivity Networks

We consider a general multi-connectivity framework, intended for ultra-reliable low-latency communications (URLLC) services, and propose a novel, preallocation-based combinatorial auction approach for the efficient allocation of channels. We compare the performance of the proposed method with several other state-of-the-art and alternative channel-allocation algorithms. The two proposed performance metrics are the capacity-based and the utility-based context. In the first case, every unit of additional capacity is regarded as beneficial for any tenant, independent of the already allocated quantity, and the main measure is the total throughput of the system. In the second case, we assume a minimal and maximal required capacity value for each tenant, and consider the implied utility values accordingly. In addition to the total system performance, we also analyze fairness and computational requirements in both contexts. We conclude that at the cost of higher but still plausible computational time, the fairness-enhanced version of the proposed preallocation based combinatorial auction algorithm outperforms every other considered method when one considers total system performance and fairness simultaneously, and performs especially well in the utility context. Therefore, the proposed algorithm may be regarded as candidate scheme for URLLC channel allocation problems, where minimal and maximal capacity requirements have to be considered

The Structure of Connectivity Functions

Graphs, matroids and polymatroids all have associated connectivity functions, and many properties of these structures follow from properties of their connectivity functions. This motivates the study of connectivity functions in general. It turns out that connectivity functions are surprisingly highly structured. We prove some interesting results about connectivity functions. In particular we show that every connectivity function is a connectivity function of a half-integral polymatroid

Structural Results for Matroids.

This dissertation solves some problems involving the structure of matroids. In Chapter 2, the class of binary matroids with no minors isomorphic to the prism graph, its dual, and the binary affine cube is completely determined. This class contains the infinite family of matroids obtained by sticking together a wheel and the Fano matroid across a triangle, and then deleting an edge of the triangle. In Chapter 3, we extend a graph result by D. W. Hall to matroids. Hall proved that if a simple, 3-connected graph has a K\sb5-minor, then it must also have a K\sb{3,3}-minor, the only exception being K\sb5 itself. We prove that if a 3-connected, binary matroid has an M(K\sb5)-minor, then it must also have a minor isomorphic to M(K\sb{3,3}) or its dual, the only exceptions being M(K\sb5), a highly symmetric 12-element matroid T\sb{12}, and T\sb{12} with any edge contracted. Chapter 4 consists of a collection of results on the intersection of circuits and cocircuits in binary matroids. In Chapter 5, we describe, in terms of excluded minors, the class of non-binary matroids with the property that a matroid is in the class if its restriction to every hyperplane is binary

Recognition Problems for Connectivity Functions

A connectivity function is a symmetric, submodular set function. Connectivity functions arise naturally from graphs, matroids and other structures. This thesis focuses mainly on recognition problems for connectivity functions, that is when a connectivity function comes from a particular type of structure. In particular we give a method for identifying when a connectivity function comes from a graph, which uses no more than a polynomial number of evaluations of the connectivity function. We also give a proof that no such method can exist for matroids