Solving Hard Graph Problems with Combinatorial Computing and Optimization

Many problems arising in graph theory are difficult by nature, and finding solutions to large or complex instances of them often require the use of computers. As some such problems are $NP$-hard or lie even higher in the polynomial hierarchy, it is unlikely that efficient, exact algorithms will solve them. Therefore, alternative computational methods are used. Combinatorial computing is a branch of mathematics and computer science concerned with these methods, where algorithms are developed to generate and search through combinatorial structures in order to determine certain properties of them. In this thesis, we explore a number of such techniques, in the hopes of solving specific problem instances of interest. Three separate problems are considered, each of which is attacked with different methods of combinatorial computing and optimization. The first, originally proposed by ErdH{o}s and Hajnal in 1967, asks to find the Folkman number $F_e(3,3;4)$, defined as the smallest order of a $K_4$-free graph that is not the union of two triangle-free graphs. A notoriously difficult problem associated with Ramsey theory, the best known bounds on it prior to this work were $19 leq F_e(3,3;4) leq 941$. We improve the upper bound to $F_e(3,3;4) leq 786$ using a combination of known methods and the Goemans-Williamson semi-definite programming relaxation of MAX-CUT. The second problem of interest is the Ramsey number $R(C_4,K_m)$, which is the smallest $n$ such that any $n$-vertex graph contains a cycle of length four or an independent set of order $m$. With the help of combinatorial algorithms, we determine $R(C_4,K_9)=30$ and $R(C_4,K_{10})=36$ using large-scale computations on the Open Science Grid. Finally, we explore applications of the well-known Lenstra-Lenstra-Lov\u27{a}sz (LLL) algorithm, a polynomial-time algorithm that, when given a basis of a lattice, returns a basis for the same lattice with relatively short vectors. The main result of this work is an application to graph domination, where certain hard instances are solved using this algorithm as a heuristic

Exploiting Symmetry in Integer Convex Optimization using Core Points

We consider convex programming problems with integrality constraints that are invariant under a linear symmetry group. To decompose such problems we introduce the new concept of core points, i.e., integral points whose orbit polytopes are lattice-free. For symmetric integer linear programs we describe two algorithms based on this decomposition. Using a characterization of core points for direct products of symmetric groups, we show that prototype implementations can compete with state-of-the-art commercial solvers, and solve an open MIPLIB problem.Comment: 15 pages; small changes according to suggestions of a referee; to appear in Operations Research Letter

Machine Learning-based Orchestration Solutions for Future Slicing-Enabled Mobile Networks

The fifth generation mobile networks (5G) will incorporate novel technologies such as network programmability and virtualization enabled by Software-Defined Networking (SDN) and Network Function Virtualization (NFV) paradigms, which have recently attracted major interest from both academic and industrial stakeholders. Building on these concepts, Network Slicing raised as the main driver of a novel business model where mobile operators may open, i.e., “slice”, their infrastructure to new business players and offer independent, isolated and self-contained sets of network functions and physical/virtual resources tailored to specific services requirements. While Network Slicing has the potential to increase the revenue sources of service providers, it involves a number of technical challenges that must be carefully addressed. End-to-end (E2E) network slices encompass time and spectrum resources in the radio access network (RAN), transport resources on the fronthauling/backhauling links, and computing and storage resources at core and edge data centers. Additionally, the vertical service requirements’ heterogeneity (e.g., high throughput, low latency, high reliability) exacerbates the need for novel orchestration solutions able to manage end-to-end network slice resources across different domains, while satisfying stringent service level agreements and specific traffic requirements. An end-to-end network slicing orchestration solution shall i) admit network slice requests such that the overall system revenues are maximized, ii) provide the required resources across different network domains to fulfill the Service Level Agreements (SLAs) iii) dynamically adapt the resource allocation based on the real-time traffic load, endusers’ mobility and instantaneous wireless channel statistics. Certainly, a mobile network represents a fast-changing scenario characterized by complex spatio-temporal relationship connecting end-users’ traffic demand with social activities and economy. Legacy models that aim at providing dynamic resource allocation based on traditional traffic demand forecasting techniques fail to capture these important aspects. To close this gap, machine learning-aided solutions are quickly arising as promising technologies to sustain, in a scalable manner, the set of operations required by the network slicing context. How to implement such resource allocation schemes among slices, while trying to make the most efficient use of the networking resources composing the mobile infrastructure, are key problems underlying the network slicing paradigm, which will be addressed in this thesis

Optimization opportunities in human in the loop computational paradigm

An emerging trend is to leverage human capabilities in the computational loop at different capacities, ranging from tapping knowledge from a richly heterogeneous pool of knowledge resident in the general population to soliciting expert opinions. These practices are, in general, termed human-in-the-loop (HITL) computations. A HITL process requires holistic treatment and optimization from multiple standpoints considering all stakeholders: a. applications, b. platforms, c. humans. In application-centric optimization, the factors of interest usually are latency (how long it takes for a set of tasks to finish), cost (the monetary or computational expenses incurred in the process), and quality of the completed tasks. Platform-centric optimization studies throughput, or revenue maximization, while human-centric optimization deals with the characteristics of the human workers, referred to as human factors, such as their skill improvement and learning, to name a few. Finally, fairness and ethical consideration are also of utmost importance in these processes./p\u3e This dissertation aims to design solutions for each of the aforementioned stakeholders. The first contribution of this dissertation is the study of recommending deployment strategies for applications consistent with task requesters’ deployment parameters. From the worker’s standpoint, this dissertation focuses on investigating online group formation where members seek to increase their learning potential via collaboration. Finally, it studies how to consolidate preferences from different workers/applications in a fair manner, such that the final order is both consistent with individual preferences and complies with a group fairness criteria. The technical contributions of this dissertation are to rigorously study these problems from theoretical standpoints, present principled algorithms with theoretical guarantees, and conduct extensive experimental analysis using large-scale real-world datasets to demonstrate their effectiveness and scalability

Best matching processes in distributed systems

The growing complexity and dynamic behavior of modern manufacturing and service industries along with competitive and globalized markets have gradually transformed traditional centralized systems into distributed networks of e- (electronic) Systems. Emerging examples include e-Factories, virtual enterprises, smart farms, automated warehouses, and intelligent transportation systems. These (and similar) distributed systems, regardless of context and application, have a property in common: They all involve certain types of interactions (collaborative, competitive, or both) among their distributed individuals—from clusters of passive sensors and machines to complex networks of computers, intelligent robots, humans, and enterprises. Having this common property, such systems may encounter common challenges in terms of suboptimal interactions and thus poor performance, caused by potential mismatch between individuals. For example, mismatched subassembly parts, vehicles—routes, suppliers—retailers, employees—departments, and products—automated guided vehicles—storage locations may lead to low-quality products, congested roads, unstable supply networks, conflicts, and low service level, respectively. This research refers to this problem as best matching, and investigates it as a major design principle of CCT, the Collaborative Control Theory. The original contribution of this research is to elaborate on the fundamentals of best matching in distributed and collaborative systems, by providing general frameworks for (1) Systematic analysis, inclusive taxonomy, analogical and structural comparison between different matching processes; (2) Specification and formulation of problems, and development of algorithms and protocols for best matching; (3) Validation of the models, algorithms, and protocols through extensive numerical experiments and case studies. The first goal is addressed by investigating matching problems in distributed production, manufacturing, supply, and service systems based on a recently developed reference model, the PRISM Taxonomy of Best Matching. Following the second goal, the identified problems are then formulated as mixed-integer programs. Due to the computational complexity of matching problems, various optimization algorithms are developed for solving different problem instances, including modified genetic algorithms, tabu search, and neighbourhood search heuristics. The dynamic and collaborative/competitive behaviors of matching processes in distributed settings are also formulated and examined through various collaboration, best matching, and task administration protocols. In line with the third goal, four case studies are conducted on various manufacturing, supply, and service systems to highlight the impact of best matching on their operational performance, including service level, utilization, stability, and cost-effectiveness, and validate the computational merits of the developed solution methodologies

Energy-efficient resource allocation in limited fronthaul capacity cloud-radio access networks

In recent years, cloud radio access networks (C-RANs) have demonstrated their role as a formidable technology candidate to address the challenging issues from the advent of Fifth Generation (5G) mobile networks. In C-RANs, the modules which are capable of processing data and handling radio signals are physically separated in two main functional groups: the baseband unit (BBU) pool consisting of multiple BBUs on the cloud, and the radio access networks (RANs) consisting of several low-power remote radio heads (RRH) whose functionality are simplified with radio transmission/reception. Thanks to the centralized computation capability of cloud computing, C-RANs enable the coordination between RRHs to significantly improve the achievable spectral efficiency to satisfy the explosive traffic demand from users. More importantly, this enhanced performance can be attained at its power-saving mode, which results in the energy-efficient C-RAN perspective. Note that such improvement can be achieved under an ideal fronthaul condition of very high and stable capacity. However, in practice, dedicated fronthaul links must remarkably be divided to connect a large amount of RRHs to the cloud, leading to a scenario of non-ideal limited fronthaul capacity for each RRH. This imposes a certain upper-bound on each user’s spectral efficiency, which limits the promising achievement of C-RANs. To fully harness the energy-efficient C-RANs while respecting their stringent limited fronthaul capacity characteristics, a more appropriate and efficient network design is essential. The main scope of this thesis aims at optimizing the green performance of C-RANs in terms of energy-efficiency under the non-ideal fronthaul capacity condition, namely energy-efficient design in limited fronthaul capacity C-RANs. Our study, via jointly determining the transmit beamforming, RRH selection, and RRH–user association, targets the following three vital design issues: the optimal trade-off between maximizing achievable sum rate and minimizing total power consumption, the maximum energy-efficiency under adaptive rate-dependent power model, the optimal joint energy-efficient design of virtual computing along with the radio resource allocation in virtualized C-RANs. The significant contributions and novelties of this work can be elaborated in the followings. Firstly, the joint design of transmit beamforming, RRH selection, and RRH–user association to optimize the trade-off between user sum rate maximization and total power consumption minimization in the downlink transmissions of C-RANs is presented in Chapter 3. We develop one powerful with high-complexity and two novel efficient low-complexity algorithms to respectively solve for a global optimal and high-quality sub-optimal solutions. The findings in this chapter show that the proposed algorithms, besides overcoming the burden to solve difficult non-convex problems within a polynomial time, also outperform the techniques in the literature in terms of convergence and achieved network performance. Secondly, Chapter 4 proposes a novel model reflecting the dependence of consumed power on the user data rate and highlights its impact through various energy-efficiency metrics in CRANs. The dominant performance of the results form Chapter 4, compared to the conventional work without adaptive rate-dependent power model, corroborates the importance of the newly proposed model in appropriately conserving the system power to achieve the most energy efficient C-RAN performance. Finally, we propose a novel model on the cloud center which enables the virtualization and adaptive allocation of computing resources according to the data traffic demand to conserve more power in Chapter 5. A problem of jointly designing the virtual computing resource together with the beamforming, RRH selection, and RRH–user association which maximizes the virtualized C-RAN energy-efficiency is considered. To cope with the huge size of the formulated optimization problem, a novel efficient with much lower-complexity algorithm compared to previous work is developed to achieve the solution. The achieved results from different evaluations demonstrate the superiority of the proposed designs compared to the conventional work

Operational Research: Methods and Applications

Throughout its history, Operational Research has evolved to include a variety of methods, models and algorithms that have been applied to a diverse and wide range of contexts. This encyclopedic article consists of two main sections: methods and applications. The first aims to summarise the up-to-date knowledge and provide an overview of the state-of-the-art methods and key developments in the various subdomains of the field. The second offers a wide-ranging list of areas where Operational Research has been applied. The article is meant to be read in a nonlinear fashion. It should be used as a point of reference or first-port-of-call for a diverse pool of readers: academics, researchers, students, and practitioners. The entries within the methods and applications sections are presented in alphabetical order. The authors dedicate this paper to the 2023 Turkey/Syria earthquake victims. We sincerely hope that advances in OR will play a role towards minimising the pain and suffering caused by this and future catastrophes