In various business and military settings, there is an expectation of on-demand delivery of supplies and services. Typically, several delivery vehicles (also called servers) carry these supplies. Requests arrive one at a time and when a request arrives, a server is assigned to this request at a cost that is proportional to the distance between the server and the request. Bad assignments will not only lead to larger costs but will also create bottlenecks by increasing delivery time. There is, therefore, a need to design decision-making algorithms that produce cost-effective assignments of servers to requests in real-time. In this thesis, we consider the online bipartite matching problem where each server can serve exactly one request. In the online minimum metric bipartite matching problem, we are provided with a set of server locations in a metric space. Requests arrive one at a time that have to be immediately and irrevocably matched to a free server. The total cost of matching all the requests to servers, also known as the online matching is the sum of the cost of all the edges in the matching. There are many well-studied models for request generation. We study the problem in the adversarial model where an adversary who knows the decisions made by the algorithm generates a request sequence to maximize ratio of the cost of the online matching and the minimum-cost matching (also called the competitive ratio). An algorithm is a-competitive if the cost of online matching is at most 'a' times the minimum cost. A recently discovered robust and deterministic online algorithm (we refer to this as the robust matching or the RM-Algorithm) was shown to have optimal competitive ratios in the adversarial model and a relatively weaker random arrival model. We extend the analysis of the RM-Algorithm in the adversarial model and show that the competitive ratio of the algorithm is sensitive to the input, i.e., for "nice" input metric spaces or "nice" server placements, the performance guarantees of the RM-Algorithm is significantly better. In fact, we show that the performance is almost optimal for any fixed metric space and server locations. / Master of Science

<div>We consider the maximum vertex-weighted matching problem (MVM), in which non-negative weights are assigned to the vertices of a graph, and the weight of a matching is the sum of the weights of the matched vertices. Vertex-weighted matchings arise in many applications, including internet advertising, facility scheduling, constraint satisfaction, the design of network switches, and computation of sparse bases for the null space or the column space of a matrix. Let m be the number of edges, n number of vertices, and D the maximum degree of a vertex in the graph. We design two exact algorithms for the MVM problem with time complexities of O(mn) and O(Dmn). The new exact algorithms use a maximum cardinality matching as an initial matching, after which the weight of the matching is increased using weight-increasing paths.</div><div><br></div><div>Although MVM problems can be solved exactly in polynomial time, exact MVM algorithms are still slow in practice for large graphs with millions and even billions of edges. Hence we investigate several approximation algorithms for MVM in this thesis. First we show that a maximum vertex-weighted matching can be approximated within an approximation ratio arbitrarily close to one, to k/(k + 1), where k is related to the length of augmenting or weight-increasing paths searched by the algorithm. We identify two main approaches for designing approximation algorithms for MVM. The first approach is direct; vertices are sorted in non-increasing order of weights, and then the algorithm searches for augmenting paths of restricted length that reach a heaviest vertex. (In this approach each vertex is processed once). The second approach repeatedly searches for augmenting paths and increasing paths, again of restricted length, until none can be found. In this second, iterative approach, a vertex may need to be processed multiple times. We design two approximation algorithms based on the direct approach with approximation ratios of 1/2 and 2/3. The time complexities of the 1/2-approximation algorithm is O(m + n log n), and that of the 2/3-approximation algorithm is O(mlogD). Employing the second approach, we design 1/2- and 2/3-approximation algorithms for MVM with time complexities of O(Dm) and O(D²m), respectively. We show that the iterative algorithm can be generalized to nd a k/(k+1)-approximate MVM with a time complexity of O(D^km). In addition, we design parallel 1/2- and 2/3-approximation algorithms for a shared memory programming model, and introduce a new technique for locking augmenting paths to avoid deadlock and related problems. </div><div><br></div><div>MVM problems may be solved using algorithms for the maximum edge-weighted matching (MEM) by assigning to each edge a weight equal to the sum of the vertex weights on its endpoints. However, our results will show that this is one way to generate MEM problems that are difficult to solve. On such problems, exact MEM algorithms may require run times that are a factor of a thousand or more larger than the time of an exact MVM algorithm. Our results show the competitiveness of the new exact algorithms by demonstrating that they outperform MEM exact algorithms. Specifically, our fastest exact algorithm runs faster than the fastest MEM implementation by a factor of 37 and 18 on geometric mean, using two different sets of weights on our test problems. In some instances, the factor can be higher than 500. Moreover, extensive experimental results show that the MVM approximation algorithm outperforms an MEM approximation algorithm with the same approximation ratio, with respect to matching weight and run time. Indeed, our results show that the MVM approximation algorithm outperforms the corresponding MEM algorithm with respect to these metrics in both serial and parallel settings.</div>

Immersion based technologies such as Augmented Reality (AR), Virtual Reality (VR) and Mixed Reality (MR), together falling under the umbrella of Extended Reality (XR) have taken the world by storm in the recent past. However, with the growing market and the increasing number of applications of XR, multiple challenges have arisen. To maintain acceptable levels of motion-to-photon latency, there is a need to serve the users with ultra low latency and with high reliability. To provide high quality rendering, these solutions have traditionally been deployed with wired connections, but severely inhibiting user mobility. Thus, the need to develop wireless solutions promising ultra low latency and high reliability emerges. Cloud/Edge based solutions promise to provide great dividends in this regard but it still remains crucial to understand how different scheduling policies perform against one another in terms of average throughput, mean system time, the number of UEs which can be serviced simultaneously etc. In this thesis, we explore how online packet scheduling policies such as first-come-first-serve, earliestdeadline-first, maximum weight scheduling etc. compare against other Quality of Experience(QoE)/ packet weight aware online scheduling policies and also against optimal offline schemes such as maximum-weighted-bipartitematching. We perform a detailed analysis of how these policies fare by studying various metrics such as the average-packet system time, competitive ratios, packet drop percentages and weight throughput, amongst others. Finally, we also explore how the introduction of multi-layered video encoding impacts XR service. Amongst the findings of the thesis, we conclude that it is possible to come up with solutions such as EDFα (which is a deadline and weight aware derivative of the earliest deadline first scheduling policy), which can either increase the weight throughput when compared to other baselines while also providing lesser packet drops and lower average system times for the scheduled packets. This algorithm can be further tuned by varying α to accordingly alter the weight throughput, system time and packet drop ratio depending on the precise user application. Additionally, we establish with the help of simulations that the introduction of multi-layered video encoding conclusively helps in reducing the average system time and eventually allows for more users to be accommodated in an XR based system at the cost of worsening video quality. / Immersionsbaserade teknologier som Augmented Reality (AR), Virtual Reality (VR) och Mixed Reality (MR), som tillsammans faller under paraplyet Extended Reality (XR) har tagit världen med storm på senare tid. Men med den växande marknaden och det ökande antalet tillämpningar av XR har flera utmaningar uppstått. För att förhindra åksjuka hos användare och för att upprätthålla acceptabla nivåer av rörelse-till-foton-latens, finns det ett behov av att betjäna användarna med ultralåg latens och med hög tillförlitlighet. För att ge högkvalitativ rendering har dessa lösningar traditionellt implementerats med trådbundna anslutningar, men de hämmar kraftigt användarens rörlighet. Därför uppstår behovet av att utveckla trådlösa lösningar som lovar ultralåg latens och hög tillförlitlighet. Moln/Edge-baserade lösningar lovar att ge stor utdelning i detta avseende, men det är fortfarande viktigt att förstå hur olika schemaläggningspolicyer fungerar mot varandra när det gäller genomsnittlig genomströmning, genomsnittlig systemtid, antalet UE:er som kan betjänas samtidigt etc. I den här avhandlingen undersöker vi hur online-paketschemaläggningspolicyer som round robin, först till kvarnförst-kvarn, tidigast-deadline-först, schemaläggning för maximal vikt etc. jämförs med andra Quality of Experience (QoE)/Viktmedvetna onlineschemaläggningspolicyer och även mot optimala offline-scheman såsom maximalt viktad-bipartite-matchning. Vi utför en detaljerad analys av hur dessa policyer klarar sig genom att studera olika mätvärden, såsom den genomsnittliga paketets systemtid, konkurrensförhållanden, procentsatser för paketnedgång och viktad genomströmning, bland annat. Slutligen undersöker vi också hur introduktionen av flerskiktad videokodning påverkar XRtjänsten. Bland resultaten av avhandlingen drar vi slutsatsen att det är möjligt att komma med lösningar som EDFα (som är en deadline- och viktmedveten derivata av Earliest deadline first scheduling policy), som antingen kan öka den viktade genomströmning jämfört med andra baslinjer samtidigt som det ger mindre paketnedgångar och lägre genomsnittliga systemtider för de schemalagda paketen. Denna algoritm kan ställas in ytterligare genom att variera α för att följaktligen ändra den viktade genomströmningen, systemtiden och paketnedgångshastigheten beroende på den exakta användarapplikationen. Dessutom fastställer vi med hjälp av simuleringar att införandet av flerskiktsvideokodning definitivt hjälper till att minska den genomsnittliga systemtiden och så småningom tillåter fler användare att få plats i ett XR-baserat system.