Tight bounds on the competitive ratio on accomodating sequences for the seat reservation problem

The unit price seat reservation problem is investigated. The seat reservation problem is the problem of assigning seat numbers on-line to requests for reservations in a train traveling through $k$ stations. We are considering the version where all tickets have the same price and where requests are treated fairly, i.e., a request which can be fulfilled must be granted. For fair deterministic algorithms, we provide an asymptotically matching upper bound to the existing lower bound which states that all fair algorithms for this problem are frac{1{2-competitive on accommodating sequences, when there are at least three seats. Additionally, we give an asymptotic upper bound of frac{7{9 for fair randomized algorithms against oblivious adversaries. We also examine concrete on-line algorithms, First-Fit and Random, for the special case of two seats. Tight analyses of their performance are given

Online Multi-Coloring with Advice

We consider the problem of online graph multi-coloring with advice. Multi-coloring is often used to model frequency allocation in cellular networks. We give several nearly tight upper and lower bounds for the most standard topologies of cellular networks, paths and hexagonal graphs. For the path, negative results trivially carry over to bipartite graphs, and our positive results are also valid for bipartite graphs. The advice given represents information that is likely to be available, studying for instance the data from earlier similar periods of time.Comment: IMADA-preprint-c

Online Dominating Set

This paper is devoted to the online dominating set problem and its variants on trees, bipartite, bounded-degree, planar, and general graphs, distinguishing between connected and not necessarily connected graphs. We believe this paper represents the first systematic study of the effect of two limitations of online algorithms: making irrevocable decisions while not knowing the future, and being incremental, i.e., having to maintain solutions to all prefixes of the input. This is quantified through competitive analyses of online algorithms against two optimal algorithms, both knowing the entire input, but only one having to be incremental. We also consider the competitive ratio of the weaker of the two optimal algorithms against the other. In most cases, we obtain tight bounds on the competitive ratios. Our results show that requiring the graphs to be presented in a connected fashion allows the online algorithms to obtain provably better solutions. Furthermore, we get detailed information regarding the significance of the necessary requirement that online algorithms be incremental. In some cases, having to be incremental fully accounts for the online algorithm\u27s disadvantage

Alternative Measures for the Analysis of Online Algorithms

In this thesis we introduce and evaluate several new models for the analysis of online algorithms. In an online problem, the algorithm does not know the entire input from the beginning; the input is revealed in a sequence of steps. At each step the algorithm should make its decisions based on the past and without any knowledge about the future. Many important real-life problems such as paging and routing are intrinsically online and thus the design and analysis of online algorithms is one of the main research areas in theoretical computer science. Competitive analysis is the standard measure for analysis of online algorithms. It has been applied to many online problems in diverse areas ranging from robot navigation, to network routing, to scheduling, to online graph coloring. While in several instances competitive analysis gives satisfactory results, for certain problems it results in unrealistically pessimistic ratios and/or fails to distinguish between algorithms that have vastly differing performance under any practical characterization. Addressing these shortcomings has been the subject of intense research by many of the best minds in the field. In this thesis, building upon recent advances of others we introduce some new models for analysis of online algorithms, namely Bijective Analysis, Average Analysis, Parameterized Analysis, and Relative Interval Analysis. We show that they lead to good results when applied to paging and list update algorithms. Paging and list update are two well known online problems. Paging is one of the main examples of poor behavior of competitive analysis. We show that LRU is the unique optimal online paging algorithm according to Average Analysis on sequences with locality of reference. Recall that in practice input sequences for paging have high locality of reference. It has been empirically long established that LRU is the best paging algorithm. Yet, Average Analysis is the first model that gives strict separation of LRU from all other online paging algorithms, thus solving a long standing open problem. We prove a similar result for the optimality of MTF for list update on sequences with locality of reference. A technique for the analysis of online algorithms has to be effective to be useful in day-to-day analysis of algorithms. While Bijective and Average Analysis succeed at providing fine separation, their application can be, at times, cumbersome. Thus we apply a parameterized or adaptive analysis framework to online algorithms. We show that this framework is effective, can be applied more easily to a larger family of problems and leads to finer analysis than the competitive ratio. The conceptual innovation of parameterizing the performance of an algorithm by something other than the input size was first introduced over three decades ago [124, 125]. By now it has been extensively studied and understood in the context of adaptive analysis (for problems in P) and parameterized algorithms (for NP-hard problems), yet to our knowledge this thesis is the first systematic application of this technique to the study of online algorithms. Interestingly, competitive analysis can be recast as a particular form of parameterized analysis in which the performance of opt is the parameter. In general, for each problem we can choose the parameter/measure that best reflects the difficulty of the input. We show that in many instances the performance of opt on a sequence is a coarse approximation of the difficulty or complexity of a given input sequence. Using a finer, more natural measure we can separate paging and list update algorithms which were otherwise indistinguishable under the classical model. This creates a performance hierarchy of algorithms which better reflects the intuitive relative strengths between them. Lastly, we show that, surprisingly, certain randomized algorithms which are superior to MTF in the classical model are not so in the parameterized case, which matches experimental results. We test list update algorithms in the context of a data compression problem known to have locality of reference. Our experiments show MTF outperforms other list update algorithms in practice after BWT. This is consistent with the intuition that BWT increases locality of reference

Operational Research and Machine Learning Applied to Transport Systems

The New Economy, environmental sustainability and global competitiveness drive inno- vations in supply chain management and transport systems. The New Economy increases the amount and types of products that can be delivered directly to homes, challenging the organisation of last-mile delivery companies. To keep up with the challenges, deliv- ery companies are continuously seeking new innovations to allow them to pack goods faster and more efficiently. Thus, the packing problem has become a crucial factor and solving this problem effectively is essential for the success of good deliveries and logistics. On land, rail transportation is known to be the most eco-friendly transport system in terms of emissions, energy consumption, land use, noise levels, and quantities of people and goods that can be moved. It is difficult to apply innovations to the rail industry due to a number of reasons: the risk aversion nature, the high level of regulations, the very high cost of infrastructure upgrades, and the natural monopoly of resources in many countries. In the UK, however, in 2018 the Department for Transport published the Joint Rail Data Action Plan, opening some rail industry datasets for researching purposes. In line with the above developments, this thesis focuses on the research of machine learning and operational research techniques in two main areas: improving packing operations for logistics and improving various operations for passenger rail. In total, the research in this thesis will make six contributions as detailed below. The first contribution is a new mathematical model and a new heuristic to solve the Multiple Heterogeneous Knapsack Problem, giving priority to smaller bins and consid- ering some important container loading constraints. This problem is interesting because many companies prefer to deal with smaller bins as they are less expensive. Moreover, giving priority to filling small bins (rather than large bins) is very important in some industries, e.g. fast-moving consumer goods. The second contribution is a novel strategy to hybridize operational research with ma- chine learning to estimate if a particular packing solution is feasible in a constant O(1) computational time. Given that traditional feasibility checking for packing solutions is an NP-Hard problem, it is expected that this strategy will significantly save time and computational effort. The third contribution is an extended mathematical model and an algorithm to apply the packing problem to improving the seat reservation system in passenger rail. The problem is formulated as the Group Seat Reservation Knapsack Problem with Price on Seat. It is an extension of the Offline Group Seat Reservation Knapsack Problem. This extension introduces a profit evaluation dependent on not only the space occupied, but also on the individual profit brought by each reserved seat. The fourth contribution is a data-driven method to infer the feasible train routing strate- gies from open data in the United Kingdom rail network. Briefly, most of the UK network is divided into sections called berths, and the transition point from one berth to another is called a berth step. There are sensors at berth steps that can detect the movement when a train passes by. The result of the method is a directed graph, the berth graph, where each node represents a berth and each arc represents a berth-step. The arcs rep- resent the feasible routing strategies, i.e. where a train can move from one berth. A connected path between two berths represents a connected section of the network. The fifth contribution is a novel method to estimate the amount of time that a train is going to spend on a berth. This chapter compares two different approaches, AutoRe- gressive Moving Average with Recurrent Neural Networks, and analyse the pros and cons of each choice with statistical analyses. The method is tested on a real-world case study, one berth that represent a busy junction in the Merseyside region. The sixth contribution is an adaptive method to forecast the running time of a train journey using the Gated Recurrent Units method. The method exploits the TD’s berth information and the berth graph. The case-study adopted in the experimental tests is the train network in the Merseyside region