A survey of approximation algorithms for capacitated vehicle routing problems

Finding the shortest travelling tour of vehicles with capacity k from the depot to the customers is called the Capacity vehicle routing problem (CVRP). CVRP plays an essential position in logistics systems, and it is the most intensively studied problem in combinatorial optimization. In complexity, CVRP with k $\ge$ 3 is an NP-hard problem, and it is APX-hard as well. We already knew that it could not be approximated in metric space. Moreover, it is the first problem resisting Arora's famous approximation framework. So, whether there is, a polynomial-time (1+$\epsilon$)-approximation for the Euclidean CVRP for any $\epsilon>0$ is still an open problem. This paper will summarize the research progress from history to up-to-date developments. The survey will be updated periodically.Comment: First submissio

Approximation Algorithms for Clustering and Facility Location Problems

Facility location problems arise in a wide range of applications such as plant or warehouse location problems, cache placement problems, and network design problems, and have been widely studied in Computer Science and Operations Research literature. These problems typically involve an underlying set F of facilities that provide service, and an underlying set D of clients that require service, which need to be assigned to facilities in a cost-effective fashion. This abstraction is quite versatile and also captures clustering problems, where one typically seeks to partition a set of data points into k clusters, for some given k, in a suitable way, which themselves find applications in data mining, machine learning, and bioinformatics. Basic variants of facility location problems are now relatively well-u nderstood, but we have much-less understanding of more-sophisticated models that better model the real-world concerns. In this thesis, we focus on three models inspired by some real-world optimization scenarios. In Chapter 2, we consider mobile facility location (MFL) problem, wherein we seek to relocate a given set of facilities to destinations closer to the clients as to minimize the sum of facility-movement and client-assignment costs. This abstracts facility-location settings where one has the flexibility of moving facilities from their current locations to other destinations so as to serve clients more efficiently by reducing their assignment costs. We give the first local-search based approximation algorithm for this problem and achieve the best-known approximation guarantee. Our main result is (3+epsilon)-approximation for this problem for any constant epsilon > 0 using local search which improves the previous best guarantee of 8-approximation algorithm due to [34] based on LP-rounding. Our results extend to the weighted generalization wherein each facility i has a non-negative weight w_i and the movement cost for i is w_i times the distance traveled by i. In Chapter 3, we consider a facility-location problem that we call the minimum-load k-facility location (MLkFL), which abstracts settings where the cost of serving the clients assigned to a facility is incurred by the facility. This problem was studied under the name of min-max star cover in [32,10], who (among other results) gave bicriteria approximation algorithms for MLkFL when F=D. MLkFL is rather poorly understood, and only an O(k)-approximation is currently known for MLkFL, even for line metrics. Our main result is the first polytime approximation scheme (PTAS) for MLkFL on line metrics (note that no non-trivial true approximation of any kind was known for this metric). Complementing this, we prove that MLkFL is strongly NP-hard on line metrics. In Chapter 4, we consider clustering problems with non-uniform lower bounds and outliers, and obtain the first approximation guarantees for these problems. We consider objective functions involving the radii of open facilities, where the radius of a facility i is the maximum distance between i and a client assigned to it. We consider two problems: minimizing the sum of the radii of the open facilities, which yields the lower-bounded min-sum-of-radii with outliers (LBkSRO) problem, and minimizing the maximum radius, which yields the lower-bounded k-supplier with outliers (LBkSupO) problem. We obtain an approximation factor of 12.365 for LBkSRO, which improves to 3.83 for the non-outlier version. These also constitute the first approximation bounds for the min-sum-of-radii objective when we consider lower bounds and outliers separately. We obtain approximation factors of 5 and 3 respectively for LBkSupO and its non-outlier version. These are the first approximation results for k-supplier with non-uniform lower bounds

Combinatorial Optimization

This report summarizes the meeting on Combinatorial Optimization where new and promising developments in the field were discussed. Th

Signalisierte Netzwerkflüsse - Optimierung von Lichtsignalanlagen und Vorwegweisern und daraus resultierende Netzwerkflussprobleme

Guideposts and traffic signals are important devices for controlling inner-city traffic and their optimized operation is essential for efficient traffic flow without congestion. In this thesis, we develop a mathematical model for guideposts and traffic signals in the context of network flow theory. Guideposts lead to confluent flows where each node in the network may have at most one outgoing flow-carrying arc. The complexity of finding maximum confluent flows is studied and several polynomial time algorithms for special graph classes are developed. For traffic signal optimization, a cyclically time-expanded model is suggested which provides the possibility of the simultaneous optimization of offsets and traffic assignment. Thus, the influence of offsets on travel times can be accounted directly. The potential of the presented approach is demonstrated by simulation of real-world instances.Vorwegweiser und Lichtsignalanlagen sind wichtige Elemente zur Steuerung innerstädtischen Verkehrs und ihre optimale Nutzung ist von entscheidender Bedeutung für einen staufreien Verkehrsfluss. In dieser Arbeit werden Vorwegweiser und Lichtsignalanlagen mittels der Netzwerkflusstheorie mathematisch modelliert. Vorwegweiser führen dabei zu konfluenten Flüssen, bei denen Fluss einen Knoten des Netzwerks nur gebündelt auf einer einzigen Kante verlassen darf. Diese konfluenten Flüsse werden hinsichtlich ihrer Komplexität untersucht und es werden Polynomialzeitalgorithmen für das Finden maximaler Flüsse auf ausgewählten Graphenklassen vorgestellt. Für die Versatzzeitoptimierung von Lichtsignalanlagen wird ein zyklisch zeitexpandiertes Modell entwickelt, das die gleichzeitige Optimierung der Verkehrsumlegung ermöglicht. So kann der Einfluss geänderter Versatzzeiten auf die Fahrzeiten direkt berücksichtigt werden. Die Leistungsfähigkeit dieses Ansatzes wird mit Hilfe von Simulationen realistischer Szenarien nachgewiesen

Distributed optimization algorithms for multihop wireless networks

Recent technological advances in low-cost computing and communication hardware design have led to the feasibility of large-scale deployments of wireless ad hoc and sensor networks. Due to their wireless and decentralized nature, multihop wireless networks are attractive for a variety of applications. However, these properties also pose significant challenges to their developers and therefore require new types of algorithms. In cases where traditional wired networks usually rely on some kind of centralized entity, in multihop wireless networks nodes have to cooperate in a distributed and self-organizing manner. Additional side constraints, such as energy consumption, have to be taken into account as well. This thesis addresses practical problems from the domain of multihop wireless networks and investigates the application of mathematically justified distributed algorithms for solving them. Algorithms that are based on a mathematical model of an underlying optimization problem support a clear understanding of the assumptions and restrictions that are necessary in order to apply the algorithm to the problem at hand. Yet, the algorithms proposed in this thesis are simple enough to be formulated as a set of rules for each node to cooperate with other nodes in the network in computing optimal or approximate solutions. Nodes communicate with their neighbors by sending messages via wireless transmissions. Neither the size nor the number of messages grows rapidly with the size of the network. The thesis represents a step towards a unified understanding of the application of distributed optimization algorithms to problems from the domain of multihop wireless networks. The problems considered serve as examples for related problems and demonstrate the design methodology of obtaining distributed algorithms from mathematical optimization methods

Fuelling the zero-emissions road freight of the future: routing of mobile fuellers

The future of zero-emissions road freight is closely tied to the sufficient availability of new and clean fuel options such as electricity and Hydrogen. In goods distribution using Electric Commercial Vehicles (ECVs) and Hydrogen Fuel Cell Vehicles (HFCVs) a major challenge in the transition period would pertain to their limited autonomy and scarce and unevenly distributed refuelling stations. One viable solution to facilitate and speed up the adoption of ECVs/HFCVs by logistics, however, is to get the fuel to the point where it is needed (instead of diverting the route of delivery vehicles to refuelling stations) using "Mobile Fuellers (MFs)". These are mobile battery swapping/recharging vans or mobile Hydrogen fuellers that can travel to a running ECV/HFCV to provide the fuel they require to complete their delivery routes at a rendezvous time and space. In this presentation, new vehicle routing models will be presented for a third party company that provides MF services. In the proposed problem variant, the MF provider company receives routing plans of multiple customer companies and has to design routes for a fleet of capacitated MFs that have to synchronise their routes with the running vehicles to deliver the required amount of fuel on-the-fly. This presentation will discuss and compare several mathematical models based on different business models and collaborative logistics scenarios

LIPIcs, Volume 244, ESA 2022, Complete Volume

LIPIcs, Volume 244, ESA 2022, Complete Volum

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

LIPIcs, Volume 248, ISAAC 2022, Complete Volume

LIPIcs, Volume 248, ISAAC 2022, Complete Volum