Approximation hardness of deadline-TSP reoptimization

AbstractGiven an instance of an optimization problem together with an optimal solution, we consider the scenario in which this instance is modified locally. In graph problems, e.g., a singular edge might be removed or added, or an edge weight might be varied, etc. For a problem U and such a local modification operation, let lm-U (local-modification-U) denote the resulting problem. The question is whether it is possible to exploit the additional knowledge of an optimal solution to the original instance or not, i.e.,whether lm-U is computationally more tractable than U. While positive examples are known e.g. for metric TSP, we give some negative examples here: Metric TSP with deadlines (time windows), if a single deadline or the cost of a single edge is modified, exhibits the same lower bounds on the approximability in these local-modification versions as those currently known for the original problem

New algorithms for Steiner tree reoptimization

Reoptimization is a setting in which we are given an (near) optimal solution of a problem instance and a local modification that slightly changes the instance. The main goal is that of finding an (near) optimal solution of the modified instance. We investigate one of the most studied scenarios in reoptimization known as Steiner tree reoptimization. Steiner tree reoptimization is a collection of strongly NP-hard optimization problems that are defined on top of the classical Steiner tree problem and for which several constant-factor approximation algorithms have been designed in the last decade. In this paper we improve upon all these results by developing a novel technique that allows us to design polynomial-time approximation schemes. Remarkably, prior to this paper, no approximation algorithm better than recomputing a solution from scratch was known for the elusive scenario in which the cost of a single edge decreases. Our results are best possible since none of the problems addressed in this paper admits a fully polynomial-time approximation scheme, unless P=NP

Sequential Decision Making Schemes in Inventory and Transportation Environments

Many mathematical models exist for the simultaneous optimization of transportation and inventory functions. A simultaneous model, while giving the lowest total cost, may not be easily implemented in a firm with decentralized transportation and inventory departments. As such, this thesis studies sequential models, where the primary department is artificially given the authority to make some set of decisions prior to the decisions made by the secondary department. Some known formulations for simultaneous models are studied in an attempt to create a sequential process for the same environment. Finally, a generalized sequential approach is developed that can be applied to any transportation and inventory model with separable costs. The generalized approach allows for the full optimization of the primary departmental costs, and then sequentially allows the optimization of the secondary departmental costs subject to a maximum allowable increase in the costs of the primary department. The analysis of this sequential approach notably reveals that when the relative deviation from the optimal cost of each department is equal, a reasonable solution with respect to total cost is attained. This balance in relative deviation is defined as the fairness point solution. Differing cost scenarios are thus tested to determine the relationship between the cost ratio among departments and the performance of the fairness point solution. The fairness point solution provides an average deviation of total cost from the total optimal cost of less than 1% in four of the seven scenarios tested. Other sequential approaches are discussed and fairness with respect to these new approaches is considered.1 yea

Multi-Robot Task Assignment and Path Finding for Time-Sensitive Missions with Online Task Generation

Executing time-sensitive multi-robot missions involves two distinct problems: Multi-Robot Task Assignment (MRTA) and Multi-Agent Path Finding (MAPF). Computing safe paths that complete every task and minimize the time to mission completion, or makespan, is a significant computational challenge even for small teams. In many missions, tasks can be generated during execution which is typically handled by either recomputing task assignments and paths from scratch, or by modifying existing plans using approximate approaches. While performing task reassignment and path finding from scratch produces theoretically optimal results, the computational load makes it too expensive for online implementation. In this work, we present Time-Sensitive Online Task Assignment and Navigation (TSOTAN), a framework which can quickly incorporate online generated tasks while guaranteeing bounded suboptimal task assignment makespans. It does this by assessing the quality of partial task reassignments and only performing a complete reoptimization when the makespan exceeds a user specified suboptimality bound. Through experiments in 2D environments we demonstrate TSOTAN's ability to produce quality solutions with computation times suitable for online implementation.Comment: 7 pages, 5 figure

Online Optimization with Lookahead

The main contributions of this thesis consist of the development of a systematic groundwork for comprehensive performance evaluation of algorithms in online optimization with lookahead and the subsequent validation of the presented approaches in theoretical analysis and computational experiments

Incorporating A New Class of Uncertainty in Disaster Relief Logistics Planning

In recent years, there has been a growing interest among emergency managers in using Social data in disaster response planning. However, the trustworthiness and reliability of posted information are two of the most significant concerns, because much of the user-generated data is initially not verified. Therefore, a key tradeoff exists for emergency managers when considering whether to incorporate Social data in disaster planning efforts. By considering Social data, a larger number of needs can be identified in a shorter amount of time, potentially enabling a faster response and satisfying a class of demand that might not otherwise be discovered. However, some critical resources can be allocated to inaccurate demands in this manner. This dissertation research is dedicated to evaluating this tradeoff by creating routing plans while considering two separate streams of information: (i) unverified data describing demand that is not known with certainty, obtained from Social media platforms and (ii) verified data describing demand known with certainty, obtained from trusted traditional sources (i.e. on the ground assessment teams). These projects extend previous models in the disaster relief routing literature that address uncertainty in demand. More broadly, this research contributes to the body of literature that addresses questions surrounding the usefulness of Social data for response planning