Well-solvable special cases of the TSP : a survey

The Traveling Salesman Problem belongs to the most important and most investigated problems in combinatorial optimization. Although it is an NP-hard problem, many of its special cases can be solved efficiently. We survey these special cases with emphasis on results obtained during the decade 1985-1995. This survey complements an earlier survey from 1985 compiled by Gilmore, Lawler and Shmoys. Keywords: Traveling Salesman Problem, Combinatorial optimization, Polynomial time algorithm, Computational complexity

Scheduling Models with Additional Features: Synchronization, Pliability and Resiliency

In this thesis we study three new extensions of scheduling models with both practical and theoretical relevance, namely synchronization, pliability and resiliency. Synchronization has previously been studied for flow shop scheduling and we now apply the concept to open shop models for the first time. Here, as opposed to the traditional models, operations that are processed together all have to be started at the same time. Operations that are completed are not removed from the machines until the longest operation in their group is finished. Pliability is a new approach to model flexibility in flow shops and open shops. In scheduling with pliability, parts of the processing load of the jobs can be re-distributed between the machines in order to achieve better schedules. This is applicable, for example, if the machines represent cross-trained workers. Resiliency is a new measure for the quality of a given solution if the input data are uncertain. A resilient solution remains better than some given bound, even if the original input data are changed. The more we can perturb the input data without the solution losing too much quality, the more resilient the solution is. We also consider the assignment problem, as it is the traditional combinatorial optimization problem underlying many scheduling problems. Particularly, we study a version of the assignment problem with a special cost structure derived from the synchronous open shop model and obtain new structural and complexity results. Furthermore we study resiliency for the assignment problem. The main focus of this thesis is the study of structural properties, algorithm development and complexity. For synchronous open shop we show that for a fixed number of machines the makespan can be minimized in polynomial time. All other traditional scheduling objectives are at least as hard to optimize as in the traditional open shop model. Starting out research in pliability we focus on the most general case of the model as well as two relevant special cases. We deliver a fairly complete complexity study for all three versions of the model. Finally, for resiliency, we investigate two different questions: `how to compute the resiliency of a given solution?' and `how to find a most resilient solution?'. We focus on the assignment problem and single machine scheduling to minimize the total sum of completion times and present a number of positive results for both questions. The main goal is to make a case that the concept deserves further study

The multi-stripe travelling salesman problem

In the classical Travelling Salesman Problem (TSP), the objective function sums the costs for travelling from one city to the next city along the tour. In the q-stripe TSP with q ≥ 1, the objective function sums the costs for travelling from one city to each of the next q cities along the tour. The resulting q-stripe TSP generalizes the TSP and forms a special case of the quadratic assignment problem. We analyze the computational complexity of the q-stripe TSP for various classes of specially structured distance matrices. We derive NP-hardness results as well as polyomially solvable cases. One of our main results generalizes a well-known theorem of Kalmanson from the classical TSP to the q-stripe TSP

A Survey of Monge Properties

Monge properties play an important role in theoretical computer science. Many greedy algorithms are based on such properties, as is speedup in dynamic programming. Monge properties are simple monotonicity properties which are observed and used in various settings such as resource optimization, computational geometry, statistical sampling, computational biology and coding

A study of Monge matrices with applications to scheduling

In this study of Monge properties I summarize a few aspects of the rich material based on these properties. Monge properties are related to the theory of convexity and just as convexity is important in classical Mathematics so is the theory of Monge properties important in Computer Science. In this thesis I will summarize a few aspects of Monge properties. I compare some of the different properties and show how that they all go back to Gaspard Monge\u27s original ideas. I will highlight a number of new results in scheduling and show how Monge properties and Monge-like properties play a role. The thesis will also give results that come from implementing a linear time algorithm for scheduling which is based on Monge properties. I focus especially on batching problems, which are important for TCP/IP acknowledgement; The contribution of my thesis is a summary of important results regarding Monge as well a new implementation of a heuristic for an NP-hard scheduling problem. The main part of my program is a linear time subroutine which is based on a Monge-like property, called the product property

Dual constrained TV-based regularization on graphs

26 pagesInternational audienceAlgorithms based on Total Variation (TV) minimization are prevalent in image processing. They play a key role in a variety of applications such as image denoising, compressive sensing and inverse problems in general. In this work, we extend the TV dual framework that includes Chambolle's and Gilboa-Osher's projection algorithms for TV minimization. We use a flexible graph data representation that allows us to generalize the constraint on the projection variable. We show how this new formulation of the TV problem may be solved by means of fast parallel proximal algorithms. On denoising and deblurring examples, the proposed approach is shown not only to perform better than recent TV-based approaches, but also to perform well on arbitrary graphs instead of regular grids. The proposed method consequently applies to a variety of other inverse problems including image fusion and mesh filtering