Improved Mathematical Programming Techniques for Approximation Algorithms

不幸的是，离散优化中的大量问题在计算上是难以解决的。这些问题源于我们复杂社会所面临的问题：协调交通网络中的车辆，编写代码以创建有效的可执行程序，确定消防站或救护站的位置以缩短响应时间。更准确地说，许多这样的问题是np困难的，这意味着我们没有，也不期望有任何有效的算法来最优地解决这些问题。为了应对这一困难，我们专注于设计有效的算法来找到接近最优的解决方案。

A striking number of problems in discrete optimization are, unfortunately, computationally intractable. These problems stem from issues faced by our complex society: coordinating vehicles in a transportation network, compiling code to create efficient executable programs, determining placements of fire or ambulance stations to improve response time. More precisely, many such problems are NP-hard meaning we do not have, nor do we expect, any efficient algorithms to solve these problems optimally. To cope with this difficulty, we focus on devising efficient algorithms that find near-optimum solutions. The subject of this proposal is designing improved approximation algorithms for NP-hard optimization problems, primarily by devising and analyzing new mathematical programming approaches. The goal is to provide new polynomial-time algorithms to compute solutions whose costs are within some proven explicit bound of the optimum solution. The fact that these algorithms will be based on mathematical programs will also help articulate the connection between theoretical computing science and practical heuristics, as linear and integer programming techniques are often used to devise algorithms that perform well experimentally yet lack proven guarantees on their worst-case performance. My proposed research includes modelling discrete optimization problems as mathematical programs and then relaxing some constraints of these programs to get models that can be solved efficiently. Typically, this is a linear or semidefinite programming relaxation of an integer program. Once this relaxation is solved, the solutions are carefully rounded in a way that obtains a feasible solution to the original model while preserving the objective function value as much as possible. This is already known to be one of the most effective ways to design approximation algorithms; eight chapters of the recent book "The Design of Approximation Algorithms" by Shmoys and Williamson are devoted to this method. In particular, applications of mathematical programming techniques to vehicle routing and resource allocation problems will be investigated. In vehicle routing problems, a strong linear programming approach has recently proven useful in addressing problems with multiple vehicles and I propose to further explore these techniques to address fundamental routing problems. Additionally, I will investigate the so-called unsplittable flow problem in trees which represents the frontier of our understanding in how to approximate resource allocation and packing problems. In particular, I expect that understanding the effectiveness of a relatively new linear programming model should either lead to improved approximations or tighter lower bounds.