Approximation and online algorithms in scheduling and coloring

In the last three decades, approximation and online algorithms have become a major area of theoretical computer science and discrete mathematics. Scheduling and coloring problems are among the most popular ones for which approximation and online algorithms have been analyzed. On one hand, motivated by the well-known difficulty to obtain good lower bounds for the problems, it is particularly hard to prove results on the online and offline performance of algorithms. On the other hand, the theoretically oriented studies of approximation and online algorithms for scheduling and coloring have also impact on the development of better algorithms for real world applications. In the thesis we present approximation algorithms and online algorithms for a number of scheduling and labeling (coloring) problems. Our work in the first part of the thesis is devoted to scheduling problems with the average weighted completion time objective function, that is primarily motivated by some theoretical questions which were open for a number of recent years. Here we present a general method which leads to the design of polynomial time approximation schemes (PTASs), best possible approximation results. In contrast, our work in the second part of the thesis is motivated by practical applications. We consider a number of new labeling and scheduling problems which occur in the design of communication networks. Here we present and analyze efficient approximation and online algorithms. We use very simple techniques which do not require large computational resources

A note on scheduling to meet two min-sum objectives

International audienceWe consider a single machine scheduling problem with two min-sum objective functions: the sum of completion times and the sum of weighted completion times. We propose a simple polynomial time (1 + (1/γ), 1 + γ)-approximation algorithm, and show that for y > 1, there is no (x, y)-approximation with 1 < x < 1 + (1/γ) and 1 < y < 1 + (γ - 1)/(2 + y)

On approximating the TSP with intersecting neighborhoods

In the TSP with neighborhoods problem we are given a set of n regions (neighborhoods) in the plane, and seek to find a minimum length TSP tour that goes through all the regions. We give two approximation algorithms for the case when the regions are allowed to intersect: We give the first O(1)-factor approximation algorithm for intersecting convex fat objects of comparable diameters where we are allowed to hit each object only at a finite set of specified points. The proof follows from two packing lemmas that are of independent interest. For the problem in its most general form (but without the specified points restriction) we give a simple O(logn)-approximation algorithm

On weighted rectangle packing with large resources

Abstract We study the problem of packing a set ofÒrectangles with weights into a dedicated rectangle so that the weight of the packed rectangles is maximized. We consider the case of large resources, that is, the side length of all rectangles is at most and the side lengths of the dedicated rectangle differ by a factor of at least���, for a fixed positive��. We present an algorithm which finds a rectangle packing of weight at least (  �) of the optimum in time polynomial inÒ. As an application we show a �-approximation algorithm for packing weighted rectangles into�rectangular bins of size���