Topics in basis reduction and integer programming

A basis reduction algorithm computes a reduced basis of a lattice consisting of short and nearly orthogonal vectors. The best known basis reduction method is due to Lenstra, Lenstra and Lovász (LLL): their algorithm has been extensively used in cryptography, experimental mathematics and integer programming. Lenstra used the LLL basis reduction algorithm to show that the integer programming problem can be solved in polynomial time when the number of variables is fixed. In this thesis, we study some topics in basis reduction and integer programming. We make the following contributions. We unify the fundamental inequalities in an LLL reduced basis, which express the shortness and near orthogonality of the basis. We analyze two recent integer programming reformulation techniques which also rely on basis reduction. The reformulation methods are easy to describe. They are also successful in practice in solving several classes of hard integer programs. First, we analyze the reformulation techniques on bounded knapsack problems. The only analyses so far are for knapsack problems with a constraint vector having a certain decomposable structure. Here we do not assume any a priori structure on the constraint vector. We then analyze the reformulation techniques on bounded integer programs. We show that if the coefficients of the constraint matrix are drawn from a sufficiently large interval, then branch and bound creates at most one node at each level if applied to the reformulated instances. On the practical side, we give some numerical values as to how large the numbers should be to make sure that for 90 and 99 percent of the reformulated instances, the number of subproblems that need to be enumerated by branch and bound is at most one at each level. These values turned out to be surprisingly small when the problem size is moderate. We also analyze the solvability of the ``majority of the low density subset sum problems using the method of branch and bound when the coefficients are chosen from a large interval

Integer-ambiguity resolution in astronomy and geodesy

Recent theoretical developments in astronomical aperture synthesis have revealed the existence of integer-ambiguity problems. Those problems, which appear in the self-calibration procedures of radio imaging, have been shown to be similar to the nearest-lattice point (NLP) problems encountered in high-precision geodetic positioning, and in global navigation satellite systems. In this paper, we analyse the theoretical aspects of the matter and propose new methods for solving those NLP problems. The related optimization aspects concern both the preconditioning stage, and the discrete-search stage in which the integer ambiguities are finally fixed. Our algorithms, which are described in an explicit manner, can easily be implemented. They lead to substantial gains in the processing time of both stages. Their efficiency was shown via intensive numerical tests.Comment: 12 pages. Soumis et accept\'e pour publication dans "Astronomische Nachrichten

ASlib: A Benchmark Library for Algorithm Selection

The task of algorithm selection involves choosing an algorithm from a set of algorithms on a per-instance basis in order to exploit the varying performance of algorithms over a set of instances. The algorithm selection problem is attracting increasing attention from researchers and practitioners in AI. Years of fruitful applications in a number of domains have resulted in a large amount of data, but the community lacks a standard format or repository for this data. This situation makes it difficult to share and compare different approaches effectively, as is done in other, more established fields. It also unnecessarily hinders new researchers who want to work in this area. To address this problem, we introduce a standardized format for representing algorithm selection scenarios and a repository that contains a growing number of data sets from the literature. Our format has been designed to be able to express a wide variety of different scenarios. Demonstrating the breadth and power of our platform, we describe a set of example experiments that build and evaluate algorithm selection models through a common interface. The results display the potential of algorithm selection to achieve significant performance improvements across a broad range of problems and algorithms.Comment: Accepted to be published in Artificial Intelligence Journa