TITLE: On the Structure of Reduced Kernel Lattice Bases

SPEAKER: Karen Aardal

ABSTRACT:

Lattice-based reformulation techniques have been used successfully both theoretically and computationally. One such reformulation is obtained from the lattice $L_0=\{x\in Z^n\mid Ax = 0\}$. Some of the hard instances in the literature that have been successfully tackled by lattice-based techniques, such as market split and certain classes of knapsack instances, have randomly generated input $A$. These instances have been posed to stimulate algorithmic research. Since the considered instances are very hard even in low dimension, less experience is available for larger instances. Recently we have studied larger instances and observed that the LLL-reduced basis of $L_0$ has a specific sparse structure. In particular, this translates into a map in which some of the original variables get a ``rich'' translation into a new variable space, whereas some variables are only substituted in the new space. If an original variable is important in the sense of branching or cutting planes, this variable should be translated in a non-trivial way. In this paper we partially explain the obtained structure of the LLL-reduced basis in the case that the input matrix $A$ consists of one row $a$. Since the input is randomly generated our analysis will be probabilistic. The key ingredient is a bound on the probability that the LLL algorithm will interchange two subsequent basis vectors. It is worth noticing that computational experiments indicate that the results of this analysis seem to apply in the same way also in the general case that $A$ consists of multiple rows. Our analysis has yet to be extended to this general case. Along with our analysis we also present some computational indications that illustrate that the probabilistic analysis conforms well with the practical behavior. This is joint work with Frederik von Heymann.

BIO:

Karen Aardal obtained her PhD from C.O.R.E, Universite' Catholique de Louvain, Belgium and has since then held positions at University of Essex, UK, Georgia Tech, Atlanta, GA, USA, and at various universities in The Netherlands. Currently she holds the position of professor of Optimization at Delft University of Technology.

Her main research interest is integer programming, and in particular algebraic approaches to integer programming. She is interested in combinatorial optimization, in particular in facility location. In terms of applications she is currently involved in a large Dutch project aiming at improving all aspects of ambulance planning, and in a European project on electricity networks.

She has served on the Council of the Mathematical Optimization Society, MOS. She has been the Publications Committee Chair and the Chair of the Executive Committee of MOS. She is a member of the board of directors of INFORMS Computing Society. She is the Area Editor for "Design and analysis of algorithms" of INFORMS Journal on Computing. She also serve on the boards of Mathematical Programming Series B, Networks, EURO Journal on Computational Optimization, and RAIRO - Operations Research.