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Series: ACO Student Seminar

A fundamental result in the geometry of numbers states that any lattice free convex set in R^n has integer width bounded by a function of dimension, i.e. the so called Flatness Theorem for Convex Bodies. This result provides the theoretical basis for the polynomial solvability of Integer Programs with a fixed number of (general) integer variables. In this work, we provide a simplified proof of the Flatness Theorem with tighter constants. Our main technical contribution is a new tight bound on the smoothing parameter of a lattice, a concept developed within lattice based cryptography which enables comparisons between certain discrete distributions over integer points with associated continuous Gaussian distributions. Based on joint work with Kai-Min Chung, Feng Hao Liu, and Christopher Peikert.

Series: ACO Student Seminar

Abstract: A hitting set for a collection of sets T is a set that has a
non-empty intersection with eachset in T; the hitting set problem is
to find a hitting set of minimum cardinality. Motivated bythe fact
that there are instances of the hitting set problem where the number of
subsets to behit is large, we introduce the notion of implicit
hitting set problems. In an implicit hitting setproblem the
collection of sets to be hit is typically too large to list explicitly;
instead, an oracleis provided which, given a set H, either
determines that H is a hitting set or returns a set inT that H does
not hit. I will show a number of examples of classic implicit hitting
set problems,and give a generic algorithm for solving such problems
exactly in an online model.I will also show how this framework is
valuable in developing approximation algorithms by presenting
a simple on-line algorithm for the minimum feedback vertex set problem.
In particular, our algorithm
gives an approximation factor of 1+ 2 log(np)/(np) for the random graph
G_{n,p}.Joint work with Richard Karp, Erick Moreno-Centeno (UC, Berkeley) and Santosh Vempala (Georgia Tech).

Series: ACO Student Seminar

Billiards is a dynamical system generated by an uniform motion of a point particle (ray of light, sound, etc.) in a domain with piecewise smooth boundary. Upon reaching the boundary the particle reflected according to the law "the angle of incidence equals the angle of reflection". Billiards appear as natural models in various branches of physics. More recently this type of models were used in oceanography, operations research, computer science, etc. I'll explain on very simple examples what is a regular and what is chaotic dynamics, the mechanisms of chaos and natural measures of complexity in dynamical systems. The talk will be accessible to undergraduates.

Series: ACO Student Seminar

Local search is one of the oldest known optimization techniques. It has
been studied extensively by Newton, Euler, etc. It is known that this
technique gives the optimum solution if the function being optimized is
concave(maximization) or convex (minimization). However, in the general
case it may only produce a "locally optimum" solution. We study how to
use this technique for a class of facility location problems and give
the currently best known approximation guarantees for the problem and a
matching "locality gap".

Series: ACO Student Seminar

Santosh Vempala and I have been exploring an intriguing newapproach to convex optimization. Intuition about first-order interiorpoint methods tells us that a main impediment to quickly finding aninside track to optimal is that a convex body's boundary can get inone's way in so many directions from so many places. If the surface ofa convex body is made to be perfectly reflecting then from everyinterior vantage point it essentially disappears. Wondering about whatthis might mean for designing a new type of first-order interior pointmethod, a preliminary analysis offers a surprising and suggestiveresult. Scale a convex body a sufficient amount in the direction ofoptimization. Mirror its surface and look directly upwards fromanywhere. Then, in the distance, you will see a point that is as closeas desired to optimal. We wouldn't recommend a direct implementation,since it doesn't work in practice. However, by trial and error we havedeveloped a new algorithm for convex optimization, which we arecalling Reflex. Reflex alternates greedy random reflecting steps, thatcan get stuck in narrow reflecting corridors, with simply-biasedrandom reflecting steps that escape. We have early experimentalexperience using a first implementation of Reflex, implemented inMatlab, solving LP's (can be faster than Matlab's linprog), SDP's(dense with several thousand variables), quadratic cone problems, andsome standard NETLIB problems.

Series: ACO Student Seminar

On-line graph coloring has a rich history, with a very large number of elegant results together with a near equal number of unsolved problems. In this talk, we will briefly survey some of the classic results including: performance on k-colorable graphs and \chi-bounded classes. We will conclude with a sketch of some recent and on-going work, focusing on the analysis of First Fit on particular classes of graphs.

Series: ACO Student Seminar

The analysis of Chvatal Gomory (CG) cuts and their associated closure for
polyhedra was initiated long ago in the study of integer programming. The
classical results of Chvatal (73) and Schrijver (80) show that the Chvatal
closure of a rational polyhedron is again itself a rational polyhedron. In
this work, we show that for the class of strictly convex bodies the above
result still holds, i.e. that the Chvatal closure of a strictly convex body
is a rational polytope.This is joint work with Santanu Dey (ISyE) and Juan Pablo Vielma (IBM).

Series: ACO Student Seminar

I will describe a simple scheme for generating a valid inequality for a
stochastic integer programs from a given valid inequality for its
deterministic counterpart. Applications to stochastic lot-sizing problems
will be discussed. This is joint work with Yongpei Guan and George Nemhauser
and is based on the following two papers
(1) Y. Guan, S. Ahmed and G.L. Nemhauser. "Cutting planes for multi-stage
stochastic integer programs," Operations Research, vol.57, pp.287-298, 2009
(2)
Y. Guan, S. Ahmed and G. L. Nemhauser. "Sequential pairing of mixed integer
inequalities," Discrete Optimization, vol.4, pp.21-39, 2007
This is a joint DOS/ACO seminar.

Series: ACO Student Seminar

In 1969, Gomory introduced the master group polyhedron for pure integer programs and derives the mixed integer cut (MIC) as a facet of a special family of these polyhedra. We study the MIC in this framework, characterizing both its facets and extreme points; next, we extend our results under mappings between group polyhedra; and finally, we conclude with related open problems. No prior knowledge of algebra or the group relaxation is assumed. Terminology will be introduced as needed. Joint work with Ellis Johnson.

Series: ACO Student Seminar

Sum-Product inequalities originated in the early 80's
with the work of Erdos and Szemeredi, who showed that there exists
a constant c such that if A is a set of n integers, n sufficiently
large, then either the sumset A+A = {a+b : a,b in A} or the product
set A.A = {ab : a,b in A}, must exceed n^(1+c) in size. Since that
time the subject has exploded with a vast number of generalizations
and extensions of the basic result, which has led to many
very interesting unsolved problems (that would make great thesis
topics). In this talk I will survey some of the developments in this
fast-growing area.