Seminars and Colloquia by Series

Friday, February 24, 2012 - 13:00 , Location: Klaus 1116W , Karthekeyan Chandrasekaran , CoC, Georgia Tech , Organizer:
I will show a new approach based on the discrepancy of the constraint matrix to verify integer feasibility of polytopes. I will then use this method to show a threshold phenomenon for integer feasibility of random polytopes. The random polytope model that we consider is P(n,m,x0,R) - these are polytopes in n-dimensional space specified by m "random" tangential hyperplanes to a ball of radius R centered around the point x0. We show that there exist constants c_1 < c_2 such that with high probability, the random polytope P(n,m,x0=(0.5,...,0.5),R) is integer infeasible if R is less than c_1sqrt(log(2m/n)) and the random polytope P(n,m,x0,R) is integer feasible for every center x0 if the radius R is at least c_2sqrt(log(2m/n)). Thus, a transition from infeasibility to feasibility happens within a constant factor increase in the radius. Moreover, if the polytope contains a ball of radius Omega(log (2m/n)), then we can find an integer solution with high probability (over the input) in randomized polynomial time. This is joint work with Santosh Vempala.
Friday, February 17, 2012 - 13:00 , Location: Skiles 005 , Will Perkins , Georgia Tech, School of Mathematics , perkins@math.gatech.edu , Organizer:
I will define planted distributions of random structures and give plenty of examples in different contexts: from balls and bins, to random permutations, to random graphs and CSP's.  I will give an idea of how they are used and why they are interesting.  Then I'll focus on one particular problem: under what conditions can you distinguish a planted distribution from the standard distribution on a random structure and how can you do it?
Friday, February 10, 2012 - 13:00 , Location: TBD , Christopher Peikert , School of Computer Science , Organizer:
I'll give a high-level tour of how lattices are providing a powerful new mathematical foundation for cryptography. Lattices provide simple, fast, and highly parallel cryptoschemes that, in contrast with many of today's popular methods (like RSA and elliptic curves), even appear to remain secure against quantum computers. No background in lattices, cryptography, or quantum computers will be necessary -- you only need to know how to add and multiply vectors and matrices.
Friday, February 3, 2012 - 13:00 , Location: Executive classroom, ISyE Main Building , Daniel Dadush , Georgia Tech, School of Industrial and Systems Engineering , Organizer:
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.
Wednesday, April 28, 2010 - 13:30 , Location: ISyE Executive Classroom , Karthik Chandrasekaran , CS ACO , Organizer:
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).
Wednesday, April 14, 2010 - 13:30 , Location: Skiles 171 , Prof. Leonid Bunimovich , School of Mathematics, Georgia Tech , Organizer:
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.
Wednesday, March 31, 2010 - 13:30 , Location: ISyE Executive Classroom , Anand Louis , CS ACO, Georgia Tech , Organizer:
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".
Wednesday, March 17, 2010 - 13:30 , Location: ISyE Executive Classroom , Prof. Merrick Furst , Computer Science, Georgia Tech , Organizer:
 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.
Wednesday, February 17, 2010 - 13:30 , Location: ISyE Executive Classroom , William T. Trotter , School of Mathematics, Georgia Tech , Organizer:
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.
Wednesday, February 10, 2010 - 13:30 , Location: ISyE Executive Classroom , Daniel Dadush , ISyE ACO, Georgia Tech , Organizer:
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).

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