Seminars and Colloquia by Series

Friday, April 15, 2016 - 13:05 , Location: Skiles 005 , Daniel Blado , Georgia Tech , Organizer: Yan Wang
We examine a variant of the knapsack problem in which item sizes are random according to an arbitrary but known distribution. In each iteration, an item size is realized once the decision maker chooses and attempts to insert an item. With the aim of maximizing the expected profit, the process ends when either all items are successfully inserted or a failed insertion occurs. We investigate the strength of a particular dynamic programming based LP bound by examining its gap with the optimal adaptive policy. Our new relaxation is based on a quadratic value function approximation which introduces the notion of diminishing returns by encoding interactions between remaining items. We compare the bound to previous bounds in literature, including the best known pseudopolynomial bound, and contrast their corresponding policies with two natural greedy policies. Our main conclusion is that the quadratic bound is theoretically more efficient than the pseudopolyomial bound yet empirically comparable to it in both value and running time.
Friday, April 8, 2016 - 13:05 , Location: Skiles 005 , Samantha Petti , Georgia Tech , Organizer: Yan Wang
Motivated by neurally feasible computation, we study Boolean functions of an arbitrary number of input variables that can be realized by recursively applying a small set of functions with a constant number of inputs each. This restricted type of construction is neurally feasible since it uses little global coordination or control. Valiant’s construction of a majority function can be realized in this manner and, as we show, can be generalized to any uniform threshold function. We study the rate of convergence, finding that while linear convergence to the correct function can be achieved for any threshold using a fixed set of primitives, for quadratic convergence, the size of the primitives must grow as the threshold approaches 0 or 1. We also study finite realizations of this process, and show that the constructions realized are accurate outside a small interval near the target threshold, where the size of the construction at each level grows as the inverse square of the interval width. This phenomenon, that errors are higher closer to thresholds (and thresholds closer to the boundary are harder to represent), is also a well-known cognitive finding. Finally, we give a neurally feasible algorithm that uses recursive constructions to learn threshold functions. This is joint work with Christos Papadimitriou and Santosh Vempala.
Friday, April 1, 2016 - 13:05 , Location: Skiles 005 , Hao Huang , Emory University , Organizer: Yan Wang
A Ruzsa-Szemeredi graph is a graph on n vertices whose edge set can be partitioned into induced matchings of size cn. The study of these graphs goes back more than 35 years and has connections with number theory, combinatorics, complexity theory and information theory. In this talk we will discuss the history and some recent developments in this area. In particular, we show that when c>1/4, there can be only constantly many matchings. On the other hand, for c=1/4, the maximum number of induced matchings is logarithmic in n. This is joint work with Jacob Fox and Benny Sudakov.
Friday, March 18, 2016 - 13:05 , Location: Skiles 005 , Chi Ho Yuen , Georgia Tech , Organizer: Yan Wang
This talk aims to give a glimpse into the theory of divisors on graphs in tropical geometry, and its recent application in bijective combinatorics. I will start by introducing basic notions and results of the subject. Then I will mention some of its connections with other fields in math. Finally I will talk about my own work on how tropical geometry leads to an unexpectedly simple class of bijections between spanning trees of a graph and its sandpile group.
Friday, March 11, 2016 - 13:05 , Location: Skiles 256 , Rui Gao , Georgia Tech , Organizer: Yan Wang
Stochastic programming is a powerful approach for decision-making under uncertainty. Unfortunately, the solution may be misleading if the underlying distribution of the involved random parameters is not known exactly. In this talk, we study distributionally robust stochastic programming (DRSP), in which the decision hedges against the worst possible distribution that belongs to an ambiguity set. More specifically, we consider the DRSP with the ambiguity set comprising all distributions that are close to some reference distribution in terms of Wasserstein distance. We derive a tractable reformulation of the DRSP problem by constructing the worst-case distribution explicitly via the first-order optimality condition of the dual problem. Our approach has several theoretical and computational implications. First, using the precise characterization of the worst-case distribution, we show that the DRSP can be approximated by robust programs to arbitrary accuracy, and thus many DRSP problems become tractable with tools from robust optimization. Second, when the objective is concave in the uncertainty, the robust-program approximation is exact and equivalent to a saddle-point problem, which can be solved by a Mirror-Prox algorithm. Third, our framework can also be applied to problems other than stochastic programming, such as a class of distributionally robust transportation problems. Furthermore, we perform sensitivity analysis with respect to the radius of the Wasserstein ball, and apply our results to the newsvendor problem, two-stage linear program with uncertainty-affected recourse, and worst-case Value-at-risk analysis.
Friday, March 4, 2016 - 13:05 , Location: Skiles 256 , Ben Cousins , Georgia Tech , Organizer: Yan Wang
I will give a tour of high-dimensional sampling algorithms, both from a theoretical and applied perspective, for generating random samples from a convex body. There are many well-studied random walks to choose from, with many of them having rigorous mixing bounds which say when the random walk has converged. We then show that the techniques from theory yield state-of-the-art algorithms in practice, where we analyze various organisms by randomly sampling their metabolic networks.This work is in collaboration with Ronan Fleming, Hulda Haraldsdottir ,and Santosh Vempala.
Friday, February 26, 2016 - 13:05 , Location: Skiles 005 , Aurko Roy , Georgia Tech , Organizer: Yan Wang
We generalize the existing reduction mechanism due to Braun, Pokutta and Zink (2014)for linear programming problems and semidefinite programming problems in two ways  1) relaxing the requirement of affineness2) extending to fractional optimization problems  As applications we prove several new LP-hardness and SDP-hardnessresults, e.g., for the (non-uniform) Sparsest Cut problem with bounded treewidth on the supply graph, the Balanced Separator problem with bounded treewidth onthe demand graph, the Max Cut problem and the Matching problem on 3-regular graphs.We also provide a new, very strong Lasserre integrality gapfor the Independent Set problem, which is strictly greater than thebest known LP approximation, showing that the Lasserre hierarchydoes not always provide the tightest SDP relaxation.Joint work with Gabor Braun and Sebastian Pokutta. 
Friday, February 5, 2016 - 13:05 , Location: Skiles 005 , Emma Cohen , Georgia Tech , Organizer: Yan Wang
We consider the Widom-Rowlinson model of two types of interacting particles on $d$-regular graphs. We prove a tight upper bound on the occupancy fraction: the expected fraction of vertices occupied by a particle under a random configuration from the model.  The upper bound is achieved uniquely by unions of complete graphs on $d+1$ vertices.  As a corollary we find that $K_{d+1}$ also maximizes the normalized partition function of the Widom-Rowlinson model over the class of $d$-regular graphs, proving a conjecture of Galvin. Joint work with Will Perkins and Prasad Tetali.
Friday, January 15, 2016 - 13:00 , Location: Skiles 005 , Richard Peng , Georgia Tech , Organizer: Yan Wang
We introduce the sparsified Cholesky and sparsified multigrid algorithms for solving systems of linear equations. These algorithms accelerate Gaussian elimination by sparsifying the nonzero matrix entries created by the elimination process.We use these new algorithms to derive the first nearly linear time algorithms for solving systems of equations in connection Laplacians, a generalization of Laplacian matrices that arise in many problems inimage and signal processing.We also prove that every connection Laplacian has a linear sized approximate inverse. This is an LU factorization with a linear number of nonzero entries that is a strong approximation of the originalmatrix. Using such a factorization one can solve systems of equations in a connection Laplacian in linear time. Such a factorization was unknown even for ordinary graph Laplacians.Joint work with Rasmus Kyng, Yin Tat Lee, Sushant Sachdeva, and Daniel Spielman. Manuscript at
Friday, December 4, 2015 - 13:00 , Location: Skiles 005 , Prateek Bhakta , Georgia Tech , Organizer: Yan Wang
We consider perfect matchings of the square-octagon lattice, also known as``fortresses.''  There is a natural local Markov chain on the setof perfect matchings that is known to be ergodic.  However, unlike  Markov chains for sampling perfect matchings on the square and hexagonallattices, corresponding to domino and lozenge tilings, respectively, the seemingly relatedMarkov chain on the square-octagon lattice appears to converge slowly.  Tounderstand why, we consider a weighted version of the problem.As with domino and lozenge tilings, it will be useful to view perfectmatchings on the square-octagon lattice in terms of sets of paths and cycleson a corresponding lattice region; here, the paths and cycles lie on theCartesian lattice and are required to turn left or right at every step.  Forinput parameters $\lambda$ and $\mu$, we define the weight of a configurationto be $\lambda^{\abs{E(\sigma)}} \mu^{\abs{V(\sigma)}},$ where $E(\sigma)$ isthe total number of edges on the paths and cycles of $\sigma$ and $V(\sigma)$is the number of vertices that are not incident to any of the paths or cyclesin $\sigma$.  Weighted paths already come up in the reduction from perfectmatchings to turning lattice paths, corresponding to the case when $\lambda=1$and $\mu = 2$.First, fixing $\mu=1$, we show that there are choices of~$\lambda$ for whichthe chain converges slowly and another for which it is fast, suggesting a phasechange in the mixing time.  More precisely,the chain requires exponential time (in the size of the lattice region) when$\lambda < 1/(2\sqrt{e})$ or $\lambda >2\sqrt{e}$, while it is polynomially mixingat $\lambda = 1$. Further, we show that for $\mu>1$, the Markov chain $\m$ is slowly mixingwhen $\lambda < \sqrt{\mu}/(2\sqrt{e})$ or $\lambda > 2\mu\sqrt{e}$. These arethe first rigorous proofs explaining why the natural local Markov chain can beslow for weighted fortresses or perfect matchings on thesquare-octagon lattice.