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Series: ACO Student Seminar

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.

Series: ACO Student Seminar

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.

Series: ACO Student Seminar

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.

Series: ACO Student Seminar

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.

Series: ACO Student Seminar

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.

Series: ACO Student Seminar

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 http://arxiv.org/abs/1512.01892.

Series: ACO Student Seminar

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.

Series: ACO Student Seminar

We present a market for allocating and scheduling resources to agents who have specified budgets and need to complete specific tasks. Two important aspects required in this market are: (1) agents need specific amounts of each resource to complete their tasks, and (2) agents would like to complete their tasks as soon as possible. In incorporating these aspects, we arrive at a model that deviates substantially from market models studied so far in economics and theoretical computer science. Indeed, all known techniques developed to compute equilibria in markets in the last decade and half seem not to apply here.We give a polynomial time algorithm for computing an equilibrium using a new technique that is somewhat reminiscent of the ''ironing" procedure used in the characterization of optimal auctions by Myerson. This is inspite of the fact that the set of equilibrium prices could be non-convex; in fact it could have ''holes''. Our market model is motivated by the cloud computing marketplace. Even though this market is already huge and is projected to grow at a massive rate, it is currently run in an ad hoc manner.Joint work with: Nikhil Devanur, Jugal Garg, Ruta Mehta, Vijay V. Vazirani

Series: ACO Student Seminar

We find a good characterization for the following problem: Given a
rational row vector c and a lattice L in R^n which contains the integer
lattice Z^n, do all lattice points of L in the half-open unit cube
[0,1)^n lie on the hyperplane {x in R^n : cx = 0}? This work generalizes
a theorem due to G. K. White, which provides sufficient and necessary
conditions for a tetrahedron in R^3 with integral vertices to have no
other integral points. Our approach is based on a novel proof of White's
result using number-theoretic techniques due to Morrison and Stevens.
In this talk, we illustrate some of the ideas and describe some
applications of this problem.

Series: ACO Student Seminar

For an integer k, the Folkman number f(k) is the least integer n for which there exists a graph G on n vertices that does not contain a clique of size k and has the property that every two coloring of E(G) yields a monochromatic clique of size of size k. That is, it is the least number of vertices in a K_{k+1}-free graph that is Ramsey to K_k. A recent result of Rodl, Rucinski, and Schacht gives an upper bound on the Folkman numbers f(k) which is exponential in k. A fundamental tool in their proof is a theorem of Saxton and Thomason on hypergraph containers. This talk will give a brief history of the Folkman numbers, introduce the hypergraph container theorem, and sketch the proof of the Rodl, Rucinski, and Schacht result. Recent work with Hiep Han, Vojtech Rodl, and Mathias Schacht on two related problems concerning cycles in graphs and arithmetic progressions in subset of the integers will also be presented.