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Series: ACO Student Seminar

We solve a 20-year old problem posed by M. Yannakakis and prove that there exists no polynomial-size linear program (LP) whose associated polytope projects to the traveling salesman polytope, even if the LP is not required to be symmetric. Moreover, we prove that this holds also for the maximum cut problem and the stable set problem. These results follow from a new connection that we make between one-way quantum communication protocols and semidefinite programming reformulations of LPs. (joint work with Samuel Fiorini, Serge Massar, Hans Raj Tiwary, and Ronald de Wolf)

Series: ACO Student Seminar

In this talk I will briefly survey results on Vertex Sparsification and some of our results on Mimicking network(or Exact Cut Sparsifier). Ankur Moitra introduced the notion of vertex sparsification to construct a smaller graph which preserves the properties of a huge network that are relevant to the terminals. Given a capacitated undirected graph $G=(V,E)$ with a set of terminals $K \subset V$, a vertex cut sparsifier is a smaller graph $H=(V_H,E_H)$ that approximately(quality f>=1) preserves all the minimum cuts between the terminals. Mimicking networks are the best quality vertex cut sparsifiers i.e, with quality 1. We improve both the previous upper($2^{2^{k}}$ ) and lower bounds($k+1$) for mimicking network reducing the doubly-exponential gap between them to a single-exponential gap. 1. Given a graph $G$, we exhibit a construction of mimicking network with at most $k$'th Hosten-Morris number ($\approx 2^{{(k-1)} \choose {\lfloor {{(k-1)}/2} \rfloor}}$) of vertices (independent of size of $V$). Furthermore, we show that the construction is optimal among all {\itrestricted mimicking networks} -- a natural class of mimicking networks that are obtained by clustering vertices together. 2. There exists graphs with $k$ terminals that have no mimicking network of size smaller than $2^{\frac{k-1}{2}}$. 3. We also exhibit constructions of better mimicking networks for trees($\lfloor(\frac{3k}{2})-1\rfloor$), outerplanar graphs($5k-9$) and graphs of bounded($t$) tree-width($k 2^{(2t+1) \choose {(2t+1)/2}}$). The talk will be self-contained and with no prerequisite.

Series: ACO Student Seminar

We present a new algorithm learning the class of two-sided disjunctions in
semi-supervised PAC setting and in the active learning model. These
algorithms are efficient and have good sample complexity. By exploiting the
power of active learning we are able to find consistent, compatible
hypotheses -- a task which is computationally intractable in the
semi-supervised setting.

Series: ACO Student Seminar

A branching random walk consists of a population of individuals each of whom perform a random walk step before giving birth to a random number of offspring and dying. The offspring then perform their own independent random steps and branching. I will present classic results on the convergence of the empirical particle measure to the Gaussian distribution, then present new results on large deviations of this empirical measure. The talk will be self-contained and can serve as an introduction to both the branching random walk and large deviation theory. The format will be 40 minutes of introduction and presentation, followed by a short break and then 20 minutes of discussion of open problems for those interested.

Series: ACO Student Seminar

Sampling permutations from S_n is a fundamental problem from probability theory. The nearest neighbor transposition chain M_n is known to converge in time \Theta(n^3 \log n) in the uniform case and time \Theta(n^2) in the constant bias case, in which we put adjacent elements in order with probability p \neq 1/2 and out of order with probability 1-p. In joint work with Prateek Bhakta, Dana Randall and Amanda Streib, we consider the variable bias case where the probability of putting an adjacent pair of elements in order depends on the two elements, and we put adjacent elements x < y in order with probability p_{x,y} and out of order with probability 1-p_{x,y}. The problem of bounding the mixing rate of M_n was posed by Fill and was motivated by the Move-Ahead-One self-organizing list update algorithm. It was conjectured that the chain would always be rapidly mixing if 1/2 \leq p_{x,y} \leq 1 for all x < y, but this was only known in the case of constant bias or when p_{x,y} is equal to 1/2 or 1, a case that corresponds to sampling linear extensions of a partial order. We prove the chain is rapidly mixing for two classes: ``Choose Your Weapon,'' where we are given r_1,..., r_{n-1} with r_i \geq 1/2 and p_{x,y}=r_x for all x < y (so the dominant player chooses the game, thus fixing his or her probability of winning), and ``League Hierarchies,'' where there are two leagues and players from the A-league have a fixed probability of beating players from the B-league, players within each league are similarly divided into sub-leagues with a possibly different fixed probability, and so forth recursively. Both of these classes include permutations with constant bias as a special case. Moreover, we also prove that the most general conjecture is false. We do so by constructing a counterexample where 1/2 \leq p_{x,y} \leq 1 for all x < y, but for which the nearest neighbor transposition chain requires exponential time to converge.

Series: ACO Student Seminar

In the last 10 years, compressed sensing has arisen as an entirely new area of mathematics, combining ideas from convex programming, random matrices, theoretical computer science and many other fields. Candes (one of the originators of the area) recently spoke about two quite recent and exciting developments, but it might be interesting to revisit the fundamentals, and see where a lot of the ideas in the more recent works have developed. In this talk, I will discuss some of the earlier papers (Candes-Romberg-Tao), define the compressed sensing problem, the key restricted isometry property and how it relates to the Johnson-Lindenstrauss lemma for random projections. I'll also discuss some of the more TCS ideas such as compressed sensing through group testing, and hopefully some of the greedy algorithm ideas as well. Finally, if time allows, I'll draw parallels with other problems, such as matrix completion, phase retrieval etc. The talk will be quite elementary, requiring only a knowledge of linear algebra, and some probability.

Series: ACO Student Seminar

This work develops approximation algorithms for a stochastic
knapsack problem involving an expected utility objective. The values
of the items in the knapsack can only be sampled from an oracle, and
the objective function is a concave function of the total value of the
items in the knapsack. We will first show a polynomial number of
samples is enough to approximate the true expected value close enough.
Then we will present an algorithm that maximizes a class of submodular
function under knapsack constraint with approximation ratio better
than 1-1/e. We will also see better bounds when the concave function
is a power function. At last, if time permits, we will give an FPTAS
of the problem when the number of scenarios is fixed.

Series: ACO Student Seminar

The theory of Groebner bases is the foundation of many algorithms in computational algebra. A Groebner basis is a special generating set of an ideal of polynomials. In this expository talk, I will introduce Groebner bases and explain how they can be used in integer programming. In particular, for an integer program, we can associate an ideal whose Groebner basis gives a set of directions that takes any feasible solution to an optimal solution.

Series: ACO Student Seminar

A linear coloring of a graph is a proper coloring of the vertices of the graph so that each pair of color classes induce a union of disjoint paths. In this talk, I will prove that for every connected graph with maximum degree at most three and every assignment of lists of size four to the vertices of the graph, there exists a linear coloring such that the color of each vertex belongs to the list assigned to that vertex and the neighbors of every degree-two vertex receive different colors, unless the graph is $C_5$ or $K_{3,3}$. This confirms a conjecture raised by Esperet, Montassier, and Raspaud. Our proof is constructive and yields a linear-time algorithm to find such a coloring. This is joint work with Gexin Yu.

Series: ACO Student Seminar

Cheeger's fundamental inequality states that any edge-weighted graph has a vertex subset $S$ such that its expansion (a.k.a. conductance of $S$ or the sparsity of the cut $(S,\bar{S})$)is bounded as follows: \phi(S) = w(S,S') /w(S) \leq \sqrt{2 \lambda_2},where $w$ is the total edge weight of a subset or a cut and $\lambda_2$ is the second smallest eigenvalue of thenormalized Laplacian of the graph.We study natural generalizations of the sparsest cut in a graph: (i) a partition ofthe vertex set into $k$ parts that minimizes the sparsity of the partition (defined as the ratio of theweight of edges between parts to the total weight of edges incident to the smallest $k-1$ parts); (ii) a collection of $k$ disjoint subsets $S_1, \ldots, S_k$ that minimize $\max_{i \in [k]} \phi(S_i)$; (iii) a subset of size $O(1/k)$ of the graph with minimum expansion. Our main results are extensions of Cheeger's classical inequality to these problems via higher eigenvalues of the graph Laplacian.In particular, for the sparsest $k$-partition, we prove that the sparsity is at most $8\sqrt{\lambda_k} \log k$where $\lambda_k$ is the $k^{th}$ smallest eigenvalue of the normalized Laplacian matrix.For the $k$ sparse cuts problem we prove that there exist$ck$ disjoint subsets $S_1, \ldots, S_{(1 - \eps)k}$, such that \max_i \phi(S_i) \leq C \sqrt{\lambda_{k} \log k}where $C>0$ are suitable absolute constants; this leads to a similar bound for the small-set expansion problem, namely for any $k$, there is a subset $S$ whoseweight is at most a $\bigO(1/k)$ fraction of the total weight and $\phi(S) \le C \sqrt{\lambda_k \log k}$. The latter two results are the best possible in terms of the eigenvalues up to constant factors. Our results are derived via simple and efficient algorithms, and can themselves be viewed as generalizations of Cheeger's method.Based on joint work with Prasad Raghavendra, Prasad Tetali and Santosh Vempala.