Georgia Institute of Technology College of Sciences

In a celebrated result, Raghavendra [Rag08] showed that, assuming Unique Game Conjecture, every Max-CSP problem has a sharp approximation threshold that matches the integrality gap of a natural SDP relaxation. Raghavendra and Steurer [RS09] also gave a simple and unified framework for optimally approximating all the Max-CSPs.In this work, we consider the problem of approximating CSPs with global cardinality constraints. For example, Max-Cut is a boolean CSP where the input is a graph $G = (V,E)$ and the goal is to find a cut $S \cup \bar S = V$ that maximizes the number of crossing edges, $|E(S,\bar S)|$. The Max-Bisection problem is a variant of Max-Cut with an additional global constraint that each side of the cut has exactly half the vertices, i.e., $|S| = |V|/2$. Several other natural optimization problems like Small Set Expansion, Min Bisection and approximating Graph Expansion can be formulated as CSPs with global constraints.In this talk, I will introduce a general approach towards approximating CSPs with global constraints using SDP hierarchies. To demonstrate the approach, I will present an improved algorithm for Max-Bisection problem that achieves the following:- Given an instance of Max-Bisection with value $1-\epsilon$, the algorithm finds a bisection with value at least $1-O(\sqrt{\epsilon})$ with running time $O(n^{poly(1/\eps)})$. This approximation is near-optimal (up to constant factors in $O()$) under the Unique Games Conjecture.- Using computer-assisted proof, we show that the same algorithm also achieves a 0.85-approximation for Max-Bisection, improving on the previous bound of 0.70 (note that it is UGC-hard to approximate better than 0.878 factor). As an attempt to prove matching hardness result, we show a generic conversion from SDP integrality gap to dictatorship test for any CSP with global cardinality constraints. The talk is based on joint work with Prasad Raghavendra.

In the node-connectivity augmentation problem, we want to add a minimum number of new edges to an undirected graph to make it k-node-connected. The complexity of this question is still open, although the analogous questions of both directed and undirected edge-connectivity and directed node-connectivity augmentation are known to be polynomially solvable. I present a min-max formula and a polynomial time algorithm for the special case when the input graph is already (k-1)-connected. The formula has been conjectured by Frank and Jordan in 1994. In the first lecture, I presented previous results on the other connectivity augmentation variants. In the second part, I shall present my min-max formula and the main ideas of the proof.

We will prove an Hopf-Lax-Oleinik formula for the solutions of some Hamilton-Jacobi equations on a general metric space. Then, we will present some consequences: in particular the equivalence of the log-Sobolev inequality and the hypercontractivity property of theHamilton-Jacobi "semi-group", (and if time allows) that Talagrand’s transport-entropy inequalities in metric space are characterizedin terms of log-Sobolev inequalities restricted to the class of c-convex functions (based on a paper in collaboration with N. Gozlan and P.M. Samson).

Brownian dynamics (BD) is a computational technique for simulating the motions of molecules interacting through hydrodynamic and non-hydrodynamic forces. BD simulations are the main tool used in computational biology for studying diffusion-controlled cellular processes. This talk presents several new numerical linear algebra techniques to accelerate large BD simulations, and related Stokesian dynamics (SD) simulations. These techniques include: 1) a preconditioned Lanczos process for computing Brownian vectors from a distribution with given covariance, 2) low-rank approximations to the hydrodynamic tensor suitable for large-scale problems, and 3) a reformulation of the computations to approximate solutions to multiple time steps simultaneously, allowing the efficient use of data parallel hardware on modern computer architectures.