Georgia Institute of Technology College of Sciences

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.

A mixed integer point is a vector in $\mathbb{R}^n$ whose first $n_1$ coordinates are integer. We present necessary and sufficient conditions for the convex hull of mixed integer points contained in a general convex set to be closed. This leads to useful results for special classes of convex sets such as pointed cones and strictly convex sets. Furthermore, by using these results, we show that there exists a polynomial time algorithm to check the closedness of the convex hull of the mixed integer points contained in the feasible region of a second order conic programming problem, for the special case this region is defined by just one Lorentz cone and one rational matrix. This is joint work with Santanu Dey.

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.

Recently, there has been great interest in understanding the fundamental limits of our ability to sample from the independent sets (i.s.) of a graph. One approach involves the study of the so-called hardcore model, in which each i.s. is selected with probability proportional to some fixed activity $\lambda$ raised to the cardinality of the given i.s. It is well-known that for any fixed degree $\Delta$, there exists a critical activity $\lambda_{\Delta}$ s.t. for all activities below $\lambda_{\Delta}$, the sampled i.s. enjoys a long-range independence (a.k.a. uniqueness) property when implemented on graphs with maximum degree $\Delta$, while for all activities above $\lambda_{\Delta}$, the sampled i.s. exhibits long-range dependencies. Such phase transitions are known to have deep connections to the inherent computational complexity of the underlying combinatorial problems. In this talk, we study a family of measures which generalizes the hardcore model by taking more structural information into account, beyond just the number of nodes belonging to the i.s., with the hope of further probing the fundamental limits of what we can learn about the i.s. of a graph using only local information. In our model, the probability assigned to a given i.s. depends not only on its cardinality, but also on how many excluded nodes are adjacent to exactly $k$ nodes belonging to the i.s., for each $k$, resulting in a parameter for each $k$. We generalize the notion of critical activity to these ``neighborly measures", and give necessary and sufficient conditions for long-range independence when certain parameters satisfy a log-convexity(concavity) requirement. To better understand the phase transitions in this richer model, we view the classical critical activity as a particular point in the parameter space, and ask which directions can one move and still maintain long-range independence. We show that the set of all such ``directions of uniqueness” has a simple polyhedral description, which we use to study how moving along these directions changes the probabilities associated with the sampled i.s. We conclude by discussing implications for choosing how to sample when trying to optimize a linear function of the underlying probabilities.