Shayan Oveis Gharan

A Randomized Rounding Approach to the Traveling Salesman Problem

For some positive constant \eps_0, we give a (3/2-\eps_0)-approximation algorithm for the following problem: given a graph G_0=(V,E_0), find the shortest tour that visits every vertex at least once. This is a special case of the metric traveling salesman problem when the underlying metric is defined by shortest path distances in G_0. The result improves on the 3/2-approximation algorithm due to Christofides [C76] for this special case. Similar to Christofides, our algorithm finds a spanning tree whose cost is upper bounded by the optimum, then it finds the minimum cost Eulerian augmentation (or T-join) of that tree. The main difference is in the selection of the spanning tree. Except in certain cases where the solution of LP is nearly integral, we select the spanning tree randomly by sampling from a maximum entropy distribution defined by the linear programming relaxation. Despite the simplicity of the algorithm, the analysis builds on a variety of ideas such as properties of strongly Rayleigh measures from probability theory, graph theoretical results on the structure of near minimum cuts, and the integrality of the T-join polytope from polyhedral theory. Also, as a byproduct of our result, we show new properties of the near minimum cuts of any graph, which may be of independent interest.

Improved Cheeger's inequality: analysis of spectral partitioning algorithms through higher order spectral gap

Let φ(G) be the minimum conductance of an undirected graph G, and let 0=λ₁ ≤ λ₂ ≤ ... ≤ λ_n ≤ 2 be the eigenvalues of the normalized Laplacian matrix of G. We prove that for any graph G and any k ≥ 2, [φ(G) = O(k) l₂/√l_k,] and this performance guarantee is achieved by the spectral partitioning algorithm. This improves Cheegers inequality, and the bound is optimal up to a constant factor for any

Approximating the Expansion Profile and Almost Optimal Local Graph Clustering

k

Online Stochastic Matching: Online Actions Based on Offline Statistics

. Our result shows that the spectral partitioning algorithm is a constant factor approximation algorithm for finding a sparse cut if l_k is a constant for some constant k. This provides some theoretical justification to its empirical performance in image segmentation and clustering problems. We extend the analysis to spectral algorithms for other graph partitioning problems, including multi-way partition, balanced separator, and maximum cut.

Multiway Spectral Partitioning and Higher-Order Cheeger Inequalities

Spectral partitioning is a simple, nearly-linear time, algorithm to find sparse cuts, and the Cheeger inequalities provide a worst-case guarantee of the quality of the approximation found by the algorithm. Local graph partitioning algorithms [1], [2], [3] run in time that is nearly linear in the size of the output set, and their approximation guarantee is worse than the guarantee provided by the Cheeger inequalities by a poly-logarithmic logΩ(1) n factor. It has been an open problem to design a local graph clustering algorithm with an approximation guarantee close to the guarantee of the Cheeger inequalities and with a running time nearly linear in the size of the output. In this paper we solve this problem; we design an algorithm with the same guarantee (up to a constant factor) as the Cheeger inequality, that runs in time slightly super linear in the size of the output. This is the first sublinear (in the size of the input) time algorithm with almost the same guarantee as the Cheegers inequality. As a byproduct of our results, we prove a bicriteria approximation algorithm for the expansion profile of any graph. Let μ(S) = Σv∈S d(v) be the volume, and φ(S) := |E(S, S̅)|/μ(S), be the conductance of a set S of vertices. If there is a set of volume at most γ and conductance φ, we can find a set of volume at most γ1+ϵ and conductance V at most O(√φ/ϵ), for any ϵ >; 0. Our proof techniques also provide a simpler proof of the structural result of Arora, Barak, Steurer [4], that can be applied to irregular graphs. Our main technical tool is a lemma stating that, for any set S of vertices of a graph, a lazy t-step random walk started from a randomly chosen vertex of S, will remain entirely inside S with probability at least (1-φ(S)/2)t. The lemma also implies a new lower bound to the uniform mixing time of any finite states reversible markov chain.

Thickness and Information in Dynamic Matching Markets

We consider the online stochastic matching problem proposed by Feldman et al. [4] as a model of display ad allocation. We are given a bipartite graph; one side of the graph corresponds to a fixed set of bins and the other side represents the set of possible ball types. At each time step, a ball is sampled independently from the given distribution and it needs to be matched upon its arrival to an empty bin. The goal is to maximize the size of the matching. We present an online algorithm for this problem with a competitive ratio of 0.702. Before our result, algorithms with a competitive ratio better than 1−1/e were known under the assumption that the expected number of arriving balls of each type is integral. A key idea of the algorithm is to collect statistics about the decisions of the optimum offline solution using Monte Carlo sampling and use those statistics to guide the decisions of the online algorithm. We also show that no online algorithm can have a competitive ratio better than 0.823.

A generalization of permanent inequalities and applications in counting and optimization

A basic fact in spectral graph theory is that the number of connected components in an undirected graph is equal to the multiplicity of the eigenvalue zero in the Laplacian matrix of the graph. In particular, the graph is disconnected if and only if there are at least two eigenvalues equal to zero. Cheegers inequality and its variants provide an approximate version of the latter fact; they state that a graph has a sparse cut if and only if there are at least two eigenvalues that are close to zero. It has been conjectured that an analogous characterization holds for higher multiplicities: There are <i>k</i> eigenvalues close to zero if and only if the vertex set can be partitioned into <i>k</i> subsets, each defining a sparse cut. We resolve this conjecture positively. Our result provides a theoretical justification for clustering algorithms that use the bottom <i>k</i> eigenvectors to embed the vertices into R^<i>k</i>, and then apply geometric considerations to the embedding. We also show that these techniques yield a nearly optimal quantitative connection between the expansion of sets of size ≈ <i>n</i>/<i>k</i> and λ_<i>k</i>, the <i>k</i>th smallest eigenvalue of the normalized Laplacian, where <i>n</i> is the number of vertices. In particular, we show that in every graph there are at least <i>k</i>/2 disjoint sets (one of which will have size at most 2<i>n</i>/<i>k</i>), each having expansion at most <i>O</i>(√λ_<i>k</i> log <i>k</i>). Louis, Raghavendra, Tetali, and Vempala have independently proved a slightly weaker version of this last result. The √log <i>k</i> bound is tight, up to constant factors, for the “noisy hypercube” graphs.

Effective-Resistance-Reducing Flows, Spectrally Thin Trees, and Asymmetric TSP

We introduce a simple model of dynamic matching in networked markets, where agents arrive and depart stochastically and the composition of the trade network depends endogenously on the matching algorithm. If the planner can identify agents who are about to depart, then waiting to thicken the market substantially reduces the fraction of unmatched agents. If not, then matching agents greedily is close to optimal. We specify conditions under which local algorithms that choose the right time to match agents, but do not exploit the global network structure, are close to optimal. Finally, we consider a setting where agents have private information about their departure times and design a mechanism to elicit this information.

A New Regularity Lemma and Faster Approximation Algorithms for Low Threshold-Rank Graphs

A polynomial pΕℝ[z1,…,zn] is real stable if it has no roots in the upper-half complex plane. Gurvitss permanent inequality gives a lower bound on the coefficient of the z1z2…zn monomial of a real stable polynomial p with nonnegative coefficients. This fundamental inequality has been used to attack several counting and optimization problems. Here, we study a more general question: Given a stable multilinear polynomial p with nonnegative coefficients and a set of monomials S, we show that if the polynomial obtained by summing up all monomials in S is real stable, then we can lower bound the sum of coefficients of monomials of p that are in S. We also prove generalizations of this theorem to (real stable) polynomials that are not multilinear. We use our theorem to give a new proof of Schrijvers inequality on the number of perfect matchings of a regular bipartite graph, generalize a recent result of Nikolov and Singh, and give deterministic polynomial time approximation algorithms for several counting problems.

Almost Optimal Local Graph Clustering Using Evolving Sets

We show that the integrality gap of the natural LP relaxation of the Asymmetric Traveling Salesman Problem is polyloglog(n). In other words, there is a polynomial time algorithm that approximates the value of the optimum tour within a factor of polyloglog(n), where polyloglog(n) is a bounded degree polynomial of loglog(n). We prove this by showing that any k-edge-connected unweighted graph has a polyloglog(n)/k-thin spanning tree. Our main new ingredient is a procedure, albeit an exponentially sized convex program, that “transforms” graphs that do not admit any spectrally thin trees into those that provably have spectrally thin trees. More precisely, given a k-edge-connected graph G = (V, E) where k ≥ 7 log(n), we show that there is a matrix D that “preserves” the structure of all cuts of G such that for a set F ⊆ E that induces an Ω(k)-edge-connected graph, the effective resistance of every edge in F w.r.t. D is at most polylog(k)/k. Then, we use our extension of the seminal work of Marcus, Spielman, and Srivastava [1], fully explained in [2], to prove the existence of a polylog(k)/k-spectrally thin tree with respect to D. Such a tree is polylog(k)/k-combinatorially thin with respect to G as D preserves the structure of cuts of G.