Roee David

Random walks with the minimum degree local rule have O ( N 2 ) cover time

For a simple (unbiased) random walk on a connected graph with n vertices, the cover time (the expected number of steps it takes to visit all vertices) is at most O(n3). We consider locally biased random walks, in which the probability of traversing an edge depends on the degrees of its endpoints. We confirm a conjecture of Abdullah, Cooper and Draief [2015] that the min-degree local bias rule ensures a cover time of O(n2). For this we formulate and prove the following lemma about spanning trees. Let R(e) denote for edge e the minimum degree among its two endpoints. We say that a weight function W for the edges is feasible if it is nonnegative, dominated by R (for every edge W (e) ≤ R(e)) and the sum over all edges of the ratios W(e)/R(e) equals n − 1. For example, in trees W (e) = R(e), and in regular graphs the sum of edge weights is d(n − 1). Lemma: for every feasible W, the minimum weight spanning tree has total weight O(n). For regular graphs, a similar lemma was proved by Kahn, Linial, Nisan and Saks [1989].

Local Reconstruction of Low-Rank Matrices and Subspaces

We study the problem of reconstructing a low-rank matrix, where the input is an n × m matrix M over a field F and the goal is to reconstruct a (near-optimal) matrix M′ that is low-rank and close to M under some distance function Δ. Furthermore, the reconstruction must be local, i.e., provides access to any desired entry of M′ by reading only a few entries of the input M (ideally, independent of the matrix dimensions n and m). Our formulation of this problem is inspired by the local reconstruction framework of Saks and Seshadhri (SICOMP, 2010). Our main result is a local reconstruction algorithm for the case where Δ is the normalized Hamming distance (between matrices). Given M that is ϵ-close to a matrix of rank d<1/ϵ (together with d and ϵ), this algorithm computes with high probability a rank-d matrix M′ that is O(dϵ)-close to M. This is a local algorithm that proceeds in two phases. The preprocessing phase reads only O˜(d/ϵ3) random entries of M, and stores a small data structure. The query phase deterministically outputs a desired entry M′i,j by reading only the data structure and 2d additional entries of M. We also consider local reconstruction in an easier setting, where the algorithm can read an entire matrix column in a single operation. When Δ is the normalized Hamming distance between vectors, we derive an algorithm that runs in polynomial time by applying our main result for matrix reconstruction. For comparison, when Δ is the truncated Euclidean distance and F=ℝ, we analyze sampling algorithms by using statistical learning tools. A preliminary version of this paper appears appears in ECCC, see: http://eccc.hpi-web.de/report/2015/128/

On the effect of randomness on planted 3-coloring models

We present the hosted coloring framework for studying al- gorithmic and hardness results for the k-coloring problem. There is a class H of host graphs. One selects a graph H ∈ H and plants in it a balanced k-coloring (by partitioning the vertex set into k roughly equal parts, and removing all edges within each part). The resulting graph G is given as input to a polynomial time algorithm that needs to k-color G (any legal k-coloring would do – the algorithm is not required to recover the planted k-coloring). Earlier planted models correspond to the case that H is the class of all n-vertex d-regular graphs, a member H ∈ H is chosen at random, and then a balanced k-coloring is planted at random. Blum and Spencer [1995] designed algorithms for this model when d = n δ (for 0 < δ ≤ 1), and Alon and Kahale [1997] managed to do so even when d is a sufficiently large constant. The new aspect in our framework is that it need not in- volve randomness. In one model within the framework (with k = 3) H is a d regular spectral expander (meaning that ex- cept for the largest eigenvalue of its adjacency matrix, every other eigenvalue has absolute value much smaller than d) chosen by an adversary, and the planted 3-coloring is ran- dom. We show that the 3-coloring algorithm of Alon and Kahale [1997] can be modified to apply to this case. In an- other model H is a random d-regular graph but the planted balanced 3-coloring is chosen by an adversary, after seeing H. We show that for a certain range of average degrees somewhat below √ n, finding a 3-coloring is NP-hard. To- gether these results (and other results that we have) help clarify which aspects of randomness in the planted coloring model are the key to successful 3-coloring algorithms.

Connectivity of Random High Dimensional Geometric Graphs

We consider graphs obtained by placing n points at random on a unit sphere in ℝ d , and connecting two points by an edge if they are close to each other (e.g., the angle at the origin that their corresponding unit vectors make is at most π/3). We refer to these graphs as geometric graphs. We also consider a complement family of graphs in which two points are connected by an edge if they are far away from each other (e.g., the angle is at least 2π/3). We refer to these graphs as anti-geometric graphs. The families of graphs that we consider come up naturally in the context of semidefinite relaxations of graph optimization problems such as graph coloring.