Cristiane M. Sato

Sparse Sums of Positive Semidefinite Matrices

Many fast graph algorithms begin by preprocessing the graph to improve its sparsity. A common form of this is spectral sparsification, which involves removing and reweighting the edges of the graph while approximately preserving its spectral properties. This task has a more general linear algebraic formulation in terms of approximating sums of rank-one matrices. This article considers a more general task of approximating sums of symmetric, positive semidefinite matrices of arbitrary rank. We present two deterministic, polynomial time algorithms for solving this problem. The first algorithm applies the pessimistic estimators of Wigderson and Xiao, and the second involves an extension of the method of Batson, Spielman, and Srivastava. These algorithms have several applications, including sparsifiers of hypergraphs, sparse solutions to semidefinite programs, sparsifiers of unique games, and graph sparsifiers with various auxiliary constraints.

On the Longest Paths and the Diameter in Random Apollonian Networks

We consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. After n−3 steps, we obtain a random triangulated plane graph with n vertices, which is called a Random Apollonian Network (RAN). We show that asymptotically almost surely (a.a.s.) every path in a RAN has length o(n), refuting a conjecture of Frieze and Tsourakakis. We also show that a RAN always has a path of length (2n−5)log2/log3, and that the expected length of its longest path is Ω(n0.88). Finally, we prove that a.a.s. the diameter of a RAN is asymptotic to clogn, where c≈1.668 is the solution of an explicit equation.

On longest paths and diameter in random apollonian networks

We consider the following iterative construction of a random planar triangulation. Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. After n – 3 steps, we obtain a random triangulated plane graph with n vertices, which is called a Random Apollonian Network (RAN). We show that asymptotically almost surely (a.a.s.) a longest path in a RAN has length o(n), refuting a conjecture of Frieze and Tsourakakis. We also show that a RAN always has a cycle (and thus a path) of length , and that the expected length of its longest cycles (and paths) is . Finally, we prove that a.a.s. the diameter of a RAN is asymptotic to , where is the solution of an explicit equation.

Arboricity and spanning-tree packing in random graphs

We study the arboricity A and the maximum number T of edge-disjoint spanning trees of the classical random graph G (n, p). For all p(n) ∈ [0, 1], we show that, with high probability, T is precisely the minimum between δ and bm/(n − 1)c, where δ is the smallest degree of the graph and m denotes the number of edges. Moreover, we explicitly determine a sharp threshold value for p such that: above this threshold, T equals bm/(n−1)c and A equals dm/(n−1)e; and below this threshold, T equals δ, and we give a two-value concentration result for the arboricity A in that range. Finally, we include a stronger version of these results in the context of the random graph process where the edges are sequentially added one by one. A direct application of our result gives a sharp threshold for the maximum load being at most k in the two-choice load balancing problem, where k →∞.

Asymptotic enumeration of sparse 2‐connected graphs

We determine an asymptotic formula for the number of labelled 2-connected (simple) graphs on n vertices and m edges, provided that m - n →∞ and m = O(nlog n) as n →∞. This is the entire range of m not covered by previous results. The proof involves determining properties of the core and kernel of random graphs with minimum degree at least 2. The case of 2-edge-connectedness is treated similarly. We also obtain formulae for the number of 2-connected graphs with given degree sequence for most (“typical”) sequences. Our main result solves a problem of Wright from 1983.

Packing arborescences in random digraphs

We study the problem of packing arborescences in the random digraph

A transition of limiting distributions of large matchings in random graphs

\mathcal D(n,p)

Arboricity and spanning-tree packing in random graphs with an application to load balancing

, where each possible arc is included uniformly at random with probability

On the robustness of random k-cores

p=p(n)

Asymptotic enumeration of sparse connected 3-uniform hypergraphs

. Let