arXiv subjects

Cristiane M. Sato

Publications and source records attributed to Cristiane M. Sato.

12 recordsLinked to original sources

arXiv2026

Maximum Cuts and Fractional Cut Covers: A Computational Study of a Randomized Semidefinite Programming Approach

We present experimental work on a primal-dual framework simultaneously approximating maximum cut and weighted fractional cut-covering instances. In this primal-dual framework, we solve a semidefinite programming (SDP) relaxation to either the maximum cut problem or to the weighted fractional cut-covering problem, and then independently sample a collection of cuts via the random-hyperplane technique. We then simultaneously certify the approximate optimality of a cut and a fractional cut cover. We present several implementations which reliably achieve the celebrated Goemans and Williamson approximation ratio of $α_{\mathrm{GW}} \approx 0.878$ for both optimization problems simultaneously, after $\lceil 128 \ln m \rceil$ samples, a number significantly smaller than the best theoretical bounds. This is the first experimental work approximating the weighted fractional cut-covering problem, and we deliver robust and repeatable results despite the use of randomized algorithms and floating-point arithmetic. Careful pre-processing of instances and post-processing of numeric results allow for good empirical outcomes with both first-order and second-order SDP solvers. Nearly optimal SDP solutions are suitably perturbed to ensure better probabilistic and numerical behavior. Our experiments deviate from theory by using a linear programming (LP) solver to compute fractional cut covers. For most instances studied, LP solving produces certifiably better results than the theoretical algorithm after $\lceil 128 \ln m \rceil$ samples. All our experiments strictly follow a unified pipeline which explicitly documents all parameters used in each run.

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arXiv2025

A Primal-Dual Extension of the Goemans--Williamson Algorithm for the Weighted Fractional Cut-Covering Problem

We study a weighted generalization of the fractional cut-covering problem, which we relate to the maximum cut problem via antiblocker and gauge duality. This relationship allows us to introduce a semidefinite programming (SDP) relaxation whose solutions may be rounded into fractional cut covers by sampling via the random hyperplane technique. We then provide a $1/α_{\scriptscriptstyle \mathrm{GW}}$-approximation algorithm for the weighted fractional cut-covering problem, where $α_{\scriptscriptstyle \mathrm{GW}} \approx 0.878$ is the approximation factor of the celebrated Goemans--Williamson algorithm for the maximum cut problem. Nearly optimal solutions of the SDPs in our duality framework allow one to consider instances of the maximum cut and the fractional cut-covering problems as primal-dual pairs, where cuts and fractional cut covers simultaneously certify each other's approximation quality. We exploit this relationship to introduce new combinatorial certificates for both problems, as well as a randomized polynomial-time algorithm for producing such certificates. In~particular, we~show how the Goemans--Williamson algorithm implicitly approximates a weighted instance of the fractional cut-covering problem, and how our new algorithm explicitly approximates a weighted instance of the maximum cut problem. We conclude by discussing the role played by geometric representations of graphs in our results, and by proving our algorithms and analyses to be optimal in several aspects.

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arXiv2024

Generalized Cuts and Grothendieck Covers: a Primal-Dual Approximation Framework Extending the Goemans--Williamson Algorithm

We provide a primal-dual framework for randomized approximation algorithms utilizing semidefinite programming (SDP) relaxations. Our framework pairs a continuum of APX-complete problems including MaxCut, Max2Sat, MaxDicut, and more generally, Max-Boolean Constraint Satisfaction and MaxQ (maximization of a positive semidefinite quadratic form over the hypercube) with new APX-complete problems which are stated as convex optimization problems with exponentially many variables. These new dual counterparts, based on what we call Grothendieck covers, range from fractional cut covering problems (for MaxCut) to tensor sign covering problems (for MaxQ). For each of these problem pairs, our framework transforms the randomized approximation algorithms with the best known approximation factors for the primal problems to randomized approximation algorithms for their dual counterparts with reciprocal approximation factors which are tight with respect to the Unique Games Conjecture. For each APX-complete pair, our algorithms solve a single SDP relaxation and generate feasible solutions for both problems which also provide approximate optimality certificates for each other. Our work utilizes techniques from areas of randomized approximation algorithms, convex optimization, spectral sparsification, as well as Chernoff-type concentration results for random matrices.

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arXiv2017

Packing arborescences in random digraphs

We study the problem of packing arborescences in the random digraph $\mathcal D(n,p)$, where each possible arc is included uniformly at random with probability $p=p(n)$. Let $λ(\mathcal D(n,p))$ denote the largest integer $λ\geq 0$ such that, for all $0\leq \ell\leq λ$, we have $\sum_{i=0}^{\ell-1} (\ell-i)|\{v: d^{in}(v) = i\}| \leq \ell$. We show that the maximum number of arc-disjoint arborescences in $\mathcal D(n,p)$ is $λ(\mathcal D(n,p))$ a.a.s. We also give tight estimates for $λ(\mathcal D(n,p))$ depending on the range of $p$.

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arXiv2015

On the maximum density of fixed strongly connected subtournaments

We study the density of fixed strongly connected subtournaments on 5 vertices in large tournaments. We determine the maximum density asymptotically for five tournaments as well as unique extremal sequences for each tournament. As a byproduct we also characterize tournaments that are recursive blow-ups of a 3-cycle as tournaments that avoid three specific tournaments of size 5.

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arXiv2015

A transition of limiting distributions of large matchings in random graphs

We study the asymptotic distribution of the number of matchings of size $\ell=\ell(n)$ in $G(n,p)$ for a wide range of $p=p(n)\in (0,1)$ and for every $1\le \ell\le \lfloor n/2\rfloor$. We prove that this distribution changes from normal to log-normal as $\ell$ increases, and we determine the critical value of $\ell$, as a function of $n$ and $p$, at which the transition of the limiting distribution occurs.

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arXiv2014

Asymptotic enumeration of sparse connected 3-uniform hypergraphs

We derive an asymptotic formula for the number of connected 3-uniform hypergraphs with vertex set $[N]$ and $M$ edges for $M=N/2+R$ as long as $R$ satisfies $R = o(N)$ and $R=ω(N^{1/3}\ln^{2} N)$. This almost completely fills the gap in the range of $M$ for which the formula is known. We approach the problem using an `inside-out' approach of an earlier paper of Pittel and the second author, for connected graphs. A key part of the method uses structural components of connected hypergraphs called cores and kernels. These are structural components of connected hypergraphs. Our results also give information on the numbers of them with a given number of vertices and edges, and hence their typical size in random connected $3$-uniform hypergraphs with $N$ vertices and $M$ edges, for the range of $M$ we consider.

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arXiv2013

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

We study the arboricity A and the maximum number T of edge-disjoint spanning trees of the Erdos-Renyi random graph G(n,p). For all p(n) in [0,1], we show that, with high probability, T is precisely the minimum between delta and floor(m/(n-1)), where delta 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 floor(m/(n-1)) and A equals ceiling(m/(n-1)); and below this threshold, T equals delta, 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 goes to infinity.

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arXiv2013

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)^{log 2/log 3}, and that the expected length of its longest path is Omega(n^0.88). Finally, we prove that a.a.s. the diameter of a RAN is asymptotic to c log n, where c \approx 1.668 is the solution of an explicit equation.

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arXiv2012

On the robustness of random k-cores

The k-core of a graph is its maximal subgraph with minimum degree at least k. In this paper, we address robustness questions about k-cores. Given a k-core, remove one edge uniformly at random and find its new k-core. We are interested in how many vertices are deleted from the original k-core to find the new one. This can be seem as a measure of robustness of the original k-core. We prove that, if the initial k-core is chosen uniformly at random from the k-cores with n vertices and m edges, its robustness depends essentially on its average degree c. We prove that, if c converges to k, then the new k-core is empty with probability 1+o(1). We define a constant c(k)' such that when k+epsilon < c < c(k)'- epsilon, the new k-core is empty with probability bounded away from zero and, if c > c(k)'+ psi with psi = omega(n^{-1/4}), psi(n) > 0 and c is bounded, then the probability that the new k-core has less than n-h(n) vertices goes to zero, for every h(n) = omega(1/psi).

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arXiv2011

Sparse Sums of Positive Semidefinite Matrices

Recently there has been much interest in "sparsifying" sums of rank one matrices: modifying the coefficients such that only a few are nonzero, while approximately preserving the matrix that results from the sum. Results of this sort have found applications in many different areas, including sparsifying graphs. In this paper we consider the more general problem of sparsifying sums of positive semidefinite matrices that have arbitrary rank. We give several algorithms for solving this problem. The first algorithm is based on the method of Batson, Spielman and Srivastava (2009). The second algorithm is based on the matrix multiplicative weights update method of Arora and Kale (2007). We also highlight an interesting connection between these two algorithms. Our algorithms have numerous applications. We show how they can be used to construct graph sparsifiers with auxiliary constraints, sparsifiers of hypergraphs, and sparse solutions to semidefinite programs.

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arXiv2011

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 $m-n\to\infty$ and $m=O(n\log n)$ as $n\to\infty$. 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.

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