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Register a new userOn the cut dimension of a graph
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Troy Lee
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Let $G = (V,w)$ be a weighted undirected graph with $m$ edges. The cut dimension of $G$ is the dimension of the span of the characteristic vectors of the minimum cuts of $G$, viewed as vectors in ${0,1}^m$. For every $n ge 2$ we show that the cut dimension of an $n$-vertex graph is at most $2n-3$, and construct graphs realizing this bound. The cut dimension was recently defined by Graur et al. cite{GPRW20}, who show that the maximum cut dimension of an $n$-vertex graph is a lower bound on the number of cut queries needed by a deterministic algorithm to solve the minimum cut problem on $n$-vertex graphs. For every $nge 2$, Graur et al. exhibit a graph on $n$ vertices with cut dimension at least $3n/2 -2$, giving the first lower bound larger than $n$ on the deterministic cut query complexity of computing mincut. We observe that the cut dimension is even a lower bound on the number of emph{linear} queries needed by a deterministic algorithm to solve mincut, where a linear query can ask any vector $x in mathbb{R}^{binom{n}{2}}$ and receives the answer $w^T x$. Our results thus show a lower bound of $2n-3$ on the number of linear queries needed by a deterministic algorithm to solve minimum cut on $n$-vertex graphs, and imply that one cannot show a lower bound larger than this via the cut dimension. We further introduce a generalization of the cut dimension which we call the $ell_1$-approximate cut dimension. The $ell_1$-approximate cut dimension is also a lower bound on the number of linear queries needed by a deterministic algorithm to compute minimum cut. It is always at least as large as the cut dimension, and we construct an infinite family of graphs on $n=3k+1$ vertices with $ell_1$-approximate cut dimension $2n-2$, showing that it can be strictly larger than the cut dimension.
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A stable cut of a graph is a cut whose weight cannot be increased by changing the side of a single vertex. Equivalently, a cut is stable if all vertices have the (weighted) majority of their neighbors on the other side. In this paper we study Min Stable Cut, the problem of finding a stable cut of minimum weight, which is closely related to the Price of Anarchy of the Max Cut game. Since this problem is NP-hard, we study its complexity on graphs of low treewidth, low degree, or both. We show that the problem is weakly NP-hard on severely restricted trees, so bounding treewidth alone cannot make it tractable. We match this with a pseudo-polynomial DP algorithm running in time $(Deltacdot W)^{O(tw)}n^{O(1)}$, where $tw$ is the treewidth, $Delta$ the maximum degree, and $W$ the maximum weight. On the other hand, bounding $Delta$ is also not enough, as the problem is NP-hard for unweighted graphs of bounded degree. We therefore parameterize Min Stable Cut by both $tw+Delta$ and obtain an FPT algorithm running in time $2^{O(Delta tw)}(n+log W)^{O(1)}$. Our main result is to provide a reduction showing that both aforementioned algorithms are essentially optimal, even if we replace treewidth by pathwidth: if there exists an algorithm running in $(nW)^{o(pw)}$ or $2^{o(Delta pw)}(n+log W)^{O(1)}$, then the ETH is false. Complementing this, we show that we can obtain an FPT approximation scheme parameterized by treewidth, if we consider almost-stable solutions. Motivated by these mostly negative results, we consider Unweighted Min Stable Cut. Here our results already imply a much faster exact algorithm running in time $Delta^{O(tw)}n^{O(1)}$. We show that this is also probably essentially optimal: an algorithm running in $n^{o(pw)}$ would contradict the ETH.
We give improved pseudorandom generators (PRGs) for Lipschitz functions of low-degree polynomials over the hypercube. These are functions of the form psi(P(x)), where P is a low-degree polynomial and psi is a function with small Lipschitz constant. PRGs for smooth functions of low-degree polynomials have received a lot of attention recently and play an important role in constructing PRGs for the natural class of polynomial threshold functions. In spite of the recent progress, no nontrivial PRGs were known for fooling Lipschitz functions of degree O(log n) polynomials even for constant error rate. In this work, we give the first such generator obtaining a seed-length of (log n)tilde{O}(d^2/eps^2) for fooling degree d polynomials with error eps. Previous generators had an exponential dependence on the degree. We use our PRG to get better integrality gap instances for sparsest cut, a fundamental problem in graph theory with many applications in graph optimization. We give an instance of uniform sparsest cut for which a powerful semi-definite relaxation (SDP) first introduced by Goemans and Linial and studied in the seminal work of Arora, Rao and Vazirani has an integrality gap of exp(Omega((log log n)^{1/2})). Understanding the performance of the Goemans-Linial SDP for uniform sparsest cut is an important open problem in approximation algorithms and metric embeddings and our work gives a near-exponential improvement over previous lower bounds which achieved a gap of Omega(log log n).
MapReduce (and its open source implementation Hadoop) has become the de facto platform for processing large data sets. MapReduce offers a streamlined computational framework by interleaving sequential and parallel computation while hiding underlying system issues from the programmer. Due to the popularity of MapReduce, there have been attempts in the theoretical computer science community to understand the power and limitations of the MapReduce framework. In the most widely studied MapReduce models each machine has memory sub-linear in the input size to the problem, hence cannot see the entire input. This restriction places many limitations on algorithms that can be developed for the model; however, the current understanding of these restrictions is still limited. In this paper, our goal is to work towards understanding problems which do not admit efficient algorithms in the MapReduce model. We study the basic question of determining if a graph is connected or not. We concentrate on instances of this problem where an algorithm is to determine if a graph consists of a single cycle or two disconnected cycles. In this problem, locally every part of the graph is similar and the goal is to determine the global structure of the graph. We consider a natural class of algorithms that can store/process/transfer the information only in the form of paths and show that no randomized algorithm cannot answer the decision question in a sub-logarithmic number of rounds. Currently, there are no absolute super constant lower bounds on the number of rounds known for any problem in MapReduce. We introduce some of the first lower bounds for a natural graph problem, albeit for a restricted class of algorithms. We believe our result makes progress towards understanding the limitations of MapReduce.
Finding cliques in random graphs and the closely related planted clique variant, where a clique of size t is planted in a random G(n,1/2) graph, have been the focus of substantial study in algorithm design. Despite much effort, the best known polynomial-time algorithms only solve the problem for t = Theta(sqrt(n)). Here we show that beating sqrt(n) would require substantially new algorithmic ideas, by proving a lower bound for the problem in the sum-of-squares (or Lasserre) hierarchy, the most powerful class of semi-definite programming algorithms we know of: r rounds of the sum-of-squares hierarchy can only solve the planted clique for t > sqrt(n)/(C log n)^(r^2). Previously, no nontrivial lower bounds were known. Our proof is formulated as a degree lower bound in the Positivstellensatz algebraic proof system, which is equivalent to the sum-of-squares hierarchy. The heart of our (average-case) lower bound is a proof that a certain random matrix derived from the input graph is (with high probability) positive semidefinite. Two ingredients play an important role in this proof. The first is the classical theory of association schemes, applied to the average and variance of that random matrix. The second is a new large deviation inequality for matrix-valued polynomials. Our new tail estimate seems to be of independent interest and may find other applications, as it generalizes both the estimates on real-valued polynomials and on sums of independent random matrices.
This tutorial review provides a guiding reference to researchers who want to have an overview of the large body of literature about graph spanners. It reviews the current literature covering various research streams about graph spanners, such as different formulations, sparsity and lightness results, computational complexity, dynamic algorithms, and applications. As an additional contribution, we offer a list of open problems on graph spanners.
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