Near Optimal Bounds for Steiner Trees in the Hypercube (original) (raw)

Erdős–Ko–Rado for Random Hypergraphs: Asymptotics and Stability

Combinatorics, Probability and Computing, 2017

We investigate the asymptotic version of the Erdős–Ko–Rado theorem for the random k-uniform hypergraph mathcalH\mathcal{H}mathcalH k (n, p). For 2⩽k(n) ⩽ n/2, let N=binomnkN=\binom{n}kN=binomnk and D=binomn−kkD=\binom{n-k}kD=binomn−kk . We show that with probability tending to 1 as n → ∞, the largest intersecting subhypergraph of mathcalH\mathcal{H}mathcalH has size (1+o(1))p\ffrac kn Nforanyfor anyforanyp\gg \ffrac nk\ln^2\biggl(\ffrac nk\biggr)D^{-1}.$$ This lower bound on p is asymptotically best possible for k = Θ(n). For this range of k and p, we are able to show stability as well. A different behaviour occurs when k = o(n). In this case, the lower bound on p is almost optimal. Further, for the small interval D −1 ≪ p ⩽ (n/k)1−ϵ D −1, the largest intersecting subhypergraph of mathcalH\mathcal{H}mathcalH k (n, p) has size Θ(ln(pD)ND −1), provided that kggsqrtnlnnk \gg \sqrt{n \ln n}kggsqrtnlnn . Together with previous work of Balogh, Bohman and Mubayi, these results settle the asymptotic size of the largest intersecting family in mathcalH\mathcal{H}mathcalH k , for essentially all values of p...

On k-Connectivity and Minimum Vertex Degree in Random s-Intersection Graphs

2015 Proceedings of the Twelfth Workshop on Analytic Algorithmics and Combinatorics (ANALCO), 2014

Random s-intersection graphs have recently received much interest in a wide range of application areas. Broadly speaking, a random s-intersection graph is constructed by first assigning each vertex a set of items in some random manner, and then putting an undirected edge between all pairs of vertices that share at least s items (the graph is called a random intersection graph when s = 1). A special case of particular interest is a uniform random s-intersection graph, where each vertex independently selects the same number of items uniformly at random from a common item pool. Another important case is a binomial random s-intersection graph, where each item from a pool is independently assigned to each vertex with the same probability. Both models have found numerous applications thus far including cryptanalysis, and modeling recommender systems, secure sensor networks, online social networks, trust networks and smallworld networks (uniform random s-intersection graphs), as well as clustering analysis, classification, and the design of integrated circuits (binomial random s-intersection graphs). In this paper, for binomial/uniform random sintersection graphs, we present results related to kconnectivity and minimum vertex degree. Specifically, we derive the asymptotically exact probabilities and zeroone laws for the following three properties: (i) k-vertexconnectivity, (ii) k-edge-connectivity and (iii) the property of minimum vertex degree being at least k.

The de Bruijn-Erdos Theorem for hypergraphs

2010

We also give an absolute lower bound cp(n,r)geqnchooser/q+r−1chooser\cp(n,r) \geq {n \choose r}/{q + r - 1 \choose r}cp(n,r)geqnchooser/q+r−1chooser when n=q2+q+r−1n = q^2 + q + r - 1n=q2+q+r−1, and for each rrr characterize the finitely many configurations achieving equality with the lower bound. Finally we note the connection of cp(n,r)\cp(n,r)cp(n,r) to extremal graph theory, and determine some new asymptotically sharp bounds for the Zarankiewicz problem.

Random subgraphs of finite graphs: III. The phase transition for the nnn-cube

2004

We study random subgraphs of the nnn-cube 0,1n\{0,1\}^n0,1n, where nearest-neighbor edges are occupied with probability ppp. Let pc(n)p_c(n)pc(n) be the value of ppp for which the expected cluster size of a fixed vertex attains the value lambda2n/3\lambda 2^{n/3}lambda2n/3, where lambda\lambdalambda is a small positive constant. Let epsilon=n(p−pc(n))\epsilon=n(p-p_c(n))epsilon=n(p−pc(n)). In two previous papers, we showed that the largest cluster inside a scaling window given by ∣epsilon∣=Theta(2−n/3)|\epsilon|=\Theta(2^{-n/3})∣epsilon∣=Theta(2−n/3) is of size Theta(22n/3)\Theta(2^{2n/3})Theta(22n/3), below this scaling window it is at most 2(log2)nepsilon−22(\log2) n\epsilon^{-2}2(log2)nepsilon−2, and above this scaling window it is at most O(epsilon2n)O(\epsilon 2^n)O(epsilon2n). In this paper, we prove that for p−pc(n)geqe−cn1/3p - p_c(n) \geq e^{-cn^{1/3}}p−pc(n)geqe−cn1/3 the size of the largest cluster is at least Theta(epsilon2n)\Theta(\epsilon 2^n)Theta(epsilon2n), which is of the same order as the upper bound. This provides an understanding of the phase transition that goes far beyond that obtained by previous authors. The proof is based on a method that has come to be known as ``sprinkling,'' and relies heavily on the specific geometry of the nnn-cube.

An introduction to the theory of random graphs

2015

This thesis provides an introduction to the fundamentals of random graph theory. The study starts introduces the two fundamental building blocks of random graph theory, namely discrete probability and graph theory. The study starts by introducing relevant concepts probability commonly used in random graph theory-these include concentration inequalities such as Chebyshev's inequality and Chernoff's inequality. Moreover we proceed by introducing central concepts in graph theory, which will underpin the later discussion. In particular we provide results such as Mycielski's construction of a family of triangle-free graphs with high chromatic number and results in Ramsey theory. Next we introduce the concept of a random graph and present two of the most famous proofs in graph theory using the theory random graphs. These include the proof of the fact that there are graphs with arbitrarily high girth and chromatic number, and a bound on the Ramsey number R(k, k). Finally we conclude by introducing the notion of a threshold function for a monotone graph property and we present proofs for the threhold functions of certain properties.

On subexponential running times for approximating directed Steiner tree and related problems

arXiv (Cornell University), 2018

This paper concerns proving almost tight (super-polynomial) running times, for achieving desired approximation ratios for various problems. To illustrate the question we study, let us consider the Set-Cover problem with n elements and m sets. Now we specify our goal to approximate Set-Cover to a factor of (1 − α) ln n, for a given parameter 0 < α < 1. What is the best possible running time for achieving such approximation ratio? This question was answered implicitly in the work of Moshkovitz [Theory of Computing, 2015]: Assuming both the Projection Games Conjecture (PGC) and the Exponential-Time Hypothesis (ETH), any ((1 − α) ln n)-approximation algorithm for Set-Cover must run in time at least 2 n c•α , for some small constant 0 < c < 1. We study the questions along this line. Our first contribution is in strengthening the above result. We show that under ETH and PGC the running time requires for any ((1 − α) ln n)-approximation algorithm for Set-Cover is essentially 2 n α. This (almost) settles the question since our lower bound matches the best known running time of 2 O(n α) for approximating Set-Cover to within a factor (1 − α) ln n given by Cygan et al. [IPL, 2009]. Our result is tight up to the constant multiplying the n α terms in the exponent. The lower bound of Set-Cover applies to all of its generalization, e.g., Group-Steiner-Tree, Directed-Steiner-Tree, Covering-Steiner-Tree and Connected-Polymatroid. We show that, surprisingly, in almost exponential running time, these problems reduce to Set-Cover. Specifically, we complement our lower bound by presenting an (1−α) ln n approximation algorithm for all aforementioned problems that runs in time 2 n α •log n • poly(m). We further study the approximation ratio in the regime of log 2−δ n for Group-Steiner-Tree and Covering-Steiner-Tree. Chekuri and Pal [FOCS, 2005] showed that Group-Steiner-Tree admits (log 2−α n)-approximation in time exp(2 log α+o(1) n), for any parameter 0 < α < 1. We show the running time lower bound of Group-Steiner-Tree: any (log 2−α n)-approximation algorithm for Group-Steiner-Tree must run in time at least exp((1 + o(1))log α−ǫ n), for any constant ǫ > 0, unless the ETH is false. Our result follows by analyzing the hardness construction of Group-Steiner-Tree due to the work of Halperin and Krauthgamer [STOC, 2003]. The same lower and upper bounds hold for Covering-Steiner-Tree.

Lower bounds for combinatorial problems on graphs

Journal of Algorithms, 1985

Nontrivial lower bounds are given for the computation time of various combinatorial problems on graphs under a linear or algebraic decision tree model. An SI(n310g n) bound is obtained for a "travelling salesman problem" on a weighted complete graph of n vertices. Moreover it is shown that the same bound is valid for the "subgraph detection problem" with respect to property P if P is hereditary and determined by components. Thus an Q(n310gn) bound is established in a unified way for a rather large class of problems. 8 1985 Academic PRSS, hc.