Results 1  10
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12
Local resilience and Hamiltonicity MakerBreaker games in random regular graphs
 Combinatorics, Probability, and Computing
"... For an increasing monotone graph property P the local resilience of a graph G with respect to P is the minimal r for which there exists of a subgraph H ⊆ G with all degrees at most r such that the removal of the edges of H from G creates a graph that does not possesses P. This notion, which was impl ..."
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Cited by 14 (6 self)
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For an increasing monotone graph property P the local resilience of a graph G with respect to P is the minimal r for which there exists of a subgraph H ⊆ G with all degrees at most r such that the removal of the edges of H from G creates a graph that does not possesses P. This notion, which was implicitly studied for some adhoc properties, was recently treated in a more systematic way in a paper by Sudakov and Vu. Most research conducted with respect to this distance notion focused on the Binomial random graph model G(n, p) and some families of pseudorandom graphs with respect to several graph properties such as containing a perfect matching and being Hamiltonian, to name a few. In this paper we continue to explore the local resilience notion, but turn our attention to random and pseudorandom regular graphs of constant degree. We investigate the local resilience of the typical random dregular graph with respect to edge and vertex connectivity, containing a perfect matching, and being Hamiltonian. In particular we prove that for every positive ε and large enough values of d with high probability the local resilience of the random dregular graph, Gn,d, with respect to being Hamiltonian is at least (1−ε)d/6. We also prove that for the Binomial random graph model G(n, p), for every positive ε> 0 and large enough values of K, if p> K ln n n then with high probability the local resilience of G(n, p) with respect to being Hamiltonian is at least (1 − ε)np/6. Finally, we apply similar techniques to Positional Games and prove that if d is large enough then with high probability a typical random dregular graph G is such that in the unbiased MakerBreaker game played on the edges of G, Maker has a winning strategy to create a Hamilton cycle. 1
Random regular graphs of nonconstant degree: edge distribution and applications
, 2006
"... In this work we show that with high probability the chromatic number of a graph sampled from the random regular graph model Gn,d for d = o(n 1/5) is concentrated in two consecutive values, thus extending a previous result of Achlioptas and Moore. This concentration phenomena is very similar to that ..."
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Cited by 10 (5 self)
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In this work we show that with high probability the chromatic number of a graph sampled from the random regular graph model Gn,d for d = o(n 1/5) is concentrated in two consecutive values, thus extending a previous result of Achlioptas and Moore. This concentration phenomena is very similar to that of the binomial random graph model G(n, p) with p = d. Our proof is largely based on ideas of Alon n and Krivelevich who proved this twopoint concentration result for G(n, p) for p = n −δ where δ> 1/2. The main tool used to derive such a result is a careful analysis of the distribution of edges in Gn,d, relying both on the switching technique and on bounding the probability of exponentially small events in the configuration model. 1
Changepoint detection over graphs with the spectral scan statistic. Arxiv preprint arXiv:1206.0773
, 2012
"... We consider the changepoint detection problem of deciding, based on noisy measurements, whether an unknown signal over a given graph is constant or is instead piecewise constant over two induced subgraphs of relatively low cut size. We analyze the corresponding generalized likelihood ratio (GLR) st ..."
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Cited by 10 (5 self)
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We consider the changepoint detection problem of deciding, based on noisy measurements, whether an unknown signal over a given graph is constant or is instead piecewise constant over two induced subgraphs of relatively low cut size. We analyze the corresponding generalized likelihood ratio (GLR) statistic and relate it to the problem of finding a sparsest cut in a graph. We develop a tractable relaxation of the GLR statistic based on the combinatorial Laplacian of the graph, which we call the spectral scan statistic, and analyze its properties. We show how its performance as a testing procedure depends directly on the spectrum of the graph, and use this result to explicitly derive its asymptotic properties on few graph topologies. Finally, we demonstrate both theoretically and by simulations that the spectral scan statistic can outperform naive testing procedures based on edge thresholding and χ2 testing. 1
Small spectral gap in the combinatorial Laplacian implies Hamiltonian
, 2006
"... We consider the spectral and algorithmic aspects of the problem of finding a Hamiltonian cycle in a graph. We show that a sufficient condition for a graph being Hamiltonian is that the eigenvalues of the combinatorial Laplacian are sufficiently close to the average degree of the graph. An algorithm ..."
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Cited by 6 (1 self)
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We consider the spectral and algorithmic aspects of the problem of finding a Hamiltonian cycle in a graph. We show that a sufficient condition for a graph being Hamiltonian is that the eigenvalues of the combinatorial Laplacian are sufficiently close to the average degree of the graph. An algorithm is given for the problem of finding a Hamiltonian cycle in graphs with bounded spectral gaps which has complexity of order n c ln n. 1
On minimal perimeter polyminoes
 In The 13th International Conference on Discrete Geometry for Computer Imagery (DGCI2006
, 2006
"... Abstract. This paper explores proofs of the isoperimetric inequality for 4connected shapes on the integer grid Z2, and its geometric meaning. Pictorially, we discuss ways to place a maximal number unit square tiles on a chess board so that the shape they form has a minimal number of unit square nei ..."
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Cited by 2 (2 self)
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Abstract. This paper explores proofs of the isoperimetric inequality for 4connected shapes on the integer grid Z2, and its geometric meaning. Pictorially, we discuss ways to place a maximal number unit square tiles on a chess board so that the shape they form has a minimal number of unit square neighbors. Previous works have shown that “digital spheres” have a minimum of neighbors for their area. We here characterize all shapes that are optimal and show that they are all close to being digital spheres. In addition, we show a similar result when the 8connectivity metric is assumed (i.e. connectivity through vertices or edges, instead of edge connectivity as in 4connectivity). 1
Edge Isoperimetric
"... Recall that in the last lecture we looked at the problem of isoperimetric inequalities in the hypercube, Qn. Our notion of boundary was that of vertex boundary, defined by δ(G) = {v ̸ ∈ S  v ∼ u, u ∈ S}. We found that when trying to minimize the vertex boundary while holding the size of our set S ..."
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Recall that in the last lecture we looked at the problem of isoperimetric inequalities in the hypercube, Qn. Our notion of boundary was that of vertex boundary, defined by δ(G) = {v ̸ ∈ S  v ∼ u, u ∈ S}. We found that when trying to minimize the vertex boundary while holding the size of our set S fixed, the (near) Hamming balls characterize those S with minimal vertex boundary. In this lecture, we consider a different notion of boundary based on edges. We define the edge boundary of S to be ∂(S) = {{u, v}  u ∈ S, v ̸ ∈ S}. This is exactly the set of edges required to disconnect S from any vertex not in S. 1.1 Four Problems We shall look at four related isoperimetric problems which all utilize the concept of edge boundary. 1.1.1 Question 1 We begin with the simplest question: Given a fixed positive integer m, what is the smallest edge boundary for a set of m vertices? We may formalize this question by defining g(m) = min
NEW ROUNDING TECHNIQUES FOR THE DESIGN AND ANALYSIS OF APPROXIMATION ALGORITHMS
, 2014
"... We study two of the most central classical optimization problems, namely the Traveling Salesman problems and Graph Partitioning problems and develop new approximation algorithms for them. We introduce several new techniques for rounding a fractional solution of a continuous relaxation of these probl ..."
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We study two of the most central classical optimization problems, namely the Traveling Salesman problems and Graph Partitioning problems and develop new approximation algorithms for them. We introduce several new techniques for rounding a fractional solution of a continuous relaxation of these problems into near optimal integral solutions. The two most notable of those are the maximum entropy rounding by sampling method and a novel use of higher eigenvectors of graphs.
Vertex Percolation on Expander Graphs
, 710
"... We say that a graph G = (V, E) on n vertices is a βexpander for some constant β> 0 if every U ⊆ V of cardinality U  ≤ n 2 satisfies NG(U)  ≥ βU  where NG(U) denotes the neighborhood of U. In this work we explore the process of deleting vertices of a βexpander independently at random wit ..."
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We say that a graph G = (V, E) on n vertices is a βexpander for some constant β> 0 if every U ⊆ V of cardinality U  ≤ n 2 satisfies NG(U)  ≥ βU  where NG(U) denotes the neighborhood of U. In this work we explore the process of deleting vertices of a βexpander independently at random with probability n −α for some constant α> 0, and study the properties of the resulting graph. Our main result states that as n tends to infinity, the deletion process performed on a βexpander graph of bounded degree will result with high probability in a graph composed of a giant component containing n − o(n) vertices that is in itself an expander graph, and constant size components. We proceed by applying the main result to expander graphs with a positive spectral gap. In the particular case of (n, d, λ)graphs, that are such expanders, we compute the values of α, under additional constraints on the graph, for which with high probability the resulting graph will stay connected, or will be composed of a giant component and isolated vertices. As a graph sampled from the uniform probability space of dregular graphs with high probability is an expander and meets the additional constraints, this result strengthens a recent result due to Greenhill, Holt and Wormald about vertex percolation on random dregular graphs. We conclude by showing that performing the above described deletion process on graphs that expand sublinear sets by an unbounded expansion ratio, with high probability results in a connected expander graph. 1
Vertex Percolation on Expander Graphs Sonny BenShimon
, 710
"... We say that a graph G = (V, E) on n vertices is a βexpander for some constant β> 0 if every U ⊆ V of cardinality U  ≤ n satisfies NG(U)  ≥ βU  where NG(U) denotes the neighborhood of U. We 2 explore the process of uniformly at random deleting vertices of a βexpander with probability n − ..."
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We say that a graph G = (V, E) on n vertices is a βexpander for some constant β> 0 if every U ⊆ V of cardinality U  ≤ n satisfies NG(U)  ≥ βU  where NG(U) denotes the neighborhood of U. We 2 explore the process of uniformly at random deleting vertices of a βexpander with probability n −α for some constant α> 0. Our main result implies that as n tends to infinity, the deletion process performed on a βexpander graph of bounded degree will result with high probability in a graph composed of a giant component containing n − o(n) vertices which is itself an expander graph, and small constant size components. We proceed by applying the main result to expander graphs with a positive spectral gap. In the particular case of (n, d, λ)graphs, which are such expanders, we compute the values of α, under additional constraints on the graph, for which with high probability the resulting graph will stay connected, or will be composed of a giant component and isolated vertices. As a graph sampled from the uniform probability space of dregular graphs with hight probability meets all of these constraints, this result strengthens a recent result due to Greenhill, Holt, and Wormald [6] who prove a similar theorem for Gn,d. We conclude by showing that performing the deletion process with the prescribed deletion probability on expander graphs that expand sublinear sets by an unbounded expansion ratio, with high probability results in an expander graph. 1