7 research outputs found

    On giant components and treewidth in the layers model

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    Given an undirected nn-vertex graph G(V,E)G(V,E) and an integer kk, let Tk(G)T_k(G) denote the random vertex induced subgraph of GG generated by ordering VV according to a random permutation Ο€\pi and including in Tk(G)T_k(G) those vertices with at most kβˆ’1k-1 of their neighbors preceding them in this order. The distribution of subgraphs sampled in this manner is called the \emph{layers model with parameter} kk. The layers model has found applications in studying β„“\ell-degenerate subgraphs, the design of algorithms for the maximum independent set problem, and in bootstrap percolation. In the current work we expand the study of structural properties of the layers model. We prove that there are 33-regular graphs GG for which with high probability T3(G)T_3(G) has a connected component of size Ξ©(n)\Omega(n). Moreover, this connected component has treewidth Ξ©(n)\Omega(n). This lower bound on the treewidth extends to many other random graph models. In contrast, T2(G)T_2(G) is known to be a forest (hence of treewidth~1), and we establish that if GG is of bounded degree then with high probability the largest connected component in T2(G)T_2(G) is of size O(log⁑n)O(\log n). We also consider the infinite two-dimensional grid, for which we prove that the first four layers contain a unique infinite connected component with probability 11

    On Percolation and NP-Hardness

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    The edge-percolation and vertex-percolation random graph models start with an arbitrary graph G, and randomly delete edges or vertices of G with some fixed probability. We study the computational hardness of problems whose inputs are obtained by applying percolation to worst-case instances. Specifically, we show that a number of classical N P-hard graph problems remain essentially as hard on percolated instances as they are in the worst-case (assuming NP !subseteq BPP). We also prove hardness results for other NP-hard problems such as Constraint Satisfaction Problems, where random deletions are applied to clauses or variables. We focus on proving the hardness of the Maximum Independent Set problem and the Graph Coloring problem on percolated instances. To show this we establish the robustness of the corresponding parameters alpha(.) and Chi(.) to percolation, which may be of independent interest. Given a graph G, let G\u27 be the graph obtained by randomly deleting edges of G. We show that if alpha(G) is small, then alpha(G\u27) remains small with probability at least 0.99. Similarly, we show that if Chi(G) is large, then Chi(G\u27) remains large with probability at least 0.99

    On Giant Components and Treewidth in the Layers Model

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    Given an undirected nn-vertex graph G(V,E)G(V,E) and an integer kk, let Tk(G)T_k(G) denote the random vertex induced subgraph of GG generated by ordering VV according to a random permutation Ο€\pi and including in Tk(G)T_k(G) those vertices with at most kβˆ’1k-1 of their neighbors preceding them in this order. The distribution of subgraphs sampled in this manner is called the \emph{layers model with parameter} kk. The layers model has found applications in studying β„“\ell-degenerate subgraphs, the design of algorithms for the maximum independent set problem, and in bootstrap percolation. In the current work we expand the study of structural properties of the layers model. We prove that there are 33-regular graphs GG for which with high probability T3(G)T_3(G) has a connected component of size Ξ©(n)\Omega(n). Moreover, this connected component has treewidth Ξ©(n)\Omega(n). This lower bound on the treewidth extends to many other random graph models. In contrast, T2(G)T_2(G) is known to be a forest (hence of treewidth~1), and we establish that if GG is of bounded degree then with high probability the largest connected component in T2(G)T_2(G) is of size O(log⁑n)O(\log n). We also consider the infinite two-dimensional grid, for which we prove that the first four layers contain a unique infinite connected component with probability 11

    Infinite and Giant Components in the Layers Percolation Model

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    In this work we continue the investigation launched in \cite{feige2013layers} of the structural properties of the structural properties of the \emph{Layers model}, a dependent percolation model. Given an undirected graph G=(V,E)G=(V,E) and an integer kk, let Tk(G)T_k(G) denote the random vertex-induced subgraph of GG, generated by ordering VV according to Uniform[0,1][0,1] i.i.d.\mathrm{i.i.d.} clocks and including in Tk(G)T_k(G) those vertices with at most kβˆ’1k-1 of their neighbors having a faster clock. The distribution of subgraphs sampled in this manner is called the \emph{layers model with parameter} kk. The layers model has found applications in the study of β„“\ell-degenerate subgraphs, the design of algorithms for the maximum independent set problem and in the study of bootstrap percolation. We prove that every infinite locally finite tree TT with no leaves, satisfying that the degree of the vertices grow sub-exponentially in their distance from the root, T3(T)T_3(T) a.s.\mathrm{a.s.} has an infinite connected component. In contrast, we show that for any locally finite graph GG, a.s.\mathrm{a.s.} every connected component of T2(G)T_2(G) is finite. We also consider random graphs with a given degree sequence and show that if the minimal degree is at least 3 and the maximal degree is bounded, then w.h.p.\mathrm{w.h.p.} T3T_3 has a giant component. Finally, we also consider Zd{\Z}^{d} and show that if dd is sufficiently large, then a.s.\mathrm{a.s.} T4(Zd)T_4(\Z^d) contains an infinite cluster
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