On random k‐out subgraphs of large graphs

We consider random subgraphs of a fixed graph with large minimum degree. We fix a positive integer k and let G_k be the random subgraph where each independently chooses k random neighbors, making kn edges in all. When the minimum degree then G_k is k‐connected w.h.p. for ; Hamiltonian for k sufficiently large. When , then G_k has a cycle of length for . By w.h.p. we mean that the probability of non‐occurrence can be bounded by a function (or ) where . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 143–157, 2017     

Local reconstruction of low‐rank matrices and subspaces

We study the problem of reconstructing a low‐rank matrix, where the input is an n × m matrix M over a field and the goal is to reconstruct a (near‐optimal) matrix that is low‐rank and close to M under some distance function Δ. Furthermore, the reconstruction must be local, i.e., provides access to any desired entry of by reading only a few entries of the input M (ideally, independent of the matrix dimensions n and m). Our formulation of this problem is inspired by the local reconstruction framework of Saks and Seshadhri (SICOMP, 2010). Our main result is a local reconstruction algorithm for the case where Δ is the normalized Hamming distance (between matrices). Given M that is ‐close to a matrix of rank (together with d and ), this algorithm computes with high probability a rank‐d matrix that is ‐close to M. This is a local algorithm that proceeds in two phases. The preprocessing phase reads only random entries of M, and stores a small data structure. The query phase deterministically outputs a desired entry by reading only the data structure and 2d additional entries of M. We also consider local reconstruction in an easier setting, where the algorithm can read an entire matrix column in a single operation. When Δ is the normalized Hamming distance between vectors, we derive an algorithm that runs in polynomial time by applying our main result for matrix reconstruction. For comparison, when Δ is the truncated Euclidean distance and , we analyze sampling algorithms by using statistical learning tools. A preliminary version of this paper appears appears in ECCC, see: http://eccc.hpi-web.de/report/2015/128/ © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 607–630, 2017     

The time of graph bootstrap percolation

Graph bootstrap percolation, introduced by Bollobás in 1968, is a cellular automaton defined as follows. Given a “small” graph H and a “large” graph , in consecutive steps we obtain from G_t by adding to it all new edges e such that contains a new copy of H. We say that G percolates if for some , we have G_t = K_n. For H = K_r, the question about the size of the smallest percolating graphs was independently answered by Alon, Frankl and Kalai in the 1980's. Recently, Balogh, Bollobás and Morris considered graph bootstrap percolation for and studied the critical probability , for the event that the graph percolates with high probability. In this paper, using the same setup, we determine, up to a logarithmic factor, the critical probability for percolation by time t for all © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 143–168, 2017     

First passage percolation and escape strategies

Consider first passage percolation on with passage times given by i.i.d. random variables with common distribution F. Let be the time from u to v for a path π and the minimal time among all paths from u to v. We ask whether or not there exist points and a semi‐infinite path such that for all n. Necessary and sufficient conditions on F are given for this to occur. When the support of F is unbounded, we also obtain results on the number of edges with large passage time used by geodesics. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 414–423, 2015     

On induced Ramsey numbers for k‐uniform hypergraphs

Let denote the complete k‐uniform k‐partite hypergraph with classes of size t and the complete k‐uniform hypergraph of order s. One can show that the Ramsey number for and satisfies when t = s^o(1) as s →∞. The main part of this paper gives an analogous result for induced Ramsey numbers: Let be an arbitrary k‐partite k‐uniform hypergraph with classes of size t and an arbitrary k‐graph of order s. We use the probabilistic method to show that the induced Ramsey number (i.e. the smallest n for which there exists a hypergraph such that any red/blue coloring of yields either an induced red copy of or an induced blue copy of ) satisfies . © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 5–20, 2016     

Eigenvalue confinement and spectral gap for random simplicial complexes

We consider the adjacency operator of the Linial‐Meshulam model for random simplicial complexes on n vertices, where each d‐cell is added independently with probability p to the complete ‐skeleton. Under the assumption , we prove that the spectral gap between the smallest eigenvalues and the remaining eigenvalues is with high probability. This estimate follows from a more general result on eigenvalue confinement. In addition, we prove that the global distribution of the eigenvalues is asymptotically given by the semicircle law. The main ingredient of the proof is a Füredi‐Komlós‐type argument for random simplicial complexes, which may be regarded as sparse random matrix models with dependent entries. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 506–537, 2017     

Uniform multicommodity flows in the hypercube with random edge‐capacities

We give two results for multicommodity flows in the d‐dimensional hypercube with independent random edge‐capacities distributed like a random variable C where . Firstly, with high probability as , the network can support simultaneous multicommodity flows of volume close to between all antipodal vertex pairs. Secondly, with high probability, the network can support simultaneous multicommodity flows of volume close to between all vertex pairs. Both results are best possible. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 437–463, 2017     

Generalized PageRank on directed configuration networks

This paper studies the distribution of a family of rankings, which includes Google's PageRank, on a directed configuration model. In particular, it is shown that the distribution of the rank of a randomly chosen node in the graph converges in distribution to a finite random variable that can be written as a linear combination of i.i.d. copies of the attracting endogenous solution to a stochastic fixed‐point equation of the form where is a real‐valued vector with , and the are i.i.d. copies of , independent of . Moreover, we provide precise asymptotics for the limit , which when the in‐degree distribution in the directed configuration model has a power law imply a power law distribution for with the same exponent. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 237–274, 2017     

The cutoff phenomenon for random birth and death chains

For any distribution π on , we study elements drawn at random from the set of tridiagonal stochastic matrices K satisfying for all . These matrices correspond to birth and death chains with stationary distribution π. We analyze an algorithm for sampling from and use results from this analysis to draw conclusions about the Markov chains corresponding to typical elements of . Our main interest is in determining when certain sequences of random birth and death chains exhibit the cutoff phenomenon. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 287–321, 2017     

σ‐algebras for quasirandom hypergraphs

We examine the correspondence between the various notions of quasirandomness for k‐uniform hypergraphs and σ‐algebras related to measurable hypergraphs. This gives a uniform formulation of most of the notions of quasirandomness for dense hypergraphs which have been studied, with each notion of quasirandomness corresponding to a σ‐algebra defined by a collection of subsets of . We associate each notion of quasirandomness with a collection of hypergraphs, the ‐adapted hypergraphs, so that G is quasirandom exactly when it contains roughly the correct number of copies of each ‐adapted hypergraph. We then identify, for each , a particular ‐adapted hypergraph with the property that if G contains roughly the correct number of copies of then G is quasirandom in the sense of . This generalizes recent results of Kohayakawa, Nagle, Rödl, and Schacht; Conlon, Hàn, Person, and Schacht; and Lenz and Mubayi giving this result for some particular notions of quasirandomness. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 114–139, 2017     

Almost‐spanning universality in random graphs

A graph G is said to be ‐universal if it contains every graph on at most n vertices with maximum degree at most Δ. It is known that for any and any natural number Δ there exists such that the random graph G(n, p) is asymptotically almost surely ‐universal for . Bypassing this natural boundary, we show that for the same conclusion holds when . © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 50, 380–393, 2017     

Four random permutations conjugated by an adversary generate Sn with high probability

We prove a conjecture dating back to a 1978 paper of D.R. Musser [11], namely that four random permutations in the symmetric group S_n generate a transitive subgroup with probability for some independent of n, even when an adversary is allowed to conjugate each of the four by a possibly different element of . In other words, the cycle types already guarantee generation of a transitive subgroup; by a well known argument, this implies generation of A_n or except for probability as . The analysis is closely related to the following random set model. A random set is generated by including each independently with probability . The sumset is formed. Then at most four independent copies of are needed before their mutual intersection is no longer infinite. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 49, 409–428, 2016     

The asphericity of random 2‐dimensional complexes

We study random 2‐dimensional complexes in the Linial–Meshulam model and prove that for the probability parameter satisfying a random 2‐complex Y contains several pairwise disjoint tetrahedra such that the 2‐complex Z obtained by removing any face from each of these tetrahedra is aspherical. Moreover, we prove that the obtained complex Z satisfies the Whitehead conjecture, i.e. any subcomplex is aspherical. This implies that Y is homotopy equivalent to a wedge where Z is a 2‐dimensional aspherical simplicial complex. We also show that under the assumptions where c > 3 and , the complex Z is genuinely 2‐dimensional and in particular, it has sizable 2‐dimensional homology; it follows that in the indicated range of the probability parameter p the cohomological dimension of the fundamental group of a random 2‐complex equals 2. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 46, 261–273, 2015     

Oscillations in the height of the Yule tree and application to the binary search tree

For a particular case of a branching random walk with lattice support, namely the Yule branching random walk, we prove that the distribution of the centred maximum oscillates around a distribution corresponding to a critical travelling wave in the following sense: there exist continuous functions and such that: where and is the height of the Yule tree. We also shows that similar oscillations occur for , when f is in a large class of functions. This process is classically related to the binary search tree, thus yielding analogous results for the height and for the saturation level of the binary search tree. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 90–120, 2017     

The threshold for combs in random graphs

For let denote the tree consisting of an ‐vertex path with disjoint ‐vertex paths beginning at each of its vertices. An old conjecture says that for any the threshold for the random graph to contain is at . Here we verify this for with any fixed . In a companion paper, using very different methods, we treat the complementary range, proving the conjecture for (with ). © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 794–802, 2016     

The Brownian plane with minimal neck baby universe

For each , let be a uniform rooted quadrangulation, endowed with an appropriate measure, of size n conditioned to have r(n) vertices in its root block. We prove that for a suitable function r(n), after rescaling graph distance by converges to a random pointed non‐compact metric measure space , in the local Gromov‐Hausdorff‐Prokhorov topology. The space is built by identifying a uniform point of the Brownian map with the distinguished point of the Brownian plane. © 2017 Wiley Periodicals, Inc. Random Struct. Alg., 51, 729–752, 2017     

Clairvoyant embedding in one dimension

Let v, w be infinite 0‐1 sequences, and a positive integer. We say that is ‐embeddable in , if there exists an increasing sequence of integers with , such that , for all . Let and be coin‐tossing sequences. We will show that there is an with the property that is ‐embeddable into with positive probability. This answers a question that was open for a while. The proof generalizes somewhat the hierarchical method of an earlier paper of the author on dependent percolation. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 520–560, 2015     

New methods to bound the critical probability in fractal percolation

We study the critical probability p_c(M) in two‐dimensional M‐adic fractal percolation. To find lower bounds, we compare fractal percolation with site percolation. Fundamentally new is the construction of a computable increasing sequence that converges to p_c(M). We prove that and . For the upper bounds, we introduce an iterative random process on a finite alphabet , which is easier to analyze than the original process. We show that and . © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 710–730, 2015     

On independent sets in random graphs

The independence number of a sparse random graph G(n,m) of average degree d = 2m/n is well‐known to be with high probability, with in the limit of large d. Moreover, a trivial greedy algorithm w.h.p. finds an independent set of size , i.e., about half the maximum size. Yet in spite of 30 years of extensive research no efficient algorithm has emerged to produce an independent set with size for any fixed (independent of both d and n). In this paper we prove that the combinatorial structure of the independent set problem in random graphs undergoes a phase transition as the size k of the independent sets passes the point . Roughly speaking, we prove that independent sets of size form an intricately rugged landscape, in which local search algorithms seem to get stuck. We illustrate this phenomenon by providing an exponential lower bound for the Metropolis process, a Markov chain for sampling independent sets. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 436–486, 2015     

A domination algorithm for {0,1}‐instances of the travelling salesman problem

We present an approximation algorithm for ‐instances of the travelling salesman problem which performs well with respect to combinatorial dominance. More precisely, we give a polynomial‐time algorithm which has domination ratio . In other words, given a ‐edge‐weighting of the complete graph on vertices, our algorithm outputs a Hamilton cycle of with the following property: the proportion of Hamilton cycles of whose weight is smaller than that of is at most . Our analysis is based on a martingale approach. Previously, the best result in this direction was a polynomial‐time algorithm with domination ratio for arbitrary edge‐weights. We also prove a hardness result showing that, if the Exponential Time Hypothesis holds, there exists a constant such that cannot be replaced by in the result above. © 2015 Wiley Periodicals, Inc. Random Struct. Alg., 48, 427–453, 2016