On percolation and ‐hardness

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 complexity of problems whose inputs are obtained by applying percolation to worst‐case instances. Specifically, we show that a number of classical ‐hard problems on graphs remain essentially as hard on percolated instances as they are in the worst‐case (assuming ). We also prove hardness results for other ‐hard problems such as Constraint Satisfaction Problems and Subset‐Sum, with suitable definitions of random deletions. Along the way, we establish that for any given graph G the independence number and the chromatic number are robust to percolation in the following sense. Given a graph G, let be the graph obtained by randomly deleting edges of G with some probability . We show that if is small, then remains small with probability at least 0.99. Similarly, we show that if is large, then remains large with probability at least 0.99. We believe these results are of independent interest.     

Swendsen‐Wang algorithm on the mean‐field Potts model

We study the q‐state ferromagnetic Potts model on the n‐vertex complete graph known as the mean‐field (Curie‐Weiss) model. We analyze the Swendsen‐Wang algorithm which is a Markov chain that utilizes the random cluster representation for the ferromagnetic Potts model to recolor large sets of vertices in one step and potentially overcomes obstacles that inhibit single‐site Glauber dynamics. Long et al. studied the case q = 2, the Swendsen‐Wang algorithm for the mean‐field ferromagnetic Ising model, and showed that the mixing time satisfies: (i) for , (ii) for , (iii) for , where β_c is the critical temperature for the ordered/disordered phase transition. In contrast, for there are two critical temperatures that are relevant. We prove that the mixing time of the Swendsen‐Wang algorithm for the ferromagnetic Potts model on the n‐vertex complete graph satisfies: (i) for , (ii) for , (iii) for , and (iv) for . These results complement refined results of Cuff et al. on the mixing time of the Glauber dynamics for the ferromagnetic Potts model.     

An optimal (ϵ,δ)‐randomized approximation scheme for the mean of random variables with bounded relative variance

Randomized approximation algorithms for many #P‐complete problems (such as the partition function of a Gibbs distribution, the volume of a convex body, the permanent of a {0,1}‐matrix, and many others) reduce to creating random variables X₁,X₂,… with finite mean μ and standard deviation σ such that μ is the solution for the problem input, and the relative standard deviation |σ/μ| ≤ c for known c. Under these circumstances, it is known that the number of samples from the {X_i} needed to form an (?,δ)‐approximation that satisfies is at least . We present here an easy to implement (?,δ)‐approximation that uses samples. This achieves the same optimal running time as other estimators, but without the need for extra conditions such as bounds on third or fourth moments.     

Edge‐coloring linear hypergraphs with medium‐sized edges

Motivated by the Erdos?‐Faber‐Lovász (EFL) conjecture for hypergraphs, we consider the list edge coloring of linear hypergraphs. We show that if the hyper‐edge sizes are bounded between i and inclusive, then there is a list edge coloring using colors. The dependence on n in the upper bound is optimal (up to the value of C_i,?).     

A characterization of constant‐sample testable properties

We characterize the set of properties of Boolean‐valued functions on a finite domain that are testable with a constant number of samples (x,f(x)) with x drawn uniformly at random from . Specifically, we show that a property is testable with a constant number of samples if and only if it is (essentially) a k‐part symmetric property for some constant k, where a property is k‐part symmetric if there is a partition of such that whether satisfies the property is determined solely by the densities of f on . We use this characterization to show that symmetric properties are essentially the only graph properties and affine‐invariant properties that are testable with a constant number of samples and that for every constant , monotonicity of functions on the d‐dimensional hypergrid is testable with a constant number of samples.     

The height of depth‐weighted random recursive trees

In this paper, we introduce a model of depth‐weighted random recursive trees, created by recursively joining a new leaf to an existing vertex . In this model, the probability of choosing depends on its depth in the tree. In particular, we assume that there is a function such that if has depth then its probability of being chosen is proportional to . We consider the expected value of the diameter of this model as determined by , and for various increasing we find expectations that range from polylogarithmic to linear.     

An extension of the Erdős‐Tetali theorem

Given a sequence , let r_??,h(n) denote the number of ways n can be written as the sum of h elements of ??. Fixing h ≥ 2, we show that if f is a suitable real function (namely: locally integrable, O‐regularly varying and of positive increase) satisfying then there must exist with for which r_{??,h + ?}(n) = Θ(f(n)^{h + ?}/n) for all ? ≥ 0. Furthermore, for h = 2 this condition can be weakened to . The proof is somewhat technical and the methods rely on ideas from regular variation theory, which are presented in an appendix with a view towards the general theory of additive bases. We also mention an application of these ideas to Schnirelmann's method.     

Diameter in ultra‐small scale‐free random graphs

It is well known that many random graphs with infinite variance degrees are ultra‐small. More precisely, for configuration models and preferential attachment models where the proportion of vertices of degree at least k is approximately k^{?(τ ? 1)} with τ ∈ (2,3), typical distances between pairs of vertices in a graph of size n are asymptotic to and , respectively. In this paper, we investigate the behavior of the diameter in such models. We show that the diameter is of order precisely when the minimal forward degree d_fwd of vertices is at least 2. We identify the exact constant, which equals that of the typical distances plus . Interestingly, the proof for both models follows identical steps, even though the models are quite different in nature.     

Partial resampling to approximate covering integer programs†

We consider column‐sparse covering integer programs, a generalization of set cover. We develop a new rounding scheme based on the partial resampling variant of the Lovász Local Lemma developed by Harris and Srinivasan. This achieves an approximation ratio of , where a_min is the minimum covering constraint and is the maximum ?₁‐norm of any column of the covering matrix A (whose entries are scaled to lie in [0, 1]). With additional constraints on the variable sizes, we get an approximation ratio of (where is the maximum number of nonzero entries in any column of A). These results improve asymptotically over results of Srinivasan and results of Kolliopoulos and Young. We show nearly‐matching lower bounds. We also show that the rounding process leads to negative correlation among the variables.     

On the cycle space of a random graph

Write for the cycle space of a graph G, for the subspace of spanned by the copies of the κ‐cycle in G, for the class of graphs satisfying , and for the class of graphs each of whose edges lies in a . We prove that for every odd and , so the 's of a random graph span its cycle space as soon as they cover its edges. For κ = 3 this was shown in [6].     

Rainbow structures in locally bounded colorings of graphs

We study approximate decompositions of edge‐colored quasirandom graphs into rainbow spanning structures: an edge‐coloring of a graph is locally ‐bounded if every vertex is incident to at most edges of each color, and is (globally) ‐bounded if every color appears at most times. Our results imply the existence of: (1) approximate decompositions of properly edge‐colored into rainbow almost‐spanning cycles; (2) approximate decompositions of edge‐colored into rainbow Hamilton cycles, provided that the coloring is ‐bounded and locally ‐bounded; and (3) an approximate decomposition into full transversals of any array, provided each symbol appears times in total and only times in each row or column. Apart from the logarithmic factors, these bounds are essentially best possible. We also prove analogues for rainbow ‐factors, where is any fixed graph. Both (1) and (2) imply approximate versions of the Brualdi‐Hollingsworth conjecture on decompositions into rainbow spanning trees.     

On the discrepancy of random low degree set systems

Motivated by the celebrated Beck‐Fiala conjecture, we consider the random setting where there are n elements and m sets and each element lies in t randomly chosen sets. In this setting, Ezra and Lovett showed an discrepancy bound when n ≤ m and an O(1) bound when n ? m^t. In this paper, we give a tight bound for the entire range of n and m, under a mild assumption that . The result is based on two steps. First, applying the partial coloring method to the case when and using the properties of the random set system we show that the overall discrepancy incurred is at most . Second, we reduce the general case to that of using LP duality and a careful counting argument.     

Packing and counting arbitrary Hamilton cycles in random digraphs

We prove packing and counting theorems for arbitrarily oriented Hamilton cycles in (n, p) for nearly optimal p (up to a factor). In particular, we show that given t = (1 ? o(1))np Hamilton cycles C₁,…,C_t, each of which is oriented arbitrarily, a digraph ～(n, p) w.h.p. contains edge disjoint copies of C₁,…,C_t, provided . We also show that given an arbitrarily oriented n‐vertex cycle C, a random digraph ～(n, p) w.h.p. contains (1 ± o(1))n!pⁿ copies of C, provided .     

On polynomial approximations to AC

Classical approximation results show that any circuit of size and depth has an ‐error probabilistic polynomial over the reals of degree . We improve this upper bound to , which is much better for small values of . We then use this result to show that ‐wise independence fools circuits of size and depth up to error at most , improving on Tal's strengthening of Braverman's result that ‐wise independence suffices. To our knowledge, this is the first PRG construction for that achieves optimal dependence on the error . We also prove lower bounds on the best polynomial approximations to . We show that any polynomial approximating the function on bits to a small constant error must have degree at least . This result improves exponentially on a result of Meka, Nguyen, and Vu (Theory Comput. 2016).     

Counting restricted orientations of random graphs

We count orientations of avoiding certain classes of oriented graphs. In particular, we study , the number of orientations of the binomial random graph in which every copy of is transitive, and , the number of orientations of containing no strongly connected copy of . We give the correct order of growth of and up to polylogarithmic factors; for orientations with no cyclic triangle, this significantly improves a result of Allen, Kohayakawa, Mota, and Parente. We also discuss the problem for a single forbidden oriented graph, and state a number of open problems and conjectures.     

Random perfect graphs

We investigate the asymptotic structure of a random perfect graph P_n sampled uniformly from the set of perfect graphs on vertex set . Our approach is based on the result of Prömel and Steger that almost all perfect graphs are generalised split graphs, together with a method to generate such graphs almost uniformly. We show that the distribution of the maximum of the stability number and clique number is close to a concentrated distribution L(n) which plays an important role in our generation method. We also prove that the probability that P_n contains any given graph H as an induced subgraph is asymptotically 0 or or 1. Further we show that almost all perfect graphs are 2‐clique‐colorable, improving a result of Bacsó et al. from 2004; they are almost all Hamiltonian; they almost all have connectivity equal to their minimum degree; they are almost all in class one (edge‐colorable using Δ colors, where Δ is the maximum degree); and a sequence of independently and uniformly sampled perfect graphs of increasing size converges almost surely to the graphon .     

On the threshold problem for Latin boxes

Let m ≤ n ≤ k. An m × n × k 0‐1 array is a Latin box if it contains exactly m n ones, and has at most one 1 in each line. As a special case, Latin boxes in which m = n = k are equivalent to Latin squares. Let be the distribution on m × n × k 0‐1 arrays where each entry is 1 with probability p, independently of the other entries. The threshold question for Latin squares asks when contains a Latin square with high probability. More generally, when does support a Latin box with high probability? Let ε > 0. We give an asymptotically tight answer to this question in the special cases where n = k and , and where n = m and . In both cases, the threshold probability is . This implies threshold results for Latin rectangles and proper edge‐colorings of K_n,n.     

Asymptotics in percolation on high‐girth expanders

We consider supercritical bond percolation on a family of high‐girth ‐regular expanders. The previous study of Alon, Benjamini and Stacey established that its critical probability for the appearance of a linear‐sized (“giant”) component is . Our main result recovers the sharp asymptotics of the size and degree distribution of the vertices in the giant and its 2‐core at any . It was further shown in the previous study that the second largest component, at any , has size at most for some . We show that, unlike the situation in the classical Erd?s‐Rényi random graph, the second largest component in bond percolation on a regular expander, even with an arbitrarily large girth, can have size for arbitrarily close to 1. Moreover, as a by‐product of that construction, we answer negatively a question of Benjamini on the relation between the diameter of a component in percolation on expanders and the existence of a giant component. Finally, we establish other typical features of the giant component, for example, the existence of a linear path.     

Phase transitions in graphs on orientable surfaces

Let be the orientable surface of genus and denote by the class of all graphs on vertex set with edges embeddable on . We prove that the component structure of a graph chosen uniformly at random from features two phase transitions. The first phase transition mirrors the classical phase transition in the Erd?s‐Rényi random graph chosen uniformly at random from all graphs with vertex set and edges. It takes place at , when the giant component emerges. The second phase transition occurs at , when the giant component covers almost all vertices of the graph. This kind of phenomenon is strikingly different from and has only been observed for graphs on surfaces.