Let @math be a fixed integer and let @math be the number of @math -colourings of the graph @math . For certain values of the average degree, the random variable @math is known to be concentrated in the sense that @math converges to @math in probability [Achlioptas and Coja-Oghlan: FOCS 2008]. In the present paper we prove a significantly stronger concentration result. Namely, we show that for a wide range of average degrees, @math converges to @math in probability for any diverging function @math . For @math exceeding a certain constant @math this result covers all average degrees up to the so-called condensation phase transition, and this is best possible. As an application, we show that the experiment of choosing a @math -colouring of the random graph @math uniformly at random is contiguous with respect to the so-called "planted model".
We expect that the present approach of combining the second moment method with small subgraph conditioning can be applied successfully to a variety of other random constraint problems. Immediate examples that spring to mind include random @math -NAESAT or random @math -XORSAT, random hypergraph @math -colourability or, more generally, the family of problems studied in @cite_19 . (On the other hand, we expect that in problems such as random @math -SAT the logarithm of the number of satisfying assignments exhibits stronger fluctuations, due to a lack of symmetry.)
@cite_19
1818081266
Many NP-complete constraint satisfaction problems appear to undergo a “phase transition” from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment (expectation) of the number of solutions tends to zero. We show that in the case of certain symmetric constraints, considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold. Specifically, we prove that the threshold for both random hypergraph 2-colorability (Property B) and random Not-All-Equal @math -SAT is @math . As a corollary, we establish that the threshold for random @math -SAT is of order @math , resolving a long-standing open problem. Over the past decade, physicists have developed deep but non-rigorous techniques for studying phase transitions in discrete structures. Recently, their ideas have been harnessed to obtain improved rigorous results on the phase transitions in binary problems such as random @math -SAT or @math -NAESAT (e.g., Coja-Oghlan and Panagiotou: STOC 2013). However, these rigorous arguments, typically centered around the second moment method, do not extend easily to problems where there are more than two possible values per variable. The single most intensely studied example of such a problem is random graph @math -coloring. Here we develop a novel approach to the second moment method in this problem. This new method, inspired by physics conjectures on the geometry of the set of @math -colorings, allows us to establish a substantially improved lower bound on the @math -colorability threshold. The new lower bound is within an additive @math of a simple first-moment upper bound and within @math of the physics conjecture. By comparison, the best previous lower bound left a gap of about @math , unbounded in terms of the number of colors [Achlioptas, Naor: STOC 2004]. Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with @math variables and @math clauses, there is a value of @math beyond which the CSP will be unsatisfiable with high probability. Strong refutation is the problem of certifying that no variable assignment satisfies more than a constant fraction of clauses; this is the natural algorithmic problem in the unsatisfiable regime (when @math ). Intuitively, strong refutation should become easier as the clause density @math grows, because the contradictions introduced by the random clauses become more locally apparent. For CSPs such as @math -SAT and @math -XOR, there is a long-standing gap between the clause density at which efficient strong refutation algorithms are known, @math , and the clause density at which instances become unsatisfiable with high probability, @math . In this paper, we give spectral and sum-of-squares algorithms for strongly refuting random @math -XOR instances with clause density @math in time @math or in @math rounds of the sum-of-squares hierarchy, for any @math and any integer @math . Our algorithms provide a smooth transition between the clause density at which polynomial-time algorithms are known at @math , and brute-force refutation at the satisfiability threshold when @math . We also leverage our @math -XOR results to obtain strong refutation algorithms for SAT (or any other Boolean CSP) at similar clause densities. Our algorithms match the known sum-of-squares lower bounds due to Grigoriev and Schonebeck, up to logarithmic factors. Additionally, we extend our techniques to give new results for certifying upper bounds on the injective tensor norm of random tensors. This paper considers stochastic optimization problems whose objective functions involve powers of random variables. For example, consider the classic Stochastic lp Load Balancing Problem (SLBp): There are @math machines and @math jobs, and known independent random variables @math decribe the load incurred on machine @math if we assign job @math to it. The goal is to assign each jobs to machines in order to minimize the expected @math -norm of the total load on the machines. While convex relaxations represent one of the most powerful algorithmic tools, in problems such as SLBp the main difficulty is to capture the objective function in a way that only depends on each random variable separately. We show how to capture @math -power-type objectives in such separable way by using the @math -function method, introduced by Lata a to relate the moment of sums of random variables to the individual marginals. We show how this quickly leads to a constant-factor approximation for very general subset selection problem with @math -moment objective. Moreover, we give a constant-factor approximation for SLBp, improving on the recent @math -approximation of [, SODA 18]. Here the application of the method is much more involved. In particular, we need to sharply connect the expected @math -norm of a random vector with the @math -moments of its marginals (machine loads), taking into account simultaneously the different scales of the loads that are incurred by an unknown assignment. Denote by an @math -component a connected @math -uniform hypergraph with @math edges and @math vertices. We prove that the expected number of creations of @math -component during a random hypergraph process tends to 1 as @math and @math tend to @math with the total number of vertices @math such that @math . Under the same conditions, we also show that the expected number of vertices that ever belong to an @math -component is approximately @math . As an immediate consequence, it follows that with high probability the largest @math -component during the process is of size @math . Our results give insight about the size of giant components inside the phase transition of random hypergraphs. We consider the problem of @math -colouring a random @math -uniform hypergraph with @math vertices and @math edges, where @math , @math , @math remain constant as @math tends to infinity. Achlioptas and Naor showed that the chromatic number of a random graph in this setting, the case @math , must have one of two easily computable values as @math tends to infinity. We give a complete generalisation of this result to random uniform hypergraphs. Survey Propagation is an algorithm designed for solving typical instances of random constraint satisfiability problems. It has been successfully tested on random 3-SAT and random @math graph 3-coloring, in the hard region of the parameter space. Here we provide a generic formalism which applies to a wide class of discrete Constraint Satisfaction Problems. We study the following one-person game against a random graph process: the Player's goal is to @math -colour a random sequence of edges @math of a complete graph on @math vertices, avoiding a monochromatic triangle for as long as possible. The game is over when a monochromatic triangle is created. The online version of the game requires that the Player should colour each edge as it comes, before looking at the next edge.While it is not hard to prove that the expected length of this game is about @math , the proof of the upper bound suggests the following relaxation: instead of colouring online, the random graph is generated in only two rounds, and the Player colours the edges of the first round before the edges of the second round are thrown in. Given the size of the first round, how many edges can there be in the second round for the Player to be likely to win? In the extreme case, when the first round consists of a random graph with @math edges, where @math is a positive constant, we show that the Player can win with high probability only if constantly many edges are generated in the second round. We determine the asymptotics of the independence number of the random @math -regular graph for all @math . It is highly concentrated, with constant-order fluctuations around @math for explicit constants @math and @math . Our proof rigorously confirms the one-step replica symmetry breaking heuristics for this problem, and we believe the techniques will be more broadly applicable to the study of other combinatorial properties of random graphs. In a coalescing random walk, a set of particles make independent discrete-time random walks on a graph. Whenever one or more particles meet at a vertex, they unite to form a single particle, which then continues a random walk through the graph. Let @math be an undirected and connected graph with @math vertices and @math edges. The coalescence time, @math , is the expected time for all particles to coalesce, when initially one particle is located at each vertex. We study the problem of bounding the coalescence time for general connected graphs and prove that @math . Here @math is the second eigenvalue of the transition matrix of the random walk. To avoid problems arising from, e.g., lack of coalescence on bipartite graphs, we assume the random walk can be made lazy if required. The value of @math is given by @math , where @math is the degree of vertex @math , and @math is the average degree. The parame... Let @math be a uniformly distributed random @math -SAT formula with @math variables and @math clauses. We present a polynomial time algorithm that finds a satisfying assignment of @math with high probability for constraint densities @math , where @math . Previously no efficient algorithm was known to find satisfying assignments with a nonvanishing probability beyond @math [A. Frieze and S. Suen, J. Algorithms, 20 (1996), pp. 312-355]. Let @math be a uniformly distributed random @math -SAT formula with @math variables and @math clauses. For clauses variables ratio @math the formula @math is satisfiable with high probability. However, no efficient algorithm is known to provably find a satisfying assignment beyond @math with a non-vanishing probability. Non-rigorous statistical mechanics work on @math -CNF led to the development of a new efficient "message passing algorithm" called [M ', Science 2002]. Experiments conducted for @math suggest that the algorithm finds satisfying assignments close to @math . However, in the present paper we prove that the basic version of Survey Propagation Guided Decimation fails to solve random @math -SAT formulas efficiently already for @math with @math almost a factor @math below @math . We study properties of multiple random walks on a graph under various assumptions of interaction between the particles. To give precise results, we make the analysis for random regular graphs. The cover time of a random walk on a random @math -regular graph was studied in [C. Cooper and A. Frieze, SIAM J. Discrete Math., 18 (2005), pp. 728-740], where it was shown with high probability (whp) that for @math the cover time is asymptotic to @math , where @math . In this paper we prove the following (whp) results, arising from the study of multiple random walks on a random regular graph @math . For @math independent walks on @math , the cover time @math is asymptotic to @math , where @math is the cover time of a single walk. For most starting positions, the expected number of steps before any of the walks meet is @math . If the walks can communicate when meeting at a vertex, we show that, for most starting positions, the expected time for @math walks to broadcast a single piece of information to each other is asymptotic to @math as @math . We also establish properties of walks where there are two types of particles, predator and prey, or where particles interact when they meet at a vertex by coalescing or by annihilating each other. For example, the expected extinction time of @math explosive particles ( @math even) tends to @math as @math . The case of @math coalescing particles, where one particle is initially located at each vertex, corresponds to a voter model defined as follows: Initially each vertex has a distinct opinion, and at each step each vertex changes its opinion to that of a random neighbor. The expected time for a unique opinion to emerge is the same as the expected time for all the particles to coalesce, which is asymptotic to @math . Combining results from the predator-prey and multiple random walk models allows us to compare expected detection times of all prey in the following scenarios: Both the predator and the prey move randomly, the prey moves randomly and the predators stay fixed, and the predators move randomly and the prey stays fixed. In all cases, with @math predators and @math prey the expected detection time is @math , where @math is the @math th harmonic number. Let @math be a nontrivial @math -ary predicate. Consider a random instance of the constraint satisfaction problem @math on @math variables with @math constraints, each being @math applied to @math randomly chosen literals. Provided the constraint density satisfies @math , such an instance is unsatisfiable with high probability. The problem is to efficiently find a proof of unsatisfiability. We show that whenever the predicate @math supports a @math - probability distribution on its satisfying assignments, the sum of squares (SOS) algorithm of degree @math (which runs in time @math ) refute a random instance of @math . In particular, the polynomial-time SOS algorithm requires @math constraints to refute random instances of CSP @math when @math supports a @math -wise uniform distribution on its satisfying assignments. Together with recent work of [LRS15], our result also implies that polynomial-size semidefinite programming relaxation for refutation requires at least @math constraints. Our results (which also extend with no change to CSPs over larger alphabets) subsume all previously known lower bounds for semialgebraic refutation of random CSPs. For every constraint predicate @math , they give a three-way hardness tradeoff between the density of constraints, the SOS degree (hence running time), and the strength of the refutation. By recent algorithmic results of [AOW15] and [RRS16], this full three-way tradeoff is , up to lower-order factors. A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph @math has a rainbow Hamiltonian path. Although this conjecture turned out to be false, it was widely believed that such a colouring always contains a rainbow cycle of length almost @math . In this paper, improving on several earlier results, we confirm this by proving that every properly edge-coloured @math has a rainbow cycle of length @math . One of the main ingredients of our proof, which is of independent interest, shows that a random subgraph of a properly edge-coloured @math formed by the edges of a random set of colours has a similar edge distribution as a truly random graph with the same edge density. In particular it has very good expansion properties. This paper presents a distributed O(1)-approximation algorithm, with expected- @math running time, in the @math model for the metric facility location problem on a size- @math clique network. Though metric facility location has been considered by a number of researchers in low-diameter settings, this is the first sub-logarithmic-round algorithm for the problem that yields an O(1)-approximation in the setting of non-uniform facility opening costs. In order to obtain this result, our paper makes three main technical contributions. First, we show a new lower bound for metric facility location, extending the lower bound of B a (ICALP 2005) that applies only to the special case of uniform facility opening costs. Next, we demonstrate a reduction of the distributed metric facility location problem to the problem of computing an O(1)-ruling set of an appropriate spanning subgraph. Finally, we present a sub-logarithmic-round (in expectation) algorithm for computing a 2-ruling set in a spanning subgraph of a clique. Our algorithm accomplishes this by using a combination of randomized and deterministic sparsification. For input @math , let @math denote the set of outputs that are the "legal" answers for a computational problem @math . Suppose @math and members of @math are so large that there is not time to read them in their entirety. We propose a model of local computation algorithms which for a given input @math , support queries by a user to values of specified locations @math in a legal output @math . When more than one legal output @math exists for a given @math , the local computation algorithm should output in a way that is consistent with at least one such @math . Local computation algorithms are intended to distill the common features of several concepts that have appeared in various algorithmic subfields, including local distributed computation, local algorithms, locally decodable codes, and local reconstruction. We develop a technique, based on known constructions of small sample spaces of @math -wise independent random variables and Beck's analysis in his algorithmic approach to the Lov ' a sz Local Lemma, which under certain conditions can be applied to construct local computation algorithms that run in polylogarithmic time and space. We apply this technique to maximal independent set computations, scheduling radio network broadcasts, hypergraph coloring and satisfying @math -SAT formulas. We design new polynomials for representing threshold functions in three different regimes: probabilistic polynomials of low degree, which need far less randomness than previous constructions, polynomial threshold functions (PTFs) with "nice" threshold behavior and degree almost as low as the probabilistic polynomials, and a new notion of probabilistic PTFs where we combine the above techniques to achieve even lower degree with similar "nice" threshold behavior. Utilizing these polynomial constructions, we design faster algorithms for a variety of problems: @math Offline Hamming Nearest (and Furthest) Neighbors: Given @math red and @math blue points in @math -dimensional Hamming space for @math , we can find an (exact) nearest (or furthest) blue neighbor for every red point in randomized time @math or deterministic time @math . These also lead to faster MAX-SAT algorithms for sparse CNFs. @math Offline Approximate Nearest (and Furthest) Neighbors: Given @math red and @math blue points in @math -dimensional @math or Euclidean space, we can find a @math -approximate nearest (or furthest) blue neighbor for each red point in randomized time near @math . @math SAT Algorithms and Lower Bounds for Circuits With Linear Threshold Functions: We give a satisfiability algorithm for @math circuits with a subquadratic number of linear threshold gates on the bottom layer, and a subexponential number of gates on the other layers, that runs in deterministic @math time. This also implies new circuit lower bounds for threshold circuits. We also give a randomized @math -time SAT algorithm for subexponential-size @math circuits, where the top @math gate and middle @math gates have @math fan-in. This article offers a simplified approach to the distribution theory of randomly weighted averages or @math -means @math , for a sequence of i.i.d.random variables @math , and independent random weights @math with @math and @math . The collection of distributions of @math , indexed by distributions of @math , is shown to encode Kingman's partition structure derived from @math . For instance, if @math has Bernoulli @math distribution on @math , the @math th moment of @math is a polynomial function of @math which equals the probability generating function of the number @math of distinct values in a sample of size @math from @math : @math . This elementary identity illustrates a general moment formula for @math -means in terms of the partition structure associated with random samples from @math , first developed by Diaconis and Kemperman (1996) and Kerov (1998) in terms of random permutations. As shown by Tsilevich (1997) if the partition probabilities factorize in a way characteristic of the generalized Ewens sampling formula with two parameters @math , found by Pitman (1992), then the moment formula yields the Cauchy-Stieltjes transform of an @math mean. The analysis of these random means includes the characterization of @math -means, known as Dirichlet means, due to Von Neumann (1941), Watson (1956) and Cifarelli and Regazzini (1990) and generalizations of L 'evy's arcsine law for the time spent positive by a Brownian motion, due to Darling (1949) Lamperti (1958) and Barlow, Pitman and Yor (1989). Random instances of constraint satisfaction problems (CSPs) appear to be hard for all known algorithms when the number of constraints per variable lies in a certain interval. Contributing to the general understanding of the structure of the solution space of a CSP in the satisfiable regime, we formulate a set of technical conditions on a large family of random CSPs and prove bounds on three most interesting thresholds for the density of such an ensemble: namely, the satisfiability threshold, the threshold for clustering of the solution space, and the threshold for an appropriate reconstruction problem on the CSPs. The bounds become asymptoticlally tight as the number of degrees of freedom in each clause diverges. The families are general enough to include commonly studied problems such as random instances of Not-All-Equal SAT, k-XOR formulae, hypergraph 2-coloring, and graph k-coloring. An important new ingredient is a condition involving the Fourier expansion of clauses, which characterizes the class of ...