For a fixed number d > 0 and n large let G(n, d/n) be the random graph on n vertices in which any two vertices are connected with probability d/n independently. The problem of determining the chromatic number of G(n, d/n) goes back to the famous 1960 article of Erdős and Rényi that started the theory of random graphs [Magayar Tud. Akad. Mat. Kutato Int. Kozl. 5 (1960) 17-61]. Progress culminated in the landmark paper of Achlioptas and Naor [Ann. Math. 162 (2005) 1333-1349], in which they calculate the chromatic number precisely for all d in a set S ⊂ (0, ∞) of asymptotic density limz→∞ 1 z z 0 1 S = 1 2 , and up to an additive error of one for the remaining d.Here we obtain a near-complete answer by determining the chromatic number of G(n, d/n) for all d in a set of asymptotic density 1. Mathematics Subject Classification: 05C80 (primary), 05C15 (secondary) Yet [1] is a pure existence result that does not provide any clue as to the location of d k−col . In a landmark paper Achlioptas and Naor [6] proved via the "second moment method" that lim inf n→∞ d k−col (n) ≥ d k,AN = 2(k − 1) ln(k − 1) = 2k ln k − 2 ln k − 2 + o k (1).(1.1)Here and throughout, o k (1) denotes a term that tends to zero in the limit of large k. By comparison, a naive application of the union bound shows that lim supRecently [14], a more sophisticated union bound argument was used to proveThus, the gap between the lower bound (1.1) and the upper bound (1.3) on d k−col (n) is about ln k + 1, an expression that diverges as k gets large. By improving the lower bound, the following theorem reduces this gap to a small absolute constant of 2 ln 2 − 1 + o k (1) ≈ 0.39.Theorem 1.
Based on a non-rigorous formalism called the "cavity method", physicists have put forward intriguing predictions on phase transitions in discrete structures. One of the most remarkable ones is that in problems such as random k-SAT or random graph k-coloring, very shortly before the threshold for the existence of solutions there occurs another phase transition called condensation [Krzakala et al., PNAS 2007]. The existence of this phase transition appears to be intimately related to the difficulty of proving precise results on, e.g., the k-colorability threshold as well as to the performance of message passing algorithms. In random graph k-coloring, there is a precise conjecture as to the location of the condensation phase transition in terms of a distributional fixed point problem. In this paper we prove this conjecture for k exceeding a certain constant k 0 . Mathematics Subject Classification: 05C80 (primary), 05C15 (secondary) 9
Estimating the leading principal components of data, assuming they are sparse, is a central task in modern high-dimensional statistics. Many algorithms were developed for this sparse PCA problem, from simple diagonal thresholding to sophisticated semidefinite programming (SDP) methods. A key theoretical question is under what conditions can such algorithms recover the sparse principal components? We study this question for a single-spike model with an ℓ0sparse eigenvector, in the asymptotic regime as dimension p and sample size n both tend to infinity. Amini and Wainwright [Ann. Statist. 37 (2009) 2877-2921] proved that for sparsity levels k ≥ Ω(n/ log p), no algorithm, efficient or not, can reliably recover the sparse eigenvector. In contrast, for k ≤ O( n/ log p), diagonal thresholding is consistent. It was further conjectured that an SDP approach may close this gap between computational and information limits. We prove that when k ≥ Ω( √ n), the proposed SDP approach, at least in its standard usage, cannot recover the sparse spike. In fact, we conjecture that in the single-spike model, no computationally-efficient algorithm can recover a spike of ℓ0-sparsity k ≥ Ω( √ n). Finally, we present empirical results suggesting that up to sparsity levels k = O( √ n), recovery is possible by a simple covariance thresholding algorithm.
Abstract. The problem of (approximately) counting the number of triangles in a graph is one of the basic problems in graph theory. In this paper we study the problem in the streaming model. We study the amount of memory required by a randomized algorithm to solve this problem. In case the algorithm is allowed one pass over the stream, we present a best possible lower bound of Ω(m) for graphs G with m edges on n vertices. If a constant number of passes is allowed, we show a lower bound of Ω(m/T ), T the number of triangles. We match, in some sense, this lower bound with a 2-pass O(m/T 1/3 )-memory algorithm that solves the problem of distinguishing graphs with no triangles from graphs with at least T triangles. We present a new graph parameter ρ(G) -the triangle density, and conjecture that the space complexity of the triangles problem is Ω(m/ρ(G)). We match this by a second algorithm that solves the distinguishing problem using O(m/ρ(G))-memory.
In this paper we analyze the performance of Warning Propagation, a popular message passing algorithm. We show that for 3CNF formulas drawn from a certain distribution over random satisfiable 3CNF formulas, commonly referred to as the planted-assignment distribution, running Warning Propagation in the standard way (run message passing until convergence, simplify the formula according to the resulting assignment, and satisfy the remaining subformula, if necessary, using a simple "off the shelf" heuristic) results in a satisfying assignment when the clause-variable ratio is a sufficiently large constant.
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