Random k-SAT and the Power of Two Choices

We study an Achlioptas-process version of the random k-SAT process: a bounded number of k-clauses are drawn uniformly at random at each step, and exactly one added to the growing formula according to a particular rule. We prove the existence of a rule that shifts the satisfiability threshold. This extends a well-studied area of probabilistic combinatorics (Achlioptas processes) to random CSP's. In particular, while a rule to delay the 2-SAT threshold was known previously, this is the first proof of a rule to shift the threshold of k-SAT for k >= 3. We then propose a gap decision problem based upon this semi-random model. The aim of the problem is to investigate the hardness of the random k-SAT decision problem, as opposed to the problem of finding an assignment or certificate of unsatisfiability. Finally, we discuss connections to the study of Achlioptas random graph processes.Comment: 13 page

Explosive Percolation: Unusual Transitions of a Simple Model

In this paper we review the recent advances on explosive percolation, a very sharp phase transition first observed by Achlioptas et al. (Science, 2009). There a simple model was proposed, which changed slightly the classical percolation process so that the emergence of the spanning cluster is delayed. This slight modification turns out to have a great impact on the percolation phase transition. The resulting transition is so sharp that it was termed explosive, and it was at first considered to be discontinuous. This surprising fact stimulated considerable interest in "Achlioptas processes". Later work, however, showed that the transition is continuous (at least for Achlioptas processes on Erdos networks), but with very unusual finite size scaling. We present a review of the field, indicate open "problems" and propose directions for future research.Comment: 27 pages, 4 figures, Review pape

Hamiltonicity thresholds in Achlioptas processes

In this paper we analyze the appearance of a Hamilton cycle in the following random process. The process starts with an empty graph on n labeled vertices. At each round we are presented with K=K(n) edges, chosen uniformly at random from the missing ones, and are asked to add one of them to the current graph. The goal is to create a Hamilton cycle as soon as possible. We show that this problem has three regimes, depending on the value of K. For K=o(\log n), the threshold for Hamiltonicity is (1+o(1))n\log n /(2K), i.e., typically we can construct a Hamilton cycle K times faster that in the usual random graph process. When K=\omega(\log n) we can essentially waste almost no edges, and create a Hamilton cycle in n+o(n) rounds with high probability. Finally, in the intermediate regime where K=\Theta(\log n), the threshold has order n and we obtain upper and lower bounds that differ by a multiplicative factor of 3.Comment: 23 page

Avoiding small subgraphs in Achlioptas processes

For a fixed integer r, consider the following random process. At each round, one is presented with r random edges from the edge set of the complete graph on n vertices, and is asked to choose one of them. The selected edges are collected into a graph, which thus grows at the rate of one edge per round. This is a natural generalization of what is known in the literature as an Achlioptas process (the original version has r=2), which has been studied by many researchers, mainly in the context of delaying or accelerating the appearance of the giant component. In this paper, we investigate the small subgraph problem for Achlioptas processes. That is, given a fixed graph H, we study whether there is an online algorithm that substantially delays or accelerates a typical appearance of H, compared to its threshold of appearance in the random graph G(n, M). It is easy to see that one cannot accelerate the appearance of any fixed graph by more than the constant factor r, so we concentrate on the task of avoiding H. We determine thresholds for the avoidance of all cycles C_t, cliques K_t, and complete bipartite graphs K_{t,t}, in every Achlioptas process with parameter r >= 2.Comment: 43 pages; reorganized and shortene