5,532 research outputs found
Ramsey games with giants
The classical result in the theory of random graphs, proved by Erdos and
Renyi in 1960, concerns the threshold for the appearance of the giant component
in the random graph process. We consider a variant of this problem, with a
Ramsey flavor. Now, each random edge that arrives in the sequence of rounds
must be colored with one of R colors. The goal can be either to create a giant
component in every color class, or alternatively, to avoid it in every color.
One can analyze the offline or online setting for this problem. In this paper,
we consider all these variants and provide nontrivial upper and lower bounds;
in certain cases (like online avoidance) the obtained bounds are asymptotically
tight.Comment: 29 pages; minor revision
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
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
On the Power of Choice for k-Colorability of Random Graphs
In an r-choice Achlioptas process, random edges are generated r at a time, and an online strategy is used to select one of them for inclusion in a graph. We investigate the problem of whether such a selection strategy can shift the k-colorability transition; that is, the number of edges at which the graph goes from being k-colorable to non-k-colorable.
We show that, for k ? 9, two choices suffice to delay the k-colorability threshold, and that for every k ? 2, six choices suffice
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