85 research outputs found
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
Fast construction on a restricted budget
We introduce a model of a controlled random graph process. In this model, the
edges of the complete graph are ordered randomly and then revealed, one
by one, to a player called Builder. He must decide, immediately and
irrevocably, whether to purchase each observed edge. The observation time is
bounded by parameter , and the total budget of purchased edges is bounded by
parameter . Builder's goal is to devise a strategy that, with high
probability, allows him to construct a graph of purchased edges possessing a
target graph property , all within the limitations of observation
time and total budget. We show the following: (a) Builder has a strategy to
achieve minimum degree at the hitting time for this property by purchasing
at most edges for an explicit ; and a strategy to achieve it
(slightly) after the threshold for minimum degree by purchasing at most
edges (which is optimal); (b) Builder has a strategy to
create a Hamilton cycle if either and , or and , for some
; similar results hold for perfect matching; (c) Builder has
a strategy to create a copy of a given -vertex tree if , and this is optimal; and (d) For or
, Builder has a strategy to create a copy of a cycle of length
if , and this is optimal.Comment: 20 pages, 2 figure
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
Recommended from our members
Combinatorics and Probability
The main theme of this workshop was the use of probabilistic methods in combinatorics and theoretical computer science. Although these methods have been around for decades, they are being refined all the time: they are getting more and more sophisticated and powerful. Another theme was the study of random combinatorial structures, either for their own sake, or to tackle extremal questions. Both themes were richly represented at the workshop, with many recent exciting results presented by the lecturers
Connectivity Thresholds for Bounded Size Rules
In an Achlioptas process, starting with a graph that has n vertices and no
edge, in each round edges are drawn uniformly at random, and using
some rule exactly one of them is chosen and added to the evolving graph. For
the class of Achlioptas processes we investigate how much impact the rule has
on one of the most basic properties of a graph: connectivity. Our main results
are twofold. First, we study the prominent class of bounded size rules, which
select the edge to add according to the component sizes of its vertices,
treating all sizes larger than some constant equally. For such rules we provide
a fine analysis that exposes the limiting distribution of the number of rounds
until the graph gets connected, and we give a detailed picture of the dynamics
of the formation of the single component from smaller components. Second, our
results allow us to study the connectivity transition of all Achlioptas
processes, in the sense that we identify a process that accelerates it as much
as possible
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
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
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