15 research outputs found
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
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
Packing Hamilton Cycles Online
It is known that w.h.p. the hitting time for the random
graph process to have minimum degree coincides with the hitting time
for edge disjoint Hamilton cycles. In this paper we prove an online
version of this property. We show that, for a fixed integer , if
random edges of are presented one by one then w.h.p. it is possible to
color the edges online with colors so that at time ,
each color class is Hamiltonian.Comment: Minor change
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