20 research outputs found
Thresholds in Random Motif Graphs
We introduce a natural generalization of the Erd\H{o}s-R\'enyi random graph
model in which random instances of a fixed motif are added independently. The
binomial random motif graph is the random (multi)graph obtained by
adding an instance of a fixed graph on each of the copies of in the
complete graph on vertices, independently with probability . We
establish that every monotone property has a threshold in this model, and
determine the thresholds for connectivity, Hamiltonicity, the existence of a
perfect matching, and subgraph appearance. Moreover, in the first three cases
we give the analogous hitting time results; with high probability, the first
graph in the random motif graph process that has minimum degree one (or two) is
connected and contains a perfect matching (or Hamiltonian respectively).Comment: 19 page
On the trace of random walks on random graphs
We study graph-theoretic properties of the trace of a random walk on a random
graph. We show that for any there exists such that the
trace of the simple random walk of length on the
random graph for is, with high probability,
Hamiltonian and -connected. In the special case (i.e.
when ), we show a hitting time result according to which, with high
probability, exactly one step after the last vertex has been visited, the trace
becomes Hamiltonian, and one step after the last vertex has been visited for
the 'th time, the trace becomes -connected.Comment: 32 pages, revised versio
Karp's patching algorithm on random perturbations of dense digraphs
We consider the following question. We are given a dense digraph with
minimum in- and out-degree at least , where is a constant.
We then add random edges to to create a digraph . Here an edge
is placed independently into with probability where
is a small positive constant. The edges of are given edge
costs , where is an independent copy of the exponential
mean one random variable i.e. . Let
be the associated cost matrix where
if . We show that w.h.p. the patching
algorithm of Karp finds a tour for the asymmetric traveling salesperson problem
that is asymptotically equal to that of the associated assignment problem.
Karp's algorithm runs in polynomial time.Comment: Fixed the proof of a lemm
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
Greedy maximal independent sets via local limits
The random greedy algorithm for finding a maximal independent set in a graph
has been studied extensively in various settings in combinatorics, probability,
computer science, and even chemistry. The algorithm builds a maximal
independent set by inspecting the graph's vertices one at a time according to a
random order, adding the current vertex to the independent set if it is not
connected to any previously added vertex by an edge.
In this paper, we present a natural and general framework for calculating the
asymptotics of the proportion of the yielded independent set for sequences of
(possibly random) graphs, involving a notion of local convergence. We use this
framework both to give short and simple proofs for results on previously
studied families of graphs, such as paths and binomial random graphs, and to
study new ones, such as random trees and random planar graphs.
We conclude our work by analysing the random greedy algorithm more closely
when the base graph is a tree. We show that in expectation, the cardinality of
a random greedy independent set in the path is no larger than that in any other
tree of the same order.Comment: 26 pages. This is an extended and revised version of a conference
version presented at the 31st International Conference on Probabilistic,
Combinatorial and Asymptotic Methods for the Analysis of Algorithms
(AofA2020
Greedy Maximal Independent Sets via Local Limits
The random greedy algorithm for finding a maximal independent set in a graph has been studied extensively in various settings in combinatorics, probability, computer science - and even in chemistry. The algorithm builds a maximal independent set by inspecting the vertices of the graph one at a time according to a random order, adding the current vertex to the independent set if it is not connected to any previously added vertex by an edge.
In this paper we present a natural and general framework for calculating the asymptotics of the proportion of the yielded independent set for sequences of (possibly random) graphs, involving a useful notion of local convergence. We use this framework both to give short and simple proofs for results on previously studied families of graphs, such as paths and binomial random graphs, and to study new ones, such as random trees.
We conclude our work by analysing the random greedy algorithm more closely when the base graph is a tree. We show that in expectation, the cardinality of a random greedy independent set in the path is no larger than that in any other tree of the same order