129 research outputs found
A linear optimization technique for graph pebbling
Graph pebbling is a network model for studying whether or not a given supply
of discrete pebbles can satisfy a given demand via pebbling moves. A pebbling
move across an edge of a graph takes two pebbles from one endpoint and places
one pebble at the other endpoint; the other pebble is lost in transit as a
toll. It has been shown that deciding whether a supply can meet a demand on a
graph is NP-complete. The pebbling number of a graph is the smallest t such
that every supply of t pebbles can satisfy every demand of one pebble. Deciding
if the pebbling number is at most k is \Pi_2^P-complete. In this paper we
develop a tool, called the Weight Function Lemma, for computing upper bounds
and sometimes exact values for pebbling numbers with the assistance of linear
optimization. With this tool we are able to calculate the pebbling numbers of
much larger graphs than in previous algorithms, and much more quickly as well.
We also obtain results for many families of graphs, in many cases by hand, with
much simpler and remarkably shorter proofs than given in previously existing
arguments (certificates typically of size at most the number of vertices times
the maximum degree), especially for highly symmetric graphs. Here we apply the
Weight Function Lemma to several specific graphs, including the Petersen,
Lemke, 4th weak Bruhat, Lemke squared, and two random graphs, as well as to a
number of infinite families of graphs, such as trees, cycles, graph powers of
cycles, cubes, and some generalized Petersen and Coxeter graphs. This partly
answers a question of Pachter, et al., by computing the pebbling exponent of
cycles to within an asymptotically small range. It is conceivable that this
method yields an approximation algorithm for graph pebbling
Modified Linear Programming and Class 0 Bounds for Graph Pebbling
Given a configuration of pebbles on the vertices of a connected graph , a
\emph{pebbling move} removes two pebbles from some vertex and places one pebble
on an adjacent vertex. The \emph{pebbling number} of a graph is the
smallest integer such that for each vertex and each configuration of
pebbles on there is a sequence of pebbling moves that places at least
one pebble on .
First, we improve on results of Hurlbert, who introduced a linear
optimization technique for graph pebbling. In particular, we use a different
set of weight functions, based on graphs more general than trees. We apply this
new idea to some graphs from Hurlbert's paper to give improved bounds on their
pebbling numbers.
Second, we investigate the structure of Class 0 graphs with few edges. We
show that every -vertex Class 0 graph has at least
edges. This disproves a conjecture of Blasiak et al. For diameter 2 graphs, we
strengthen this lower bound to , which is best possible. Further, we
characterize the graphs where the bound holds with equality and extend the
argument to obtain an identical bound for diameter 2 graphs with no cut-vertex.Comment: 19 pages, 8 figure
Constructions for the optimal pebbling of grids
In [C. Xue, C. Yerger: Optimal Pebbling on Grids, Graphs and Combinatorics]
the authors conjecture that if every vertex of an infinite square grid is
reachable from a pebble distribution, then the covering ratio of this
distribution is at most . First we present such a distribution with
covering ratio , disproving the conjecture. The authors in the above paper
also claim to prove that the covering ratio of any pebble distribution is at
most . The proof contains some errors. We present a few interesting
pebble distributions that this proof does not seem to cover and highlight some
other difficulties of this topic
Critical Pebbling Numbers of Graphs
We define three new pebbling parameters of a connected graph , the -,
-, and -critical pebbling numbers. Together with the pebbling number, the
optimal pebbling number, the number of vertices and the diameter of the
graph, this yields 7 graph parameters. We determine the relationships between
these parameters. We investigate properties of the -critical pebbling
number, and distinguish between greedy graphs, thrifty graphs, and graphs for
which the -critical pebbling number is .Comment: 26 page
Approximating Cumulative Pebbling Cost Is Unique Games Hard
The cumulative pebbling complexity of a directed acyclic graph is defined
as , where the minimum is taken over all
legal (parallel) black pebblings of and denotes the number of
pebbles on the graph during round . Intuitively, captures
the amortized Space-Time complexity of pebbling copies of in parallel.
The cumulative pebbling complexity of a graph is of particular interest in
the field of cryptography as is tightly related to the
amortized Area-Time complexity of the Data-Independent Memory-Hard Function
(iMHF) [AS15] defined using a constant indegree directed acyclic
graph (DAG) and a random oracle . A secure iMHF should have
amortized Space-Time complexity as high as possible, e.g., to deter brute-force
password attacker who wants to find such that . Thus, to
analyze the (in)security of a candidate iMHF , it is crucial to
estimate the value but currently, upper and lower bounds for
leading iMHF candidates differ by several orders of magnitude. Blocki and Zhou
recently showed that it is -Hard to compute , but
their techniques do not even rule out an efficient
-approximation algorithm for any constant . We
show that for any constant , it is Unique Games hard to approximate
to within a factor of .
(See the paper for the full abstract.)Comment: 28 pages, updated figures and corrected typo
The Optimal Rubbling Number of Ladders, Prisms and M\"obius-ladders
A pebbling move on a graph removes two pebbles at a vertex and adds one
pebble at an adjacent vertex. Rubbling is a version of pebbling where an
additional move is allowed. In this new move, one pebble each is removed at
vertices and adjacent to a vertex , and an extra pebble is added at
vertex . A vertex is reachable from a pebble distribution if it is possible
to move a pebble to that vertex using rubbling moves. The optimal rubbling
number is the smallest number needed to guarantee a pebble distribution of
pebbles from which any vertex is reachable. We determine the optimal
rubbling number of ladders (), prisms () and
M\"oblus-ladders
Pebbling in Semi-2-Trees
Graph pebbling is a network model for transporting discrete resources that
are consumed in transit. Deciding whether a given configuration on a particular
graph can reach a specified target is -complete, even for diameter
two graphs, and deciding whether the pebbling number has a prescribed upper
bound is -complete. Recently we proved that the pebbling number
of a split graph can be computed in polynomial time. This paper advances the
program of finding other polynomial classes, moving away from the large tree
width, small diameter case (such as split graphs) to small tree width, large
diameter, continuing an investigation on the important subfamily of chordal
graphs called -trees. In particular, we provide a formula, that can be
calculated in polynomial time, for the pebbling number of any semi-2-tree,
falling shy of the result for the full class of 2-trees.Comment: Revised numerous arguments for clarity and added technical lemmas to
support proof of main theorem bette
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