182 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
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
Pebbling and Branching Programs Solving the Tree Evaluation Problem
We study restricted computation models related to the Tree Evaluation
Problem}. The TEP was introduced in earlier work as a simple candidate for the
(*very*) long term goal of separating L and LogDCFL. The input to the problem
is a rooted, balanced binary tree of height h, whose internal nodes are labeled
with binary functions on [k] = {1,...,k} (each given simply as a list of k^2
elements of [k]), and whose leaves are labeled with elements of [k]. Each node
obtains a value in [k] equal to its binary function applied to the values of
its children, and the output is the value of the root. The first restricted
computation model, called Fractional Pebbling, is a generalization of the
black/white pebbling game on graphs, and arises in a natural way from the
search for good upper bounds on the size of nondeterministic branching programs
(BPs) solving the TEP - for any fixed h, if the binary tree of height h has
fractional pebbling cost at most p, then there are nondeterministic BPs of size
O(k^p) solving the height h TEP. We prove a lower bound on the fractional
pebbling cost of d-ary trees that is tight to within an additive constant for
each fixed d. The second restricted computation model we study is a semantic
restriction on (non)deterministic BPs solving the TEP - Thrifty BPs.
Deterministic (resp. nondeterministic) thrifty BPs suffice to implement the
best known algorithms for the TEP, based on black (resp. fractional) pebbling.
In earlier work, for each fixed h a lower bound on the size of deterministic
thrifty BPs was proved that is tight for sufficiently large k. We give an
alternative proof that achieves the same bound for all k. We show the same
bound still holds in a less-restricted model, and also that gradually weaker
lower bounds can be obtained for gradually weaker restrictions on the model.Comment: Written as one of the requirements for my MSc. 29 pages, 6 figure
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