199 research outputs found
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
Width and size of regular resolution proofs
This paper discusses the topic of the minimum width of a regular resolution
refutation of a set of clauses. The main result shows that there are examples
having small regular resolution refutations, for which any regular refutation
must contain a large clause. This forms a contrast with corresponding results
for general resolution refutations.Comment: The article was reformatted using the style file for Logical Methods
in Computer Scienc
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
Reversible Simulation of Irreversible Computation by Pebble Games
Reversible simulation of irreversible algorithms is analyzed in the stylized
form of a `reversible' pebble game. While such simulations incur little
overhead in additional computation time, they use a large amount of additional
memory space during the computation. The reacheable reversible simulation
instantaneous descriptions (pebble configurations) are characterized
completely. As a corollary we obtain the reversible simulation by Bennett and
that among all simulations that can be modelled by the pebble game, Bennett's
simulation is optimal in that it uses the least auxiliary space for the
greatest number of simulated steps. One can reduce the auxiliary storage
overhead incurred by the reversible simulation at the cost of allowing limited
erasing leading to an irreversibility-space tradeoff. We show that in this
resource-bounded setting the limited erasing needs to be performed at precise
instants during the simulation. We show that the reversible simulation can be
modified so that it is applicable also when the simulated computation time is
unknown.Comment: 11 pages, Latex, Submitted to Physica
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