4,801 research outputs found
On the Relative Strength of Pebbling and Resolution
The last decade has seen a revival of interest in pebble games in the context
of proof complexity. Pebbling has proven a useful tool for studying
resolution-based proof systems when comparing the strength of different
subsystems, showing bounds on proof space, and establishing size-space
trade-offs. The typical approach has been to encode the pebble game played on a
graph as a CNF formula and then argue that proofs of this formula must inherit
(various aspects of) the pebbling properties of the underlying graph.
Unfortunately, the reductions used here are not tight. To simulate resolution
proofs by pebblings, the full strength of nondeterministic black-white pebbling
is needed, whereas resolution is only known to be able to simulate
deterministic black pebbling. To obtain strong results, one therefore needs to
find specific graph families which either have essentially the same properties
for black and black-white pebbling (not at all true in general) or which admit
simulations of black-white pebblings in resolution. This paper contributes to
both these approaches. First, we design a restricted form of black-white
pebbling that can be simulated in resolution and show that there are graph
families for which such restricted pebblings can be asymptotically better than
black pebblings. This proves that, perhaps somewhat unexpectedly, resolution
can strictly beat black-only pebbling, and in particular that the space lower
bounds on pebbling formulas in [Ben-Sasson and Nordstrom 2008] are tight.
Second, we present a versatile parametrized graph family with essentially the
same properties for black and black-white pebbling, which gives sharp
simultaneous trade-offs for black and black-white pebbling for various
parameter settings. Both of our contributions have been instrumental in
obtaining the time-space trade-off results for resolution-based proof systems
in [Ben-Sasson and Nordstrom 2009].Comment: Full-length version of paper to appear in Proceedings of the 25th
Annual IEEE Conference on Computational Complexity (CCC '10), June 201
Inapproximability of the Standard Pebble Game and Hard to Pebble Graphs
Pebble games are single-player games on DAGs involving placing and moving
pebbles on nodes of the graph according to a certain set of rules. The goal is
to pebble a set of target nodes using a minimum number of pebbles. In this
paper, we present a possibly simpler proof of the result in [CLNV15] and
strengthen the result to show that it is PSPACE-hard to determine the minimum
number of pebbles to an additive term for all , which improves upon the currently known additive constant hardness of
approximation [CLNV15] in the standard pebble game. We also introduce a family
of explicit, constant indegree graphs with nodes where there exists a graph
in the family such that using constant pebbles requires moves
to pebble in both the standard and black-white pebble games. This independently
answers an open question summarized in [Nor15] of whether a family of DAGs
exists that meets the upper bound of moves using constant pebbles
with a different construction than that presented in [AdRNV17].Comment: Preliminary version in WADS 201
Understanding Space in Proof Complexity: Separations and Trade-offs via Substitutions
For current state-of-the-art DPLL SAT-solvers the two main bottlenecks are
the amounts of time and memory used. In proof complexity, these resources
correspond to the length and space of resolution proofs. There has been a long
line of research investigating these proof complexity measures, but while
strong results have been established for length, our understanding of space and
how it relates to length has remained quite poor. In particular, the question
whether resolution proofs can be optimized for length and space simultaneously,
or whether there are trade-offs between these two measures, has remained
essentially open.
In this paper, we remedy this situation by proving a host of length-space
trade-off results for resolution. Our collection of trade-offs cover almost the
whole range of values for the space complexity of formulas, and most of the
trade-offs are superpolynomial or even exponential and essentially tight. Using
similar techniques, we show that these trade-offs in fact extend to the
exponentially stronger k-DNF resolution proof systems, which operate with
formulas in disjunctive normal form with terms of bounded arity k. We also
answer the open question whether the k-DNF resolution systems form a strict
hierarchy with respect to space in the affirmative.
Our key technical contribution is the following, somewhat surprising,
theorem: Any CNF formula F can be transformed by simple variable substitution
into a new formula F' such that if F has the right properties, F' can be proven
in essentially the same length as F, whereas on the other hand the minimal
number of lines one needs to keep in memory simultaneously in any proof of F'
is lower-bounded by the minimal number of variables needed simultaneously in
any proof of F. Applying this theorem to so-called pebbling formulas defined in
terms of pebble games on directed acyclic graphs, we obtain our results.Comment: This paper is a merged and updated version of the two ECCC technical
reports TR09-034 and TR09-047, and it hence subsumes these two report
Cluster counting: The Hoshen-Kopelman algorithm vs. spanning tree approaches
Two basic approaches to the cluster counting task in the percolation and
related models are discussed. The Hoshen-Kopelman multiple labeling technique
for cluster statistics is redescribed. Modifications for random and aperiodic
lattices are sketched as well as some parallelised versions of the algorithm
are mentioned. The graph-theoretical basis for the spanning tree approaches is
given by describing the "breadth-first search" and "depth-first search"
procedures. Examples are given for extracting the elastic and geometric
"backbone" of a percolation cluster. An implementation of the "pebble game"
algorithm using a depth-first search method is also described.Comment: LaTeX, uses ijmpc1.sty(included), 18 pages, 3 figures, submitted to
Intern. J. of Modern Physics
VLSI layouts and DNA physical mappings
We show that an important problem (-ICG) in computational biology is
equivalent to a colored version of a well-known graph layout problem (-CVS).Comment: 7 page
Pebbling Arguments for Tree Evaluation
The Tree Evaluation Problem was introduced by Cook et al. in 2010 as a
candidate for separating P from L and NL. The most general space lower bounds
known for the Tree Evaluation Problem require a semantic restriction on the
branching programs and use a connection to well-known pebble games to generate
a bottleneck argument. These bounds are met by corresponding upper bounds
generated by natural implementations of optimal pebbling algorithms. In this
paper we extend these ideas to a variety of restricted families of both
deterministic and non-deterministic branching programs, proving tight lower
bounds under these restricted models. We also survey and unify known lower
bounds in our "pebbling argument" framework
Understanding space in resolution: optimal lower bounds and exponential trade-offs
We continue the study of tradeoffs between space and length of
resolution proofs and focus on two new results:
begin{enumerate}
item
We show that length and space in resolution are uncorrelated. This
is proved by exhibiting families of CNF formulas of size that
have proofs of length but require space . Our
separation is the strongest possible since any proof of length
can always be transformed into a proof in space , and
improves previous work reported in [Nordstr"{o}m 2006, Nordstr"{o}m and
H{aa}stad 2008].
item We prove a number of trade-off results for space in the range
from constant to , most of them superpolynomial or even
exponential. This is a dramatic improvement over previous results in
[Ben-Sasson 2002, Hertel and Pitassi 2007, Nordstr"{o}m 2007].
end{enumerate}
The key to our results is the following, somewhat surprising, theorem:
Any CNF formula can be transformed by simple substitution
transformation into a new formula such that if has the right
properties, can be proven in resolution in essentially the same
length as but the minimal space needed for is lower-bounded
by the number of variables that have to be mentioned simultaneously in
any proof for . Applying this theorem to so-called pebbling
formulas defined in terms of pebble games over directed acyclic graphs
and analyzing black-white pebbling on these graphs yields our results
- …