16 research outputs found
Parameterized complexity of DPLL search procedures
We study the performance of DPLL algorithms on parameterized problems. In particular, we investigate how difficult it is to decide whether small solutions exist for satisfiability and other combinatorial problems. For this purpose we develop a Prover-Delayer game which models the running time of DPLL procedures and we establish an information-theoretic method to obtain lower bounds to the running time of parameterized DPLL procedures. We illustrate this technique by showing lower bounds to the parameterized pigeonhole principle and to the ordering principle. As our main application we study the DPLL procedure for the problem of deciding whether a graph has a small clique. We show that proving the absence of a k-clique requires n steps for a non-trivial distribution of graphs close to the critical threshold. For the restricted case of tree-like Parameterized Resolution, this result answers a question asked in [11] of understanding the Resolution complexity of this family of formulas
LIPIcs
We study space complexity and time-space trade-offs with a focus not on peak memory usage but on overall memory consumption throughout the computation. Such a cumulative space measure was introduced for the computational model of parallel black pebbling by [Alwen and Serbinenko ’15] as a tool for obtaining results in cryptography. We consider instead the non- deterministic black-white pebble game and prove optimal cumulative space lower bounds and trade-offs, where in order to minimize pebbling time the space has to remain large during a significant fraction of the pebbling. We also initiate the study of cumulative space in proof complexity, an area where other space complexity measures have been extensively studied during the last 10–15 years. Using and extending the connection between proof complexity and pebble games in [Ben-Sasson and Nordström ’08, ’11] we obtain several strong cumulative space results for (even parallel versions of) the resolution proof system, and outline some possible future directions of study of this, in our opinion, natural and interesting space measure
Proof complexity of resolution-based QBF calculi
Proof systems for quantified Boolean formulas (QBFs) provide a theoretical underpinning for the performance of important QBF solvers. However, the proof complexity of these proof systems is currently not well understood and in particular lower bound techniques are missing. In this paper we exhibit a new and elegant proof technique for showing lower bounds in QBF proof systems based on strategy extraction. This technique provides a direct transfer of circuit lower bounds to lengths of proofs lower bounds. We use our method to show the hardness of a natural class of parity formulas for Q-resolution and universal Q-resolution. Variants of the formulas are hard for even stronger systems as long-distance Q-resolution and extensions. With a completely different lower bound argument we show the hardness of the prominent formulas of Kleine Büning et al. [34] for the strong expansion-based calculus IR-calc. Our lower bounds imply new exponential separations between two different types of resolution-based QBF calculi: proof systems for CDCLbased solvers (Q-resolution, long-distance Q-resolution) and proof systems for expansion-based solvers (∀Exp+Res and its generalizations IR-calc and IRM-calc). The relations between proof systems from the two different classes were not known before
Structure and Problem Hardness: Goal Asymmetry and DPLL Proofs in<br> SAT-Based Planning
In Verification and in (optimal) AI Planning, a successful method is to
formulate the application as boolean satisfiability (SAT), and solve it with
state-of-the-art DPLL-based procedures. There is a lack of understanding of why
this works so well. Focussing on the Planning context, we identify a form of
problem structure concerned with the symmetrical or asymmetrical nature of the
cost of achieving the individual planning goals. We quantify this sort of
structure with a simple numeric parameter called AsymRatio, ranging between 0
and 1. We run experiments in 10 benchmark domains from the International
Planning Competitions since 2000; we show that AsymRatio is a good indicator of
SAT solver performance in 8 of these domains. We then examine carefully crafted
synthetic planning domains that allow control of the amount of structure, and
that are clean enough for a rigorous analysis of the combinatorial search
space. The domains are parameterized by size, and by the amount of structure.
The CNFs we examine are unsatisfiable, encoding one planning step less than the
length of the optimal plan. We prove upper and lower bounds on the size of the
best possible DPLL refutations, under different settings of the amount of
structure, as a function of size. We also identify the best possible sets of
branching variables (backdoors). With minimum AsymRatio, we prove exponential
lower bounds, and identify minimal backdoors of size linear in the number of
variables. With maximum AsymRatio, we identify logarithmic DPLL refutations
(and backdoors), showing a doubly exponential gap between the two structural
extreme cases. The reasons for this behavior -- the proof arguments --
illuminate the prototypical patterns of structure causing the empirical
behavior observed in the competition benchmarks
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
Towards Understanding and Harnessing the Potential of Clause Learning
Efficient implementations of DPLL with the addition of clause learning are
the fastest complete Boolean satisfiability solvers and can handle many
significant real-world problems, such as verification, planning and design.
Despite its importance, little is known of the ultimate strengths and
limitations of the technique. This paper presents the first precise
characterization of clause learning as a proof system (CL), and begins the task
of understanding its power by relating it to the well-studied resolution proof
system. In particular, we show that with a new learning scheme, CL can provide
exponentially shorter proofs than many proper refinements of general resolution
(RES) satisfying a natural property. These include regular and Davis-Putnam
resolution, which are already known to be much stronger than ordinary DPLL. We
also show that a slight variant of CL with unlimited restarts is as powerful as
RES itself. Translating these analytical results to practice, however, presents
a challenge because of the nondeterministic nature of clause learning
algorithms. We propose a novel way of exploiting the underlying problem
structure, in the form of a high level problem description such as a graph or
PDDL specification, to guide clause learning algorithms toward faster
solutions. We show that this leads to exponential speed-ups on grid and
randomized pebbling problems, as well as substantial improvements on certain
ordering formulas
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