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Variations of a Pebble Game on Graphs
Abstract
Two variations are examined of a one-person pebble game played on directed graphs, which has been studied as a model of register allocation. The black-white pebble game of Cook and Sethi is shown to require as many pebbles in the worst case as the normal pebble game, to within a constant factor. For another version of the pebble game, the problem of deciding whether a given number of pebbles is sufficient for a given graph is shown to be complete in polynomial space.
... Instead, what we do is to prove lower bounds directly for the blob-pebble game. This is not immediately clear how to do, since the lower bound proofs for black-white pebbling price in, for instance, [24,31,37,39] all break down for the more general blob-pebble game. We are currently able to obtain lower bounds only for the limited class of layered spreading graphs (to be defined below), a class that includes binary trees and pyramid graphs. ...
... An obvious question is whether this lower bound on clause space in terms of black-white pebbling price is true for arbitrary DAGs. In particular, does it hold for the family of DAGs {G n } ∞ n=1 in [31] of size O(n) that have maximal black-white pebbling price BW-Peb(G n ) = Ω(n/ log n) in terms of size? If it could be proven for pebbling contradictions over such graphs that pebbling price bounds clause space from below, this would immediately imply that there are k-CNF formulas refutable in small length that can be maximally complex with respect to clause space. ...
... We are currently working on this problem, but note that these DAGs in [31] seem to have much more challenging structural properties that makes it hard to lift the lower bound argument from standard black-white pebblings to blob-pebblings. ...
Most state-of-the-art satisfiability algorithms today are variants of the DPLL procedure augmented with clause learning. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field of proof complexity, the resources of time and memory correspond to the length and space of resolution proofs. There has been a long line of research trying to understand these proof complexity measures, as well as relating them to the width of proofs, i.e., the size of the largest clause in the proof, which has been shown to be intimately connected with both length and space. While strong results have been proven for length and width, our understanding of space is still quite poor. For instance, it has remained open whether the fact that a formula is provable in short length implies that it is also provable in small space (which is the case for length versus width), or whether on the contrary these measures are completely unrelated in the sense that short proofs can be arbitrarily complex with respect to space.
In this paper, we present some evidence that the true answer should be that the latter case holds and provide a possible roadmap for how such an optimal separation result could be obtained. We do this by proving a tight bound of Theta(√(n)) on the space needed for so-called pebbling contradictions over pyramid graphs of size n.
Also, continuing the line of research initiated by (Ben-Sasson 2002) into trade-offs between different proof complexity measures, we present a simplified proof of the recent length-space trade-off result in (Hertel and Pitassi 2007), and show how our ideas can be used to prove a couple of other exponential trade-offs in resolution.
... Much previous research has focused on proving lower and upper bounds on the pebbling space cost (i.e. the maximum number of pebbles used at any point in time) of pebbling a given DAG under the rules of each of these games. For all of the aforementioned pebble games (except the red-blue pebble game since it relies on a different set of parameters), any DAG can be pebbled using O(n/ log n) pebbles [GT78,HPV77,PTC76]. Furthermore, there exist DAGs for each of the games that require Ω(n/ log n) pebbles [GT78,HPV77,PTC76]. ...
... For all of the aforementioned pebble games (except the red-blue pebble game since it relies on a different set of parameters), any DAG can be pebbled using O(n/ log n) pebbles [GT78,HPV77,PTC76]. Furthermore, there exist DAGs for each of the games that require Ω(n/ log n) pebbles [GT78,HPV77,PTC76]. ...
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 n nodes where there exists a graph in the family such that using constant k 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 k pebbles with a different construction than that presented in [AdRNV17].
... Much previous research has focused on proving lower and upper bounds on the pebbling space cost (i.e. the maximum number of pebbles used at any point in time) of pebbling a given DAG under the rules of each of these games. For all of the aforementioned pebble games (except the red-blue pebble game since it relies on a different set of parameters), any DAG can be pebbled using O(n/ log n) pebbles [GT78,HPV77,PTC76]. Furthermore, there exist DAGs for each of the games that require Ω(n/ log n) pebbles [GT78,HPV77,PTC76]. ...
... For all of the aforementioned pebble games (except the red-blue pebble game since it relies on a different set of parameters), any DAG can be pebbled using O(n/ log n) pebbles [GT78,HPV77,PTC76]. Furthermore, there exist DAGs for each of the games that require Ω(n/ log n) pebbles [GT78,HPV77,PTC76]. ...
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 [4] 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 [4] in the standard pebble game. We also introduce a family of explicit, constant indegree graphs with n nodes where there exists a graph in the family such that using pebbles requires moves to pebble in both the standard and black-white pebble games. This independently answers an open question summarized in [14] of whether a family of DAGs exists that meets the upper bound of moves using constant k pebbles with a different construction than that presented in [1].
... If we take a constant number of variables per vertex and study DAGs with constant fan-in, it is easy to show that XOR-pebbling contradictions can be refuted in linear length and constant width. Using the result from [GT78] which exhibits a family of fan-in 2 DAGs {G n } ∞ n=1 of size O(n) having pebbling price Ω(n/ log n), we get the following corollary. ...
... This was not known at the time of the original theorem in[GT78]. What is needed is an explicit construction of superconcentrators of linear density, and it has since been shown in[GG81] how to do this with[AC03] presenting the currently best construction. ...
... Before presenting the proof, let us demonstrate how this theorem implies part 4 of Theorem 3.1. The following theorem, due to [GT78] shows that there exist explicit constructions of graphs with large blackwhite pebbling measure. ...
... Theorem 3.10. [GT78] For arbitrarily large n, there exist circuits G n of size n such that ...
We investigate tradeoffs of various basic complexity measures such as size, space, and width. We show examples of formulas that have optimal proofs with respect to any one of these parameters, but optimizing one parameter must cost an increase in the other. These results have implications to the efficiency (or rather, inefficiency) of some commonly used SAT solving heuristics. Our proof relies on a novel connection of the variable space of a proof to the black-white pebbling measure of an underlying graph.
... Dymond and Tompa [10] showed that the reversible pebbling number of any graph with n vertices and in-degree 2 is O(n/ log n). Gilbert and Tarjan [11] constructed matching graphs (see also Nordström's excellent survey [21]). Theorem 2.8. ...
An approximate computation of a Boolean function by a circuit or switching network is a computation in which the function is computed correctly on the majority of the inputs (rather than on all inputs). Besides being interesting in their own right, lower bounds for approximate computation have proved useful in many sub areas of complexity theory, such as cryptography and derandomization. Lower bounds for approximate computation are also known as correlation bounds or average case hardness. In this paper, we obtain the first average case monotone depth lower bounds for a function in monotone P. We tolerate errors that are asymptotically the best possible for monotone circuits. Specifically, we prove average case exponential lower bounds on the size of monotone switching networks for the GEN function. As a corollary, we separate the monotone NC hierarchy in the case of errors -- a result which was previously only known for exact computations. Our proof extends and simplifies the Fourier analytic technique due to Potechin, and further developed by Chan and Potechin. As a corollary of our main lower bound, we prove that the communication complexity approach for monotone depth lower bounds does not naturally generalize to the average case setting.
... We will mainly be interested in the the more general black-white pebble game modelling nondeterministic computation. Black-white pebbling was introduced in [18] and has been studied in [23], [26], [27], [29]–[31], [46] and other papers. Let us refer to vertices of a directed graph having indegree 0 as sources and vertices having outdegree 0 as sinks. ...
The last decade has seen a revival of interest in pebble games in the context of proof complexity. Pebbling has proven to be 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 article 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 Nordström [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 Nordström [2011].
... If we could prove Conjecture 3, we would immediately get a positive answer to Conjecture 2 as well. For it was shown in [32] 8 that there are DAGs G n of fanin 2 and size O(n) that have black-white pebbling price BW-Peb(G n ) = Θ(n/ log n). Thus, assuming Conjecture 3 and plugging in the pebbling contradictions defined over these DAGs G n , we would get a k-CNF formulafamily {F n } ∞ n=1 of size O(n) with L(F n 0) = O(n), W(F n 0) = O(1), and Sp (F n 0) = Ω(n/ log n). ...
The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of clauses kept in memory simultaneously if the proof is only allowed to infer new clauses from clauses currently in memory. Both of these measures have previously been studied and related to the resolution refutation size of unsatisfiable conjunctive normal form (CNF) formulas. Also, the minimum refutation space of a formula has been proven to be at least as large as the minimum refutation width, but it has been open whether space can be separated from width or the two measures coincide asymptotically. We prove that there is a family of k-CNF formulas for which the refutation width in resolution is constant but the refutation space is nonconstant, thus solving a problem mentioned in several previous papers.
... Here we see some results that are interesting for the comparison of both games: (2.1) For both games, it is known that if G is a DAG with indegree 2 and n vertices, then an optimal strategy from (0, 0) to ({r}, 0) for some r ~ V uses at most O(n/log n) pebbles [1], and there exists a family of graphs which needs this number of pebbles [2, 5]. (2.2) If S,, is a pyramid with m levels and root r ($5 is shown inFig. ...
The number of pebbles used in the black [black-white] pebble game corresponds to the storage requirement of the deterministic [non-deterministic] evaluation of a straight line program. Suppose a distinguished vertex of a directed acyclic graph can be pebbled with k pebbles in the black-white pebble game. Then it can be pebbled with k′≤1/2k(k−1)+1 pebbles in the black pebble game.
... Since there are DAGs Gn of fan-in 2 and size O(n) which have black-white pebbling price BW-Peb(Gn) = Θ ` n/ log n ´ (see [22]), 3 a proof of Conjecture 2 would immediately yield the corollary that there is a family of unsatisfiable k-CNF formulas ˘ Fn ¯ ∞ n=1 of size O(n) such that W ` ...
The width of a resolution proof is the maximal number of literals in any clause of the proof. The space of a proof is the maximal number of clauses kept in memory simultaneously if the proof is only allowed to infer new clauses from clauses currently in memory. Both of these measures have previously been studied and related to the resolution refutation size of unsatisfiable CNF formulas. Also, the refutation space of a formula has been proven to be at least as large as the refutation width, but it has been open whether space can be separated from width or the two measures coincide asymptotically. We prove that there is a family of k-CNF formulas for which the refutation width in resolution is constant but the refutation space is non-constant, thus solving a problem mentioned in several previous papers.
We give a significantly simplified proof of the exponential separation between regular and general resolution of Alekhnovich et al. (2007) as a consequence of a general theorem lifting proof depth to regular proof length in resolution. This simpler proof then allows us to strengthen the separation further, and to construct families of theoretically very easy benchmarks that are surprisingly hard for SAT solvers in practice.
Boolean satisfiability (SAT) solvers have improved enormously in performance over the last 10---15 years and are today an indispensable tool for solving a wide range of computational problems. However, our understanding of what makes SAT instances hard or easy in practice is still quite limited. A recent line of research in proof complexity has studied theoretical complexity measures such as length, width, and space in resolution, which is a proof system closely related to state-of-the-art conflict-driven clause learning (CDCL) SAT solvers. Although it seems like a natural question whether these complexity measures could be relevant for understanding the practical hardness of SAT instances, to date there has been very limited research on such possible connections. This paper sets out on a systematic study of the interconnections between theoretical complexity and practical SAT solver performance. Our main focus is on space complexity in resolution, and we report results from extensive experiments aimed at understanding to what extent this measure is correlated with hardness in practice. Our conclusion from the empirical data is that the resolution space complexity of a formula would seem to be a more fine-grained indicator of whether the formula is hard or easy than the length or width needed in a resolution proof. On the theory side, we prove a separation of general and tree-like resolution space, where the latter has been proposed before as a measure of practical hardness, and also show connections between resolution space and backdoor sets.
Pebble games were extensively studied in the 1970s and 1980s in a number of
different contexts. The last decade has seen a revival of interest in pebble
games coming from the field of proof complexity. Pebbling has proven to be 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. This is a survey of research in proof
complexity drawing on results and tools from pebbling, with a focus on proof
space lower bounds and trade-offs between proof size and proof space.
Most state-of-the-art satisfiability algorithms today are vari- ants of the DPLL procedure augmented with clause learn- ing. The main bottleneck for such algorithms, other than the obvious one of time, is the amount of memory used. In the field of proof complexity, the resources of time and mem- ory correspond to the length and space of resolution proofs. There has been a long line of research trying to understand these proof complexity measures, as well as relating them to the width of proofs, i.e., the size of the largest clause in the proof, which has been shown to be intimately connected with both length and space. While strong results have been proven for length and width, our understanding of space is still quite poor. For instance, it has remained open whether the fact that a formula is provable in short length implies that it is also provable in small space (which is the case for length versus width), or whether on the contrary these mea- sures are completely unrelated in the sense that short proofs can be arbitrarily complex with respect to space. In this paper, we present some evidence that the true answer should be that the latter case holds and provide a possible roadmap for how such an optimal separation result could be obtained. We do this by proving a tight bound of ( p n) on the space needed for so-called pebbling contra-
The last decade has seen a revival of interest in pebble games in the context of proof complexity. Pebbling has proven to be 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].
We study the relation between the number of pebbles used in the black and the black-white pebble game and show that the additional use of white pebbles cannot save more than a square-root and give an example in which it does save a factor 1/2.
The pebble game on directed acyclic graphs is commonly encountered as an abstract model for register allocation problems. The traditional move rule of the game asserts that one may "put a pebble on node x once all its immediate predecessors have a pebble", leaving it open whether the pebble to be placed on x should be taken from some predecessor of x or from the free pool (the strict interpretation). We show that allowing pebbles to slide along an edge as a legal move enables one to save precisely one pebble over the strict interpretation. However, in the worst case the saving may be obtained only at the cost of squaring the time needed to pebble the dag. It shows that one has to be very careful in describing properties of pebblings; the interpretation of the rules can seriously affect the results. As a main result we prove a linear to exponential time trade-off for any fixed interpretation of the rules when a single pebble is saved. There exist families of dags with indegrees 2, with the property that they can be pebbled in linear time when one more pebble than the minimum needed is available but which require exponential time when the extra pebble is dropped.
The complexity of the black-white pebbling game has remained an open problem for 30 years. In this paper we show that the black-white pebbling game is PSPACE-complete.
A number of works have looked at the relationship between length and space of resolution proofs. A notorious question has been whether the existence of a short proof implies the existence of a proof that can be verified using limited space.In this paper we resolve the question by answering it negatively in the strongest possible way. We show that there are families of 6-CNF formulas of size n, for arbitrarily large n, that have resolution proofs of length O(n) but for which any proof requires space Omega(n/log n). This is the strongest asymptotic separation possible since any proof of length O(n) can always be transformed into a proof in space O(n / log n).Our result follows by reducing the space complexity of so called pebbling formulas over a directed acyclic graph to the black-white pebbling price of the graph.The proof is somewhat simpler than previous results (in particular, those reported in [Nordstrom 2006, Nordstrom and Hastad 2008]) as it uses a slightly different flavor of pebbling formulas which allows for a rather straightforward reduction of proof space to standard black-white pebbling price.
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 O(n) that have proofs of length O(n) but require space . Our separation is the strongest possible since any proof of length O(n) 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 F can be transformed by simple substitution transformation into a new formula such that if F has the right properties, can be proven in resolution in essentially the same length as F but the minimal space needed for is lower-bounded by the number of variables that have to be mentioned simultaneously in any proof for F. 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. @InProceedings{bensasson_et_al:DSP:2008:1781, author = {Eli Ben-Sasson and Jakob Nordstr{"o}m}, title = {Understanding space in resolution: optimal lower bounds and exponential trade-offs}, booktitle = {Computational Complexity of Discrete Problems }, year = {2008}, editor = {Peter Bro Miltersen and R{"u}diger Reischuk and Georg Schnitger and Dieter van Melkebeek}, number = {08381}, series = {Dagstuhl Seminar Proceedings}, ISSN = {1862-4405}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, Germany}, address = {Dagstuhl, Germany}, URL = {http://drops.dagstuhl.de/opus/volltexte/2008/1781}, annote = {Keywords: Proof complexity, Resolution, Pebbling.} }
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