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The PSPACE-completeness of black-white pebbling
Abstract
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.
... Inspired by previous works [GLT80,HP10,Cha13b] truth values are represented using the gadget in Figure 14. Proof. ...
... This game is a strict generalization of the standard (black) pebble game, and so intuitively it should be at least as hard, but the added option of placing nondeterministic white pebbles anywhere in the graph completely destroys locality and makes the reduction in [GLT80] break down. Hertel and Pitassi [HP10] showed a PSPACE-completeness result in the nonstandard setting when unbounded (and very large) fan-in is allowed. Essentially, the large fan-in makes it possible to lock down almost all pebbles in one place at a time (namely on the predecessors of a large fan-in vertex to be pebbled) and to completely rule out any use of white pebbles, reducing the whole problem to black pebbling (although this reduction, it should be stressed, is far from trivial). ...
We consider the pebble game on DAGs with bounded fan-in introduced in [Paterson and Hewitt '70] and the reversible version of this game in [Bennett '89], and study the question of how hard it is to decide exactly or approximately the number of pebbles needed for a given DAG in these games. We prove that the problem of eciding whether s~pebbles suffice to reversibly pebble a DAG G is PSPACE-complete, as was previously shown for the standard pebble game in [Gilbert, Lengauer and Tarjan '80]. Via two different graph product constructions we then strengthen these results to establish that both standard and reversible pebbling space are PSPACE-hard to approximate to within any additive constant. To the best of our knowledge, these are the first hardness of approximation results for pebble games in an unrestricted setting (even for polynomial time). Also, since [Chan '13] proved that reversible pebbling is equivalent to the games in [Dymond and Tompa '85] and [Raz and McKenzie '99], our results apply to the Dymond--Tompa and Raz--McKenzie games as well, and from the same paper it follows that resolution depth is PSPACE-hard to determine up to any additive constant. We also obtain a multiplicative logarithmic separation between reversible and standard pebbling space. This improves on the additive logarithmic separation previously known and could plausibly be tight, although we are not able to prove this. We leave as an interesting open problem whether our additive hardness of approximation result could be strengthened to a multiplicative bound if the computational resources are decreased from polynomial space to the more common setting of polynomial time.
... Specifically, finding the minimum number of black pebbles needed to pebble a DAG in the standard pebble game is PSPACE-complete [GLT79] and finding the minimum number of black pebbles needed in the one-shot case is NP-complete [Set75]. In addition, finding the minimum number of pebbles in both the black-white and reversible pebble games have been recently shown to be both PSPACEcomplete [CLNV15,HP10]. But the result for the black-white pebble game is proven for unbounded indegree [HP10]. ...
... In addition, finding the minimum number of pebbles in both the black-white and reversible pebble games have been recently shown to be both PSPACEcomplete [CLNV15,HP10]. But the result for the black-white pebble game is proven for unbounded indegree [HP10]. A key open question in the field is whether hardness results can be obtained for constant indegree graphs for the black-white pebble game. ...
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].
... The computational complexity of several graph pebbling problems has been explored previously in various settings [GLT80,HP10]. However, we stress that the structure of a pebbling minimizing cumulative cost can be very different from the structure of a pebbling minimizing space-time cost or 2 Indeed, Alwen and Blocki [AB16b] subsequently introduced heuristics to improve their attack and demonstrated that their attacks were effective even for smaller (practical) values of n by simulating their attack against real Argon2i instances. ...
... The computational complexity of various graph pebbling has been explored previously in different settings [GLT80,HP10]. Gilbert et al. [GLT80] focused on space-complexity of the black-pebbling game. Here, the goal is to find a pebbling which minimizes the total number of pebbles on the graph at any point in time (intuitively this corresponds to minimizing the maximum space required during computation of the associated function). ...
We consider the computational complexity of finding a legal black pebbling of a DAG G=(V,E) with minimum cumulative cost. A black pebbling is a sequence of sets of nodes which must satisfy the following properties: (we start off with no pebbles on G), (every sink node was pebbled at some point) and (we can only place a new pebble on a node v if all of v's parents had a pebble during the last round). The cumulative cost of a pebbling is . The cumulative pebbling cost is an especially important security metric for data-independent memory hard functions, an important primitive for password hashing. Thus, an efficient (approximation) algorithm would be an invaluable tool for the cryptanalysis of password hash functions as it would provide an automated tool to establish tight bounds on the amortized space-time cost of computing the function. We show that such a tool is unlikely to exist. In particular, we prove the following results. (1) It is \texttt{NP}\mbox{-}\texttt{Hard} to find a pebbling minimizing cumulative cost. (2) The natural linear program relaxation for the problem has integrality gap , where n is the number of nodes in G. We conjecture that the problem is hard to approximate. (3) We show that a related problem, find the minimum size subset such that , is also \texttt{NP}\mbox{-}\texttt{Hard}. In fact, under the unique games conjecture there is no -approximation algorithm.
... He showed that this problem is PSPACEcomplete. Determining the irreversible black and black-white pebbling number are known to be PSPACE-complete on DAGs (See [5], [6]). If we restrict the irreversible black pebble game to be read-once (each node is pebbled only once), then the problem becomes NP-complete (See [11]). ...
The reversible pebble game is a combinatorial game played on rooted DAGs. This game was introduced by Bennett (1989) motivated by applications in designing space efficient reversible algorithms. Recently, Chan (2013) showed that the reversible pebble game number of any DAG is the same as its Dymond-Tompa pebble number and Raz-Mckenzie pebble number. We show, as our main result, that for any rooted directed tree T, its reversible pebble game number is always just one more than the edge rank coloring number of the underlying undirected tree U of T. It is known that given a DAG G as input, determining its reversible pebble game number is PSPACE-hard. Our result implies that the reversible pebble game number of trees can be computed in polynomial time. We also address the question of finding the number of steps required to optimally pebble various families of trees. It is known that trees can be pebbled in steps where n is the number of nodes in the tree. Using the equivalence between reversible pebble game and the Dymond-Tompa pebble game (Chan, 2013), we show that complete binary trees can be pebbled in steps, a substantial improvement over the naive upper bound of . It remains open whether complete binary trees can be pebbled in polynomial (in n) number of steps. Towards this end, we show that almost optimal (i.e., within a factor of for any constant ) pebblings of complete binary trees can be done in polynomial number of steps. We also show a time-space trade-off for reversible pebbling for families of bounded degree trees by a divide-and-conquer approach: for any constant , such families can be pebbled using pebbles in O(n) steps. This generalizes an analogous result of Kralovic (2001) for chains.
... Such length-space trade-offs have been established in restricted settings by the current authors in [11, 37] 2 but nothing 1 A proof system is said to be sequential if a proof π in the system is a sequence of lines π = {L 1 , . . . , Lτ } where each line is derived from previous lines by one of a finite set of allowed inference rules. 2 A related result, claimed in [29], has later been retracted by the authors in [30]. has been known for refutations of explicit formulas in general, unrestricted resolution. ...
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 reports
We consider the computational complexity of finding a legal black pebbling of a DAG G=(V,E) with minimum cumulative cost. A black pebbling is a sequence of sets of nodes which must satisfy the following properties: (we start off with no pebbles on G), (every sink node was pebbled at some point) and (we can only place a new pebble on a node v if all of v’s parents had a pebble during the last round). The cumulative cost of a pebbling is . The cumulative pebbling cost is an especially important security metric for data-independent memory hard functions, an important primitive for password hashing. Thus, an efficient (approximation) algorithm would be an invaluable tool for the cryptanalysis of password hash functions as it would provide an automated tool to establish tight bounds on the amortized space-time cost of computing the function. We show that such a tool is unlikely to exist in the most general case. In particular, we prove the following results.
It is to find a pebbling minimizing cumulative cost.
The natural linear program relaxation for the problem has integrality gap , where n is the number of nodes in G. We conjecture that the problem is hard to approximate.
We show that a related problem, find the minimum size subset such that , is also . In fact, under the Unique Games Conjecture there is no -approximation algorithm.
A surprising equivalence between different forms of pebble games on graphs - Dymond-Tompa pebble game (studied in [4]), Raz-McKenzie pebble game (studied in [10]) and reversible pebbling (studied in [1]) - was established recently by Chan[2]. Motivated by this equivalence, we study the reversible pebble game and establish the following results.
We give a polynomial time algorithm for computing reversible pebbling number of trees. As our main technical contribution, we show that the reversible pebbling number of any tree is exactly one more than the edge rank colouring of the underlying undirected tree.
By exploiting the connection with the Dymond-Tompa pebble game, we show that complete binary trees have optimal pebblings that take at most steps. This substantially improves the previous bound of steps.
Furthermore, we show that almost optimal (within factor for any constant ) pebblings of complete binary trees can be done in polynomial number of steps.
We also show a time-space tradeoff for reversible pebbling for families of bounded degree trees: for any constant , such families can be pebbled using pebbles in O(n) steps. This generalizes a result of Královic[7] who showed the same for chains.
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.
The equivalence problem for Kleene's regular expressions has several effective solutions, all of which are computationally inefficient. In [1], we showed that this inefficiency is an inherent property of the problem by showing that the problem of membership in any arbitrary context-sensitive language was easily reducible to the equivalence problem for regular expressions. We also showed that with a squaring abbreviation ( writing (E)2 for E×E) the equivalence problem for expressions required computing space exponential in the size of the expressions. In this paper we consider a number of similar decidable word problems from automata theory and logic whose inherent computational complexity can be precisely characterized in terms of time or space requirements on deterministic or nondeterministic Turing machines. The definitions of the word problems and a table summarizing their complexity appears in the next section. More detailed comments and an outline of some of the proofs follows in the remaining sections. Complete proofs will appear in the forthcoming papers [9, 10, 13]. In the final section we describe some open problems.
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.
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.
We construct a family (G
p
|p) of directed acyclic graphs such that any black pebble strategy forG
p
requiresp
2 pebbles whereas 3p+1 pebbles are sufficient when white pebbles are allowed.
We introduce a new way to measure the space needed in resolution refutations of CNF formulas in propositional logic. With the former definition (1994, B. H. Kleine and T. Lettman, “Aussangenlogik: Deduktion und Algorithmen, Teubner, Stuttgart) the space required for the resolution of any unsatisfiable formula in CNF is linear in the number of clauses. The new definition allows a much finer analysis of the space in the refutation, ranging from constant to linear space. Moreover, the new definition allows us to relate the space needed in a resolution proof of a formula to other well-studied complexity measures. It coincides with the complexity of a pebble game in the resolution graphs of a formula and, as we show, has relationships to the size of the refutation. We also give upper and lower bounds on the space needed for the resolution of unsatisfiable formulas. We show that Tseitin formulas associated to a certain kind of expander graphs of n nodes need resolution space n−c for some constant c. Measured on the number of clauses, this result is the best possible. We also show that the formulas expressing the general pigeonhole principle with n holes and more than n pigeons need space n+1 independent of the number of pigeons. Since a matching space upper bound of n+1 for these formulas exists, the obtained bound is exact. We also point to a possible connection between resolution space and resolution width, another measure for the complexity of resolution refutations.
It is proved that for infinitely many n there is a directed acyclic graph with vertex indegrees bounded by 2 that has a strategy of the black-white pebble game using n pebbles and for which any strategy of the black pebble game requires pebbles. This shows that there is a family of straight-line programs for which nondeterminism reduces the space required to evaluate the programs by more than any constant factor.
An intriguing question is whether (log n)2 space is enough to recognize the class of languages recognizable in deterministic polynomial time. This question has earlier been narrowed down to the storage required to recognize a particular language called SP. SP is clearly in and it has been shown that if SP has tape complexity (log n)k, then every member of has tape complexity (log n)k. This paper presents further evidence in support of the conjecture that SP cannot be recognized using storage (log n)k for any k. We have no techniques at present for proving such a statement for Turing machines in general; we prove the result for a suitably restricted device.
We examine a pebbling problem which has been used to study the storage requirements of various models of computation. Sethi has shown this problem to be NP-hard and Lingas has shown a generalization to be P-space complete. We prove the original problem P-space complete by employing a modification of Lingas's proof. The pebbling problem is one of the few examples of a P-space complete problem not exhibiting any obvious quantifier alternation.
The Prover/Delayer game is a combinatorial game that can be used to prove upper and lower bounds on the size of Tree Resolution proofs, and also perfectly characterizes the space needed to compute them. As a proof system, Tree Resolution forms the underpinnings of all DPLL-based SAT solvers, so it is of interest not only to proof complexity researchers, but also to those in the area of propositional reasoning. In this paper, we prove the PSPACE-Completeness of the Prover/Delayer game as well as the problem of predicting Tree Resolution space requirements, where space is the number of clauses that must be kept in memory simultaneously during the computation of a refuta- tion. Since in practice memory is often a limiting resource, researchers developing SAT solvers may wish to know ahead of time how much memory will be required for solving a certain formula, but the present result shows that predicting this is at least as hard as it would be to simply nd a refutation.
The pebble game on AND/OR dags which is natural generalization of a well known pebble game on dags is considered. The following problem is given: if k pebbles are enough to place a pebble on a given vertex of an AND/OR dag. It is shown that the problem is log-space complete for languages accepted in polynomial space.
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.
Please don't be shy about sending even vague pointers to people who may have complete or partial resolutions of the problems mentioned in any of the open questions columns that have appeared as earlier complexity theory columns. Though I don't give a ...
This is the sixth edition of a quarterly column the purpose of which is to provide continuing coverage of new developments in the theory of NP-completeness. The presentation is modeled on that used by M. R. Garey and myself in our book “Computers and Intractability: A Guide to the Theory of NP-Completeness,” W. H. Freeman & Co., San Francisco, 1979 (hereinafter referred to as “[G&J]”; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, cross-references will be given to that book and the list of problems (NP-complete and harder) presented there. Readers who have results they would like mentioned (NP-hardness, PSPACE-hardness, polynomial-time-solvability, etc.), or open problems they would like publicized, should send them to David S. Johnson, Room 2C-355, Bell Laboratories, Murray Hill, NJ 07974, including details, or at least sketches, of any new proofs (full papers are preferred). In the case of unpublished results, please state explicitly that you would like the results to be mentioned in the column. Comments and corrections are also welcome. For more details on the nature of the column and the form of desired submissions, see the December 1981 issue of this Journal.
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.
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.