105 research outputs found

    Inapproximability of the Standard Pebble Game and Hard to Pebble Graphs

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    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 n1/3ϵn^{1/3-\epsilon} term for all ϵ>0\epsilon > 0, 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 nn nodes where there exists a graph in the family such that using constant kk pebbles requires Ω(nk)\Omega(n^k) 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 O(nk)O(n^k) moves using constant kk pebbles with a different construction than that presented in [AdRNV17].Comment: Preliminary version in WADS 201

    On the Relative Strength of Pebbling and Resolution

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    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

    Approximating Cumulative Pebbling Cost Is Unique Games Hard

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    The cumulative pebbling complexity of a directed acyclic graph GG is defined as cc(G)=minPiPi\mathsf{cc}(G) = \min_P \sum_i |P_i|, where the minimum is taken over all legal (parallel) black pebblings of GG and Pi|P_i| denotes the number of pebbles on the graph during round ii. Intuitively, cc(G)\mathsf{cc}(G) captures the amortized Space-Time complexity of pebbling mm copies of GG in parallel. The cumulative pebbling complexity of a graph GG is of particular interest in the field of cryptography as cc(G)\mathsf{cc}(G) is tightly related to the amortized Area-Time complexity of the Data-Independent Memory-Hard Function (iMHF) fG,Hf_{G,H} [AS15] defined using a constant indegree directed acyclic graph (DAG) GG and a random oracle H()H(\cdot). A secure iMHF should have amortized Space-Time complexity as high as possible, e.g., to deter brute-force password attacker who wants to find xx such that fG,H(x)=hf_{G,H}(x) = h. Thus, to analyze the (in)security of a candidate iMHF fG,Hf_{G,H}, it is crucial to estimate the value cc(G)\mathsf{cc}(G) but currently, upper and lower bounds for leading iMHF candidates differ by several orders of magnitude. Blocki and Zhou recently showed that it is NP\mathsf{NP}-Hard to compute cc(G)\mathsf{cc}(G), but their techniques do not even rule out an efficient (1+ε)(1+\varepsilon)-approximation algorithm for any constant ε>0\varepsilon>0. We show that for any constant c>0c > 0, it is Unique Games hard to approximate cc(G)\mathsf{cc}(G) to within a factor of cc. (See the paper for the full abstract.)Comment: 28 pages, updated figures and corrected typo

    Time and Space Bounds for Reversible Simulation

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    We prove a general upper bound on the tradeoff between time and space that suffices for the reversible simulation of irreversible computation. Previously, only simulations using exponential time or quadratic space were known. The tradeoff shows for the first time that we can simultaneously achieve subexponential time and subquadratic space. The boundary values are the exponential time with hardly any extra space required by the Lange-McKenzie-Tapp method and the (log3\log 3)th power time with square space required by the Bennett method. We also give the first general lower bound on the extra storage space required by general reversible simulation. This lower bound is optimal in that it is achieved by some reversible simulations.Comment: 11 pages LaTeX, Proc ICALP 2001, Lecture Notes in Computer Science, Vol xxx Springer-Verlag, Berlin, 200

    Two-Player Graph Pebbling

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    Given a graph G with pebbles on the vertices, we define a pebbling move as removing two pebbles from a vertex u, placing one pebble on a neighbor v, and discarding the other pebble, like a toll. The pebbling number n(G) is the least number of pebbles needed so that every arrangement of n(G) pebbles can place a pebble on any vertex through a sequence of pebbling moves. We introduce a new variation on graph pebbling called two-player pebbling. In this, players called the mover and the defender alternate moves, with the stipulation that the defender cannot reverse the previous move. The mover wins only if they can place a pebble on a specified vertex and the defender wins if the mover cannot. We define n(G), analogously, as the minimum number of pebbles such that given every configuration of the n(G) pebbles and every specified vertex r, the mover has a winning strategy. First, we will investigate upper bounds for n(G) on various classes of graphs and find a certain structure for which the defender has a winning strategy, no matter how many pebbles are in a configuration. Then, we characterize winning configurations for both players on a special class of diameter 2 graphs. Finally, we show winning configurations for the mover on paths using a recursive argument

    Pebbling Arguments for Tree Evaluation

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    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 Proof Complexity: Separations and Trade-offs via Substitutions

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    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
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