Approximating Cumulative Pebbling Cost Is Unique Games Hard

Authors Jeremiah Blocki , Seunghoon Lee , Samson Zhou

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Jeremiah Blocki
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Seunghoon Lee
  • Department of Computer Science, Purdue University, West Lafayette, IN, USA
Samson Zhou
  • School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA


Part of this work was done while Samson Zhou was a postdoctoral fellow at Indiana University.

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Jeremiah Blocki, Seunghoon Lee, and Samson Zhou. Approximating Cumulative Pebbling Cost Is Unique Games Hard. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 13:1-13:27, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)


The cumulative pebbling complexity of a directed acyclic graph G is defined as cc(G) = min_P ∑_i |P_i|, where the minimum is taken over all legal (parallel) black pebblings of G and |P_i| denotes the number of pebbles on the graph during round i. Intuitively, cc(G) captures the amortized Space-Time complexity of pebbling m copies of G in parallel. The cumulative pebbling complexity of a graph G is of particular interest in the field of cryptography as cc(G) is tightly related to the amortized Area-Time complexity of the Data-Independent Memory-Hard Function (iMHF) f_{G,H} [Joël Alwen and Vladimir Serbinenko, 2015] defined using a constant indegree directed acyclic graph (DAG) G and a random oracle H(⋅). 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 x such that f_{G,H}(x) = h. Thus, to analyze the (in)security of a candidate iMHF f_{G,H}, it is crucial to estimate the value 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-Hard to compute cc(G), but their techniques do not even rule out an efficient (1+ε)-approximation algorithm for any constant ε>0. We show that for any constant c > 0, it is Unique Games hard to approximate cc(G) to within a factor of c. Along the way, we show the hardness of approximation of the DAG Vertex Deletion problem on DAGs of constant indegree. Namely, we show that for any k,ε >0 and given a DAG G with N nodes and constant indegree, it is Unique Games hard to distinguish between the case that G is (e_1, d_1)-reducible with e_1=N^{1/(1+2 ε)}/k and d_1=k N^{2 ε/(1+2 ε)}, and the case that G is (e_2, d_2)-depth-robust with e_2 = (1-ε)k e_1 and d_2= 0.9 N^{(1+ε)/(1+2 ε)}, which may be of independent interest. Our result generalizes a result of Svensson who proved an analogous result for DAGs with indegree ?(N).

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational complexity and cryptography
  • Security and privacy → Hash functions and message authentication codes
  • Cumulative Pebbling Cost
  • Approximation Algorithm
  • Unique Games Conjecture
  • γ-Extreme Depth Robust Graph
  • Superconcentrator
  • Memory-Hard Function


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