On the Computational Complexity of Minimal Cumulative Cost Graph Pebbling

Abstract

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 $$P_0,\ldots, P_t \subseteq V$$ of sets of nodes which must satisfy the following properties: $$P_0 = \emptyset $$ (we start off with no pebbles on G), $${\mathsf {sinks}} (G) \subseteq \bigcup _{j \le t} P_j$$ (every sink node was pebbled at some point) and $${\mathsf {parents}} \big (P_{i+1}\backslash P_i\big ) \subseteq P_i$$ (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 $$P_0,P_1,\ldots, P_t \subseteq V$$ is $${\mathsf {cc}} (P) = \left| P_1\right| + \ldots + \left| P_t\right| $$. 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 $${\mathtt {NP}\text {-}\mathtt {Hard}} $$ to find a pebbling minimizing cumulative cost.The natural linear program relaxation for the problem has integrality gap $$\tilde{O}(n)$$, 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 $$S\subseteq V$$ such that $${\mathsf {depth}} (G-S) \le d$$, is also $${\mathtt {NP}\text {-}\mathtt {Hard}} $$. In fact, under the Unique Games Conjecture there is no $$(2-\epsilon )$$ -approximation algorithm.