最小累积费用图卵石化的计算复杂性
On the Computational Complexity of Minimal Cumulative Cost Graph Pebbling
摘要：
我们考虑的计算复杂性，找到一个合法的黑色鹅卵石的DAG $G=(V,E)$最小的累积成本。一个黑pebbling是一个节点集序列$P_0,\ldots, P_t \subseteq V$，它必须满足以下属性：$P_0 = \emptyset$（我们开始时$G$上没有pebbles）、$\mathsf{sinks}(G) \subseteq \bigcup_{j \leq t} P_j$（每个sink节点在某个点上都被pebbles）和$\mathsf{parents}\big(P_{i+1}\backslash P_i\big) \subseteq P_i$（我们只能在一个节点$v$上放置一个新的pebbles，如果$v$的所有父节点在上一轮中都有pebbles）。pebbling $P_0,P_1,\ldots, P_t \subseteq V$的累积成本是$\mathsf{cc}(P) = | P_1| + \ldots + | P_t|$。累积pebbling成本是数据独立内存硬函数的一个特别重要的安全度量，是密码散列的一个重要原语。因此，一个有效的（近似）算法将是密码散列函数的密码分析的宝贵工具，因为它将提供一个自动化的工具来建立计算函数的摊销时空成本的严格界限。我们表明，这样的工具是不可能存在的。特别是，我们证明了以下结果。(1)这是$\texttt{NP}\mbox{-}\texttt{Hard}$找到一个卵石最小化累积成本。(2)该问题的自然线性规划松弛具有完整性间隙$\tilde{O}(n)$，其中$n$是$G$中的节点数。我们推测这个问题很难近似解决。(3)我们表明，一个相关的问题，找到最小尺寸的子集$S\subseteq V$，$\textsf{depth}(G-S) \leq d$，也是$\texttt{NP}\mbox{-}\texttt{Hard}$。事实上，在唯一对策猜想下并不存在$(2-ε)$ -近似算法。
英文摘要：
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 $G$), $\mathsf{sinks}(G) \subseteq \bigcup_{j \leq t} P_j$ (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 $\mathsf{cc}(P) = | P_1| + \ldots + | P_t|$. 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 $\tilde{O}(n)$, 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 $S\subseteq V$ such that $\textsf{depth}(G-S) \leq d$, is also $\texttt{NP}\mbox{-}\texttt{Hard}$. In fact, under the unique games conjecture there is no $(2-ε)$-approximation algorithm.
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