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A New Connection Between Node and Edge Depth Robust Graphs

Abstract

Given a directed acyclic graph (DAG) G=(V,E)G = (V,E), we say that GG is (e,d)(e,d)-depth-robust (resp. (e,d)(e,d)-edge-depth-robust) if for any set SVS \subset V (resp. SES \subseteq E) of at most Se|S| \leq e nodes (resp. edges) the graph GSG-S contains a directed path of length dd. While edge-depth-robust graphs are potentially easier to construct many applications in cryptography require node depth-robust graphs with small indegree. We create a graph reduction that transforms an (e,d)(e, d)-edge-depth-robust graph with mm edges into a (e/2,d)(e/2,d)-depth-robust graph with O(m)O(m) nodes and constant indegree. One immediate consequence of this result is the first construction of a provably (nloglognlogn,n(logn)1+loglogn)(\frac{n \log \log n}{\log n}, \frac{n}{(\log n)^{1 + \log \log n}})-depth-robust graph with constant indegree, where previous constructions for e=nloglognlogne =\frac{n \log \log n}{\log n} had d=O(n1ϵ)d = O(n^{1-\epsilon}). Our reduction crucially relies on ST-Robust graphs, a new graph property we introduce which may be of independent interest. We say that a directed, acyclic graph with nn inputs and nn outputs is (k1,k2)(k_1, k_2)-ST-Robust if we can remove any k1k_1 nodes and there exists a subgraph containing at least k2k_2 inputs and k2k_2 outputs such that each of the k2k_2 inputs is connected to all of the k2k_2 outputs. If the graph if (k1,nk1)(k_1,n-k_1)-ST-Robust for all k1nk_1 \leq n we say that the graph is maximally ST-robust. We show how to construct maximally ST-robust graphs with constant indegree and O(n)O(n) nodes. Given a family M \mathbb{M} of ST-robust graphs and an arbitrary (e,d)(e, d)-edge-depth-robust graph GG we construct a new constant-indegree graph Reduce(G,M) \mathrm{Reduce}(G, \mathbb{M}) by replacing each node in GG with an ST-robust graph from M \mathbb{M}. We also show that ST-robust graphs can be used to construct (tight) proofs-of-space and (asymptotically) improved wide-block labeling functions.

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