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Given a directed acyclic graph (DAG) G=(V,E), we say that G is (e,d)-depth-robust (resp. (e,d)-edge-depth-robust) if for any set S⊆V (resp. S⊆E) of at most |S|≤e nodes (resp. edges) the graph G−S contains a directed path of length d. 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)-edge-depth-robust graph with m edges into a (e/2,d)-depth-robust graph with O(m) nodes and constant indegree. One immediate consequence of this result is the first construction of a provably (nloglognlogn,nlogn(logn)loglogn)-depth-robust graph with constant indegree. 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 n inputs and n outputs is (k1,k2)-ST-robust if we can remove any k1 nodes and there exists a subgraph containing at least k2 inputs and k2 outputs such that each of the k2 inputs is connected to all of the k2 outputs. If the graph if (k1,n−k1)-ST-robust for all k1≤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) nodes. Given a family M of ST-robust graphs and an arbitrary (e,d)-edge-depth-robust graph G we construct a new constant-indegree graph Reduce(G,M) by replacing each node in G with an ST-robust graph from 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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The effect of adding randomly weighted edges
We consider the following question. We have a dense regular graph $$G$$ with degree $$\a n$$, where $$\a>0$$ is a constant. We add $m=o(n^2)$ random edges. The edges of the augmented graph $G(m)$ are given independent edge weights $$X(e),e\in E(G(m))$$. We estimate the minimum weight of some specified combinatorial structures. We show that in certain cases, we can obtain the same estimate as is known for the complete graph, but scaled by a factor $$\a^{-1}$$. We consider spanning trees, shortest paths and perfect matchings in (pseudo-random) bipartite graphs.
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- Award ID(s):
- 1952285
- PAR ID:
- 10320466
- Date Published:
- Journal Name:
- SIAM journal on discrete mathematics
- Volume:
- 35
- ISSN:
- 1095-7146
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1137/20M1335418
