Abstract
We study proof labeling schemes in directed networks, and ask what assumptions are necessary to be able to certify reachability-related problems such as strong connectivity, or the existence of a node from which all nodes are reachable. In contrast to undirected networks, in directed networks, having unique identifiers alone does not suffice to be able to certify all graph properties; thus, we study the effect of knowing the size of the graph, and of each node knowing its out-degree. We formalize the notion of giving the nodes initial knowledge about the network, and give tight characterizations of the types of knowledge that are necessary and sufficient to certify several reachability-related properties, or to be able to certify any graph property. For example, we show that in order to certify that the network contains a node that is reachable from all nodes, it is necessary and sufficient to have any two of the assumptions we study (unique identifiers, knowing the size, or knowing the out-degree); and to certify strong connectivity, it is necessary and sufficient to have any single assumption.
Research funded by the Israel Science Foundation, Grant No. 2801/20, and also supported by Len Blavatnik and the Blavatnik Family foundation.
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Notes
- 1.
The latter type of knowledge, which is not given in advance but rather computed by the nodes during runtime, is addressed in [1].
- 2.
A correctness proof was not given in [8], but more importantly, we must prove that the DFS can be verified even using only unidirectional communication.
- 3.
This is in some sense an extension of the argument in [4], where it is shown that if the prover cannot use the unique identifiers when choosing its proof, then the languages that can be recognized are exactly those that are closed under lifts.
- 4.
An alternative approach might be to specify the execution of the charge-distribution protocol up to some sufficiently large number of rounds, and deduce the asymptotic behavior, but this is problematic for two reasons: first, a Markov chain might take an exponential number of steps to approach its limiting behavior; and second, the convergence time depends on the size of the chain, but we would like to use our PLS in settings where the size of the graph is not necessarily known.
- 5.
This notion is closely related to covering maps and lifts [4], but those classical notions are not appropriate for directed graphs, and they also do not handle initial knowledge in the way we require here.
- 6.
Node v of course knows its own UID, but in order for v’s out-neighbors to learn \(\mathsf {UID}(v)\), it must be part of v’s label.
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Ben Shimon, Y., Fischer, O., Oshman, R. (2022). Proof Labeling Schemes for Reachability-Related Problems in Directed Graphs. In: Parter, M. (eds) Structural Information and Communication Complexity. SIROCCO 2022. Lecture Notes in Computer Science, vol 13298. Springer, Cham. https://doi.org/10.1007/978-3-031-09993-9_2
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