Abstract
We study a new model of verification of boolean predicates over distributed networks. Given a network configuration, the proof-labeling scheme (PLS) model defines a distributed proof in the form of a label that is given to each node, and all nodes locally verify that the network configuration satisfies the desired boolean predicate by exchanging labels with their neighbors. The proof size of the scheme is defined to be the maximum size of a label.
In this work, we extend this model by defining the approximate proof-labeling scheme (APLS) model. In this new model, the predicates for verification are of the form \(\psi \le \varphi \), where \(\psi , \varphi : \mathcal{F}\rightarrow \mathbb {N}\) for a family of configurations \(\mathcal{F}\). Informally, the predicates considered in this model are a comparison between two values of the configuration. As in the PLS model, nodes exchange labels in order to locally verify the predicate, and all must accept if the network satisfies the predicate. The soundness condition is relaxed with an approximation ration \(\alpha \), so that only if \(\psi > \alpha \varphi \) some node must reject.
We show that in the APLS model, the proof size can be much smaller than the proof size of the same predicate in the PLS model. Moreover, we prove that there is a tradeoff between the approximation ratio and the proof size.
K. Censor-Hillel and A. Paz—Supported by ISF individual research grant 1696/14.
M. Perry—Partially supported by Apple Graduate Fellowship.
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- 1.
Recall that W is the maximum weight of an edge in the graph. If \(W=1\), we interpret \(O(\log W)\) as O(1).
- 2.
This lower bound holds also for randomized protocols, which we do not discuss in this work.
- 3.
See Chap. 2.2 of [1]. We use \(P=\lfloor (k-2)/4 \rfloor \).
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Acknowledgment
We thank Gilad Kutiel, Seffi Naor and Dror Rawitz for discussions of the primal-dual method, and the anonymous reviewers of SIROCCO 2017 for valuable comments.
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Censor-Hillel, K., Paz, A., Perry, M. (2017). Approximate Proof-Labeling Schemes. In: Das, S., Tixeuil, S. (eds) Structural Information and Communication Complexity. SIROCCO 2017. Lecture Notes in Computer Science(), vol 10641. Springer, Cham. https://doi.org/10.1007/978-3-319-72050-0_5
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