Abstract. We tackle local distributed testing of graph properties. This framework is well suited to contexts in which data dispersed among the nodes of a network can be collected by some central authority (like in, e.g., sensor networks). In local distributed testing, each node can provide the central authority with just a few information about what it perceives from its neighboring environment, and, based on the collected information, the central authority is aiming at deciding whether or not the network satisfies some property. We analyze in depth the prominent example of checking cycle-freeness, and establish tight bounds on the amount of information to be transferred by each node to the central authority for deciding cycle-freeness. In particular, we show that distributedly testing cycle-freeness requires at least ⌈log d⌉ − 1 bits of information per node in graphs with maximum degree d, even for connected graphs. Our proof is based on a novel version of the seminal result by Naor and Stockmeyer (1995) enabling to reduce the study of certain kinds of algorithms to order-invariant algorithms, and on an appropriate use of the known fact that every free group can be linearly ordered.

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International audienceThe main objective of this paper is to provide illustrative examples of distributed computing problems for which it is possible to design tight lower bounds for quantum algorithms without having to manipulate concepts from quantum mechanics, at all. As a case study, we address the following class of 2-player problems. Alice (resp., Bob) receives a boolean x (resp., y) as input, and must return a boolean a (resp., b) as output. A game between Alice and Bob is defined by a pair (?, f) of boolean functions. The objective of Alice and Bob playing game (?, f) is, for every pair (x, y) of inputs, to output values a and b, respectively, satisfying ?(a, b) = f(x, y), in absence of any communication between the two players, but in presence of shared resources. The ability of the two players to solve the game then depends on the type of resources they share. It is known that, for the so-called CHSH game, i.e., for the game a ? b = x ? y, the ability for the players to use entangled quantum bits (qubits) helps. We show that, apart from the CHSH game, quantum correlations do not help, in the sense that, for every game not equivalent to the CHSH game, there exists a classical protocol (using shared randomness) whose probability of success is at least as large as the one of any protocol using quantum resources. This result holds for both worst case and average case analysis. It is achieved by considering a model stronger than quantum correlations, the non-signaling model, which subsumes quantum mechanics, but is far easier to handle

Given a boolean predicate on labeled networks (e.g., the network is acyclic, or the network is properly colored, etc.), deciding in a distributed manner whether a given labeled network satisfies that predicate typically consists, in the standard setting, of every node inspecting its close neighborhood, and outputting Preliminary versions of this work appeared as extended abstracts in the proceedings of the 40th International Workshop on Graph-Theoretic Concepts in Computer Science (WG 2014) [4], and of the 15th International Symposium on Stabilization, Safety, and Security of Distributed Systems (SSS 2013) [5]. P. Fraigniaud receives additional support from the ANR project DESCARTES (ANR-16-CE40-0023), and from the INRIA project GANG. D. Ilcinkas is partially supported by the ANR projects DESCARTES (ANR-16-CE40-0023) and ESTATE (ANR-16-CE25-0009). This study was carried out in the framework of the program "investment for the future" of Idex Bordeaux-CPU (ANR-10-IDEX-03-02).

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When playing the boolean game (δ, f ), two players, upon reception of respective inputs x and y, must respectively output a and b satisfying δ(a, b) = f (x, y), in absence of any communication. It is known that, for δ(a, b) = a ⊕ b, the ability for the players to use entangled quantum bits (qbits) helps. In this paper, we show that, for δ different from the exclusive-or operator, quantum correlations do not help. This result is an invitation to revisit the theory of distributed checking, a.k.a. distributed verification, currently sticked to the usage of decision functions δ based on the and-operator, hence potentially preventing us from using the potential benefit of quantum effects.