This work studies decision problems from the perspective of nondeterministic distributed algorithms. For a yes-instance there must exist a proof that can be verified with a distributed algorithm: all nodes must accept a valid proof, and at least one node must reject an invalid proof. We focus on locally checkable proofs that can be verified with a constanttime distributed algorithm. For example, it is easy to prove that a graph is bipartite: the locally checkable proof gives a 2-colouring of the graph, which only takes 1 bit per node. However, it is more difficult to prove that a graph is not bipartite-it turns out that any locally checkable proof requires Ω(log n) bits per node. In this work we classify graph problems according to their local proof complexity, i.e., how many bits per node are needed in a locally checkable proof. We establish tight or near-tight results for classical graph properties such as the chromatic number. We show that the proof complexities form a natural hierarchy of complexity classes: for many classical graph problems, the proof complexity is either 0, Θ(1), Θ(log n), or poly(n) bits per node. Among the most difficult graph properties are symmetric graphs, which require Ω(n 2) bits per node, and non-3-colourable graphs, which require Ω(n 2 / log n) bits per node-any pure graph property admits a trivial proof of size O(n 2).

No abstract

We develop a new method to prove communication lower bounds for composed functions of the form f • g n where f is any boolean function on n inputs and g is a sufficiently "hard" two-party gadget. Our main structure theorem states that each rectangle in the communication matrix of f • g n can be simulated by a nonnegative combination of juntas. This is a new formalization for the intuition that each low-communication randomized protocol can only "query" few inputs of f as encoded by the gadget g. Consequently, we characterize the communication complexity of f • g n in all known one-sided (i.e., not closed under complement) zero-communication models by a corresponding query complexity measure of f. These models in turn capture important lower bound techniques such as corruption, smooth rectangle bound, relaxed partition bound, and extended discrepancy. As applications, we resolve several open problems from prior work: We show that SBP cc (a class characterized by corruption) is not closed under intersection. An immediate corollary is that MA cc = SBP cc. These results answer questions of Klauck (CCC 2003) and Böhler et al. (JCSS 2006). We also show that approximate nonnegative rank of partial boolean matrices does not admit efficient error reduction. This answers a question of Kol et al. (ICALP 2014) for partial matrices. In subsequent work, our structure theorem has been applied to resolve the communication complexity of the Clique vs. Independent Set problem.

We prove several results which, together with prior work, provide a nearly-complete picture of the relationships among classical communication complexity classes between P and PSPACE, short of proving lower bounds against classes for which no explicit lower bounds were already known. Our article also serves as an up-to-date survey on the state of structural communication complexity.Among our new results we show that MA ⊆ ZPP NP [1] , that is, Merlin-Arthur proof systems cannot be simulated by zero-sided error randomized protocols with one NP query. Here the class ZPP NP[1] has the property that generalizing it in the slightest ways would make it contain AM ∩ coAM, for which it is notoriously open to prove any explicit lower bounds. We also prove that US ⊆ ZPP NP[1] , where US is the class whose canonically complete problem is the variant of set-disjointness where yes-instances are uniquely intersecting. We also prove that US ⊆ coDP, where DP is the class of differences of two NP sets. Finally, we explore an intriguing open issue: are rank-1 matrices inherently more powerful than rectangles in communication complexity? We prove a new separation concerning PP that sheds light on this issue and strengthens some previously known separations.

Abstract. We use critical block sensitivity, a new complexity measure introduced by Huynh and Nordström (STOC 2012), to study the communication complexity of search problems. To begin, we give a simple new proof of the following central result of Huynh and Nordström: if S is a search problem with critical block sensitivity b, then every randomised two-party protocol solving a certain two-party lift of S requires Ω(b) bits of communication. Besides simplicity, our proof has the advantage of generalising to the multi-party setting. We combine these results with new critical block sensitivity lower bounds for Tseitin and Pebbling search problems to obtain the following applications.• Monotone circuit depth: We exhibit a monotone n-variable function in NP whose monotone circuits require depth Ω(n/ log n); previously, a bound of Ω( √ n) was known (Raz and Wigderson, JACM 1992). Moreover, we prove a Θ( √ n) monotone depth bound for a function in monotone P.• Proof complexity: We prove new rank lower bounds as well as obtain the first length-space lower bounds for semi-algebraic proof systems, including Lovász-Schrijver and Lasserre (SOS) systems. In particular, these results extend and simplify the works of Beame et al. (SICOMP 2007) and Huynh and Nordström.

Abstract. Do unique node identifiers help in deciding whether a network G has a prescribed property P? We study this question in the context of distributed local decision, where the objective is to decide whether G ∈ P by having each node run a constant-time distributed decision algorithm. If G ∈ P, all the nodes should output yes; if G / ∈ P, at least one node should output no. A recent work (Fraigniaud et al., OPODIS 2012) studied the role of identifiers in local decision and gave several conditions under which identifiers are not needed. In this article, we answer their original question. More than that, we do so under all combinations of the following two critical variations on the underlying model of distributed computing: − (B): the size of the identifiers is bounded by a function of the size of the input network; as opposed to (¬B): the identifiers are unbounded. − (C): the nodes run a computable algorithm; as opposed to (¬C): the nodes can compute any, possibly uncomputable function.While it is easy to see that under (¬B, ¬C) identifiers are not needed, we show that under all other combinations there are properties that can be decided locally if and only if identifiers are present. Our constructions use ideas from classical computability theory.

No abstract

We study whether information complexity can be used to attack the long-standing open problem of proving lower bounds against Arthur-Merlin (AM) communication protocols. Our starting point is to show that-in contrast to plain randomized communication complexity-every boolean function admits an AM communication protocol where on each yesinput, the distribution of Merlin's proof leaks no information about the input and moreover, this proof is unique for each outcome of Arthur's randomness. We posit that these two properties of zero information leakage and unambiguity on yes-inputs are interesting in their own right and worthy of investigation as new avenues toward AM.• Zero-information protocols (ZAM). Our basic ZAM protocol uses exponential communication for some functions, and this raises the question of whether more efficient protocols exist. We prove that all functions in the classical space-bounded complexity classes NL and ⊕L have polynomial-communication ZAM protocols. We also prove that ZAM complexity is lower bounded by conondeterministic communication complexity.• Unambiguous protocols (UAM). Our most technically substantial result is a Ω(n) lower bound on the UAM complexity of the NP-complete set-intersection function; the proof uses information complexity arguments in a new, indirect way and overcomes the "zero-information barrier" described above. We also prove that in general, UAM complexity is lower bounded by the classic discrepancy bound, and we give evidence that it is not generally lower bounded by the classic corruption bound. * All proofs appear in the full version of this work [23].