Mika Göös

Locally checkable proofs

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 constant-time 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 ©(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, (1), (log n), or poly(n) bits per node. Among the most difficult graph properties are symmetric graphs, which require ©(n²) bits per node, and non-3-colourable graphs, which require ©(n²/log n) bits per node - any pure graph property admits a trivial proof of size O(n²).

Deterministic Communication vs. Partition Number

We show that deterministic communication complexity can be super logarithmic in the partition number of the associated communication matrix. We also obtain near-optimal deterministic lower bounds for the Clique vs. Independent Set problem, which in particular yields new lower bounds for the log-rank conjecture. All these results follow from a simple adaptation of a communication-to-query simulation theorem of Raz and McKenzie (Combinatorica 1999) together with lower bounds for the analogous query complexity questions.

Lower bounds for local approximation

In the study of deterministic distributed algorithms it is commonly assumed that each node has a unique O(log n)-bit identifier. We prove that for a general class of graph problems, local algorithms (constant-time distributed algorithms) do not need such identifiers: a port numbering and orientation is sufficient. Our result holds for so-called simple PO-checkable graph optimisation problems; this includes many classical packing and covering problems such as vertex covers, edge covers, matchings, independent sets, dominating sets, and edge dominating sets. We focus on the case of bounded-degree graphs and show that if a local algorithm finds a constant-factor approximation of a simple PO-checkable graph problem with the help of unique identifiers, then the same approximation ratio can be achieved on anonymous networks. As a corollary of our result and by prior work, we derive a tight lower bound on the local approximability of the minimum edge dominating set problem. Our main technical tool is an algebraic construction of homogeneously ordered graphs: We say that a graph is (α,r)-homogeneous if its nodes are linearly ordered so that an α fraction of nodes have pairwise isomorphic radius-r neighbourhoods. We show that there exists a finite (α,r)-homogeneous 2k-regular graph of girth at least g for any α<1 and any r, k, and g.

Communication lower bounds via critical block sensitivity

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 function on n variables 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 tight Θ(√n) monotone depth bound for a function in monotone P. This implies an average-case hierarchy theorem within monotone P similar to a result of Filmus et al. (FOCS 2013). • 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.

What can be decided locally without identifiers

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 has property P by having each node run a constant-time distributed decision algorithm. In a yes-instance all nodes should output yes, while in a no-instance at least one node should output no. Recently, Fraigniaud et al. (OPODIS 2012) gave several conditions under which identifiers are not needed, and they conjectured that identifiers are not needed in any decision problem. In the present work, we disprove the conjecture. More than that, we analyse two critical variations of the underlying model of distributed computing: (B): the size of the identifiers is bounded by a function of the size of the input network, (¬B): the identifiers are unbounded, (C): the nodes run a computable algorithm, (¬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.

Linear-in-delta lower bounds in the LOCAL model

By prior work, there is a distributed graph algorithm that finds a maximal fractional matching (maximal edge packing) in O(Δ) rounds, independently of n; here Δ is the maximum degree of the graph and n is the number of nodes in the graph. We show that this is optimal: there is no distributed algorithm that finds a maximal fractional matching in o(Δ) rounds, independently of n. Our work gives the first linear-in-Δ lower bound for a natural graph problem in the standard LOCAL model of distributed computing---prior lower bounds for a wide range of graph problems have been at best logarithmic in Δ.

Randomized Communication versus Partition Number

We show that randomized communication complexity can be superlogarithmic in the partition number of the associated communication matrix, and we obtain near-optimal randomized lower bounds for the Clique versus Independent Set problem. These results strengthen the deterministic lower bounds obtained in prior work (Göös, Pitassi, and Watson, FOCS’15). One of our main technical contributions states that information complexity when the cost is measured with respect to only 1-inputs (or only 0-inputs) is essentially equivalent to information complexity with respect to all inputs.

Randomized distributed decision

The paper tackles the power of randomization in the context of local distributed computing by analyzing the ability to “boost” the success probability of deciding a distributed language using a Monte-Carlo algorithm. We prove that, in many cases, the ability to increase the success probability for deciding distributed languages is rather limited. This contrasts with the sequential computing setting where boosting can systematically be achieved by repeating the randomized execution.

Query-to-Communication Lifting for BPP

For any n-bit boolean function f, we show that the randomized communication complexity of the composed function f o g^n, where g is an index gadget, is characterized by the randomized decision tree complexity of f. In particular, this means that many query complexity separations involving randomized models (e.g., classical vs. quantum) automatically imply analogous separations in communication complexity.

Separations in Communication Complexity Using Cheat Sheets and Information Complexity

While exponential separations are known between quantum and randomized communication complexity for partial functions (Raz, STOC 1999), the best known separation between these measures for a total function is quadratic, witnessed by the disjointness function. We give the first super-quadratic separation between quantum and randomized communication complexity for a total function, giving an example exhibiting a power 2.5 gap. We further present a 1.5 power separation between exact quantum and randomized communication complexity, improving on the previous ≈ 1.15 separation by Ambainis (STOC 2013). Finally, we present a nearly optimal quadratic separation between randomized communication complexity and the logarithm of the partition number, improving upon the previous best power 1.5 separation due to Goos, Jayram, Pitassi, and Watson. Our results are the communication analogues of separations in query complexity proved using the recent cheat sheet framework of Aaronson, Ben-David, and Kothari (STOC 2016). Our main technical results are randomized communication and information complexity lower bounds for a family of functions, called lookup functions, that generalize and port the cheat sheet framework to communication complexity.