Juho Hirvonen

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.

A lower bound for the distributed Lovász local lemma

We show that any randomised Monte Carlo distributed algorithm for the Lovász local lemma requires Omega(log log n) communication rounds, assuming that it finds a correct assignment with high probability. Our result holds even in the special case of d = O(1), where d is the maximum degree of the dependency graph. By prior work, there are distributed algorithms for the Lovász local lemma with a running time of O(log n) rounds in bounded-degree graphs, and the best lower bound before our work was Omega(log* n) rounds [Chung et al. 2014].

Distributed maximal matching: greedy is optimal

We study distributed algorithms that find a maximal matching in an anonymous, edge-coloured graph. If the edges are properly coloured with k colours, there is a trivial greedy algorithm that finds a maximal matching in k-1 synchronous communication rounds. The present work shows that the greedy algorithm is optimal in the general case: if A is a deterministic distributed algorithm that finds a maximal matching in anonymous, k-edge-coloured graphs, then there is a worst-case input in which the running time of A is at least k1 rounds. If we focus on graphs of maximum degree Δ, it is known that a maximal matching can be found in O(Δ+ log* k) rounds, and prior work implies a lower bound of Ω(polylog(Δ) + log* k) rounds. Our work closes the gap between upper and lower bounds: the complexity is Θ(Δ+ log* k) rounds. To our knowledge, this is the first linear-in-Δ lower bound for the distributed complexity of a classical graph problem.

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.

A Hierarchy of Local Decision

We extend the notion of distributed decision in the framework of distributed network computing, inspired by recent results on so-called distributed graph automata. We show that, by using distributed decision mechanisms based on the interaction between a prover and a disprover, the size of the certificates distributed to the nodes for certifying a given network property can be drastically reduced. For instance, we prove that minimum spanning tree can be certified with

Node Labels in Local Decision

O(\log n)

Non-local Probes Do Not Help with Many Graph Problems

-bit certificates in

New classes of distributed time complexity

n

Locally Optimal Load Balancing

-node graphs, with just one interaction between the prover and the disprover, while it is known that certifying MST requires

Node labels in local decision

\Omega(\log^2 n)