# Distributed Minimum Cut Approximation

Abstract:

We study the problem of computing approximate minimum edge cuts by distributed algorithms. We present two randomized approximation algorithms that both run in a standard synchronous message passing model where in each round, $O(log n)$ bits can be transmitted over every edge (a.k.a. the CONGEST model). The first algorithm is based on a simple and new approach for analyzing random edge sampling, which we call random layering technique. For any any weighted graph and any $\epsilon \in (0, 1)$, the algorithm finds a cut of size at most $O(\epsilon^{-1}\lambda)$ in $O(D) + \tilde{O}(n^{1/2 + \epsilon})$ rounds, where $\lambda$ is the minimum-cut size and the $\tilde{O}$-notation hides poly-logarithmic factors in $n$. In addition, using the outline of a centralized algorithm due to Matula [SODA '93], we present a randomized algorithm to compute a cut of size at most $(2+\epsilon)\lambda$ in $\tilde{O}((D+\sqrt{n})/\epsilon^5)$ rounds for any $\epsilon>0$. The time complexities of our algorithms almost match the $\tilde{\Omega}(D + \sqrt{n})$ lower bound of Das Sarma et al. [STOC '11], thus leading to an answer to an open question raised by Elkin [SIGACT-News '04] and Das Sarma et al. [STOC '11].
To complement our upper bound results, we also strengthen the $\tilde{\Omega}(D + \sqrt{n})$ lower bound of Das Sarma et al. by extending it to unweighted graphs. We show that the same lower bound also holds for unweighted multigraphs (or equivalently for weighted graphs in which $O(w\log n)$ bits can be transmitted in each round over an edge of weight $w$). For unweighted simple graphs, we show that computing an $\alpha$-approximate minimum cut requires time at least $\tilde{\Omega}(D + \sqrt{n}/\alpha^{1/4})$.

Guest: Mohsen Ghaffari (Electrical Engineering and Computer Science Department, Massachusetts Institute of Technology)

Host: Merav Parter

# Lower Bounds for Local Approximation

## Mika Göös, Juho Hirvonen, Jukka Suomela

Abstract: 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 $(\alpha,r)$-homogeneous if its nodes are linearly ordered so that an $\alpha$ fraction of nodes have pairwise isomorphic radius-$r$ neighbourhoods. We show that there exists a finite $(\alpha,r)$-homogeneous $2k$-regular graph of girth at least $g$ for any $\alpha < 1$ and any $r$, $k$, and $g$.

Guest: Jukka Suomela
Host: Merav Parter

# Networks Cannot Compute Their Diameter in Sublinear Time

## Silvio Frischknecht, Stephan Holzer and Roger Wattenhofer

Abstract:
We study the problem of computing the diameter of a network in a distributed way. The model of distributed computation we consider is: in each synchronous round, each node can transmit a different (but) short message to each of its neighbors. We provide an $\tilde{O}(n)$ lower bound for the number of communication rounds needed, where n denotes the number of nodes in the network. This lower bound is valid even if the diameter of the network is a small constant. We also show that a $(3/2-\epsilon)$-approximation of the diameter requires $\tilde{\Omega}(\sqrt{n} +D)$ rounds. Furthermore we use our new technique to prove an $\tilde{\Omega}(\sqrt{n} +D)$ lower bound on approximating the girth of a graph by a factor
$2-\epsilon$.

Guest: Stephan Holzer
Host: Chen Avin