Rounds vs Communication Tradeoffs for Maximal Independent Sets

09/19/2022
by   Sepehr Assadi, et al.
0

We consider the problem of finding a maximal independent set (MIS) in the shared blackboard communication model with vertex-partitioned inputs. There are n players corresponding to vertices of an undirected graph, and each player sees the edges incident on its vertex – this way, each edge is known by both its endpoints and is thus shared by two players. The players communicate in simultaneous rounds by posting their messages on a shared blackboard visible to all players, with the goal of computing an MIS of the graph. While the MIS problem is well studied in other distributed models, and while shared blackboard is, perhaps, the simplest broadcast model, lower bounds for our problem were only known against one-round protocols. We present a lower bound on the round-communication tradeoff for computing an MIS in this model. Specifically, we show that when r rounds of interaction are allowed, at least one player needs to communicate Ω(n^1/20^r+1) bits. In particular, with logarithmic bandwidth, finding an MIS requires Ω(loglogn) rounds. This lower bound can be compared with the algorithm of Ghaffari, Gouleakis, Konrad, Mitrović, and Rubinfeld [PODC 2018] that solves MIS in O(loglogn) rounds but with a logarithmic bandwidth for an average player. Additionally, our lower bound further extends to the closely related problem of maximal bipartite matching. To prove our results, we devise a new round elimination framework, which we call partial-input embedding, that may also be useful in future work for proving round-sensitive lower bounds in the presence of edge-sharing between players. Finally, we discuss several implications of our results to multi-round (adaptive) distributed sketching algorithms, broadcast congested clique, and to the welfare maximization problem in two-sided matching markets.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
03/12/2021

Robust Lower Bounds for Graph Problems in the Blackboard Model of Communication

We give lower bounds on the communication complexity of graph problems i...
research
08/04/2022

Computing Power of Hybrid Models in Synchronous Networks

During the last two decades, a small set of distributed computing models...
research
04/17/2020

Distributed Lower Bounds for Ruling Sets

Given a graph G = (V,E), an (α, β)-ruling set is a subset S ⊆ V such tha...
research
11/26/2021

An Optimal Algorithm for Finding Champions in Tournament Graphs

A tournament graph T = (V, E ) is an oriented complete graph, which can ...
research
05/19/2019

Broadcast Congested Clique: Planted Cliques and Pseudorandom Generators

We develop techniques to prove lower bounds for the BCAST(log n) Broadca...
research
05/17/2019

Separating k-Player from t-Player One-Way Communication, with Applications to Data Streams

In a k-party communication problem, the k players with inputs x_1, x_2, ...
research
02/21/2023

The Swiss Gambit

In each round of a Swiss-system tournament, players of similar score are...

Please sign up or login with your details

Forgot password? Click here to reset