Consensus time of undecided dynamics with many opinions (synchronous and asynchronous)

Determine tight upper and lower bounds on the consensus time of the undecided dynamics on an n-vertex complete graph with self-loops when the number of opinions satisfies 2 ≤ k ≤ n, under both synchronous round-based updates and asynchronous single-vertex updates. The objective is to characterize convergence rates (e.g., high-probability bounds) analogous to those established for 3-Majority and 2-Choices in this paper.

Background

The paper develops nearly tight bounds for the consensus time of 3-Majority and 2-Choices across all ranges of the number of opinions k on complete graphs. In outlining future directions, the authors highlight undecided dynamics, a well-studied protocol in distributed computing, as a natural next target for applying their technical tools.

Despite substantial attention in prior work, consensus-time guarantees for undecided dynamics with an arbitrary number of opinions (beyond k=2) have not been established, and the authors explicitly note that this remains open for both synchronous and asynchronous settings.