NP-completeness in the gossip monoid

TL;DR

This paper proves that key decision problems in gossip monoids, which model network information exchange, are NP-complete, revealing their computational complexity and impacting understanding of their structure.

Contribution

It establishes NP-completeness for important decision problems in gossip monoids, connecting algebraic properties with computational complexity.

Findings

Membership problem is NP-complete.

Deciding possible states of knowledge is NP-complete.

Results explain the difficulty in determining gossip monoid cardinalities.

Abstract

Gossip monoids form an algebraic model of networks with exclusive, transient connections in which nodes, when they form a connection, exchange all known information. They also arise naturally in pure mathematics, as the monoids generated by the set of all equivalence relations on a given finite set under relational composition. We prove that a number of important decision problems for these monoids (including the membership problem, and hence the problem of deciding whether a given state of knowledge can arise in a network of the kind under consideration) are NP-complete. As well as being of interest in their own right, these results shed light on the apparent difficulty of establishing the cardinalities of the gossip monoids: a problem which has attracted some attention in the last few years.