Structure theory of flip graphs with applications to Weak Symmetry Breaking

TL;DR

This paper develops a combinatorial structure theory of flip graphs to analyze the iterated immediate snapshot complexity of the Weak Symmetry Breaking task, revealing that minimal rounds do not grow with the number of processes.

Contribution

It introduces a purely combinatorial approach to studying flip graphs and applies this to determine bounds on IIS complexity for WSB, avoiding topological methods.

Findings

Existence of infinitely many n where WSB is solvable in 3 rounds

Minimal number of rounds does not tend to infinity as n increases

Proposes a new paradigm linking IIS complexity bounds to Diophantine equations

Abstract

This paper is devoted to advancing the theoretical understanding of the iterated immediate snapshot (IIS) complexity of the Weak Symmetry Breaking task (WSB). Our rather unexpected main theorem states that there exist infinitely many values of n, such that WSB for n~processes is solvable by a certain explicitly constructed 3-round IIS protocol. In particular, the minimal number of rounds, which an IIS protocol needs in order to solve the WSB task, does not go to infinity, when the number of processes goes to infinity. Our methods can also be used to generate such values of n. We phrase our proofs in combinatorial language, while avoiding using topology. To this end, we study a~certain class of graphs, which we call flip graphs. These graphs encode adjacency structure in certain subcomplexes of iterated standard chromatic subdivisions of a simplex. While keeping the geometric background in mind for an additional intuition, we develop the structure theory of matchings in flip graphs in a purely combinatorial way. Our bound for the IIS complexity is then a corollary of this general theory. As an afterthought of our result, we suggest to change the overall paradigm. Specifically, we think, that the bounds on the IIS complexity of solving WSB for n processes should be formulated in terms of the size of the solutions of the associated Diophantine equation, rather than in terms of the value n itself.