Homotopy Theory in Complexity of the Graph Homomorphism Problem (Invited Talk)

Abstract

I will talk about an emerging application of homotopy theory in computational complexity of combinatorial problems, more precisely homomorphism problems. Homomorphism problems appear under many names, including constraint satisfaction problems and conjunctive database queries.
A prime example of a homomorphism problem is graph colouring; by a colouring of a graph with k colours, I mean an assignment of colours to vertices under which no edge is monochromatic - this is equivalent to the existence of a homomorphism to the clique with k vertices. Graph 3-colouring is a prototypical example of an NP-complete problem. There are many variations on this problem whose complexity remains widely open. For example, although it is generally believed that colouring a 3-colourable graph with a fixed number of colours is NP-hard, only the hardness of colouring of such a graph with 5 colours is known (and shown only in 2019).
I will give an overview of several related results about variations of graph colouring that share a common theme of using a method based on tools from topological combinatorics and on ideas of Lovász [J. Comb. Theory, Ser. A, 25(3):319-324, 1978].