Topological Combinatorics and the Evasiveness Conjecture

The Kahn, Saks, and Sturtevant approach to the Evasiveness Conjecture (see the original paper here) is an epic application of pure mathematics to computer science. I’ll give an overview of the approach here, and probably try to add some more information on the problem in other posts.

tl;dr The KSS approach provides an algebraic-topological attack to a combinatorial hypothesis, and reduces a graph complexity problem to a problem of contractibility and (not) finding fixed points.

First, the Evasiveness Conjecture states that any (non-trivial) monotone graph property is evasive. In other words, if you’re trying to figure out whether an undirected n-vertex graph satisfies a certain property (e.g., whether the graph contains a triangle or is connected), and this property is monotone (meaning that if you add more edges to the graph, then it still satisfies the property), then if all you’re allowed to do is ask questions of the form “Is edge (i, j) in the graph?”, then you need to query for every single edge before you can determine whether the graph satisfies the property or not. For example, if you want to figure out whether a graph G contains a clique of size 5, then you need to know whether each of the n(n-1)/2 possible edges is in the graph or not before you can answer for certain.

Next, given any monotone graph property on n-vertex graphs, we can associate it with a simplicial complex S (basically, an n-dimensional structure formed by gluing together a bunch of hypertriangles), by taking the complex to be the set of all n-vertex graphs that don’t satisfy the property.

Kahn, Saks, and Sturtevant then prove that if a monotone graph property is not evasive, then its associated simplicial complex is contractible, and thus (by the Lefschetz Fixed-Point theorem) any auto-simplicial map on the complex (a function from the complex to itself that preserves faces) has a fixed point.

Thus, we can prove that a monotone graph property is evasive by finding a simplicial map that has no fixed point (which we can do by showing that no orbit of the map is a face of the complex). This approach has been used to prove things like the evasiveness of graph properties when the number of vertices is prime or a prime power, and the evasiveness of all bipartite graph properties.