Evasiveness of Graph Properties and Topological Fixed-Point Theorems

TL;DR

This paper reviews topological methods used to analyze the complexity of graph properties, focusing on the evasiveness conjecture and providing a self-contained tutorial on key proofs and recent developments.

Contribution

It offers a comprehensive, accessible tutorial on topological techniques applied to the evasiveness of graph properties, including central proofs and recent advances.

Findings

Topological methods provide partial results on the evasiveness conjecture.

The paper presents a self-contained account of key proofs in the field.

Recent results suggest progress towards understanding the conjecture.

Abstract

Many graph properties (e.g., connectedness, containing a complete subgraph) are known to be difficult to check. In a decision-tree model, the cost of an algorithm is measured by the number of edges in the graph that it queries. R. Karp conjectured in the early 1970s that all monotone graph properties are evasive -- that is, any algorithm which computes a monotone graph property must check all edges in the worst case. This conjecture is unproven, but a lot of progress has been made. Starting with the work of Kahn, Saks, and Sturtevant in 1984, topological methods have been applied to prove partial results on the Karp conjecture. This text is a tutorial on these topological methods. I give a fully self-contained account of the central proofs from the paper of Kahn, Saks, and Sturtevant, with no prior knowledge of topology assumed. I also briefly survey some of the more recent results on evasiveness.