1/3–2/3 Conjecture for Balancing Comparisons in DAG Sorting

Determine whether for every directed acyclic graph G (equivalently, every finite partially ordered set), there exists a balancing comparison x < y such that, among all topological orders consistent with G, the fraction in which x precedes y lies between 1/3 and 2/3.

Background

In the DAG sorting problem, the goal is to discover an unknown total order consistent with a given partial order (represented by a DAG) using binary comparisons. A central concept is a balancing comparison, a query x < y? that splits the set of remaining consistent topological orders by a constant factor.

The 1/3–2/3 conjecture posits that for every finite poset (or DAG), such a balancing comparison exists with balance parameter δ = 1/3. If true, repeatedly selecting a balancing comparison would yield an O(log T) comparison algorithm, where T is the number of topological orders consistent with the DAG. Although the paper provides an algorithm achieving O(log T) comparisons without relying on this conjecture, the conjecture itself remains a longstanding open question in poset theory and algorithms for sorting under partial information.