Judicious partitions of directed graphs

TL;DR

This paper investigates a fundamental directed graph partitioning problem extending Max Cut, establishing optimal bounds for bipartitions with many crossing edges under various minimum outdegree conditions.

Contribution

It proves tight bounds for bipartitions maximizing crossing edges in directed graphs with different minimum outdegree constraints, extending classical Max Cut results.

Findings

For outdegree ≥ 2, at least 1/6 of edges cross in each direction.

For outdegree ≥ 3, the crossing fraction increases to 1/5.

As outdegree grows, the crossing fraction approaches 1/4.

Abstract

The area of judicious partitioning considers the general family of partitioning problems in which one seeks to optimize several parameters simultaneously, and these problems have been widely studied in various combinatorial contexts. In this paper, we study essentially the most fundamental judicious partitioning problem for directed graphs, which naturally extends the classical Max Cut problem to this setting: we seek bipartitions in which many edges cross in each direction. It is easy to see that a minimum outdegree condition is required in order for the problem to be nontrivial, and we prove that every directed graph with M edges and minimum outdegree at least two admits a bipartition in which at least (1/6 + o(1))M edges cross in each direction. We also prove that if the minimum outdegree is at least three, then the constant can be increased to 1/5. If the minimum outdegree tends to infinity with N, then the constant increases to 1/4. All of these constants are best-possible, and provide asymptotic answers to a question of Alex Scott.