Balanced Judicious Partition is Fixed-Parameter Tractable

TL;DR

This paper proves that the Balanced Judicious Bipartition problem is fixed-parameter tractable, advancing the algorithmic understanding of judicious partitioning problems in extremal combinatorics.

Contribution

We establish that Balanced Judicious Bipartition is fixed-parameter tractable, providing the first FPT algorithm for this class of judicious partitioning problems.

Findings

BJB is fixed-parameter tractable (FPT).

JB is also FPT as a consequence.

Advances the algorithmic theory of judicious partitioning problems.

Abstract

The family of judicious partitioning problems, introduced by Bollob\'as and Scott to the field of extremal combinatorics, has been extensively studied from a structural point of view for over two decades. This rich realm of problems aims to counterbalance the objectives of classical partitioning problems such as Min Cut, Min Bisection and Max Cut. While these classical problems focus solely on the minimization/maximization of the number of edges crossing the cut, judicious (bi)partitioning problems ask the natural question of the minimization/maximization of the number of edges lying in the (two) sides of the cut. In particular, Judicious Bipartition (JB) seeks a bipartition that is "judicious" in the sense that neither side is burdened by too many edges, and Balanced JB also requires that the sizes of the sides themselves are "balanced" in the sense that neither of them is too large. Both of these problems were defined in the work by Bollob\'as and Scott, and have received notable scientific attention since then. In this paper, we shed light on the study of judicious partitioning problems from the viewpoint of algorithm design. Specifically, we prove that BJB is FPT (which also proves that JB is FPT).