Balanced Judicious Bipartition is Fixed-Parameter Tractable

Abstract

The family of judicious partitioning problems, introduced by Bollobás 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 Bollobás 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).

Parameterized complexity of multi-node hubs

“…To the best of our knowledge, besides our algorithm, only algorithms for Bisection and a problem called Judicious Partition [32] are based on the decomposition of [15]. Exploring the power of this decomposition more deeply might be fruitful; in this context, it is noteworthy to mention [31].…”

Perfect forests in graphs and their extensions

“…The answer is positive and the proof is trivial. In fact, it is 1 Such parameterizations were studied for many graph-theoretical and constraint satisfaction problems, see, for example, [2,6,13,17,18]. In other words, we can ask whether there is a 0-perfect forest of size at least n k − ( ∕ n k 2 + , respectively), where k is the parameter.…”

On the Parameterized Complexity of the Maximum Exposure Problem

“…[2] which is based on branching algorithms. This approach has recently gained much attraction in fixed cardinality parameterized problems [12,14,15]. We also use the result by Lenstra and Kannan [9,11], which shows that ILP optimization is fixed parameter tractable when parameterized by the number of variables.…”