Equitable Connected Partition and Structural Parameters Revisited: N-fold Beats Lenstra

Abstract

In the Equitable Connected Partition (ECP for short) problem, we are given a graph $G=(V,E)$ together with an integer $p\in\mathbb{N}$, and our goal is to find a partition of~$V$ into~$p$~parts such that each part induces a connected sub-graph of $G$ and the size of each two parts differs by at most~$1$. On the one hand, the problem is known to be NP-hard in general and W[1]-hard with respect to the path-width, the feedback-vertex set, and the number of parts~$p$ combined. On the other hand, fixed-parameter algorithms are known for parameters the vertex-integrity and the max leaf number.

In this work, we systematically study ECP with respect to various structural restrictions of the underlying graph and provide a clear dichotomy of its parameterised complexity. Specifically, we show that the problem is in FPT when parameterized by the modular-width and the distance to clique. Next, we prove W[1]-hardness with respect to the distance to cluster, the $4$-path vertex cover number, the distance to disjoint paths, and the feedback-edge set, and NP-hardness for constant shrub-depth graphs. Our hardness results are complemented by matching algorithmic upper-bounds: we give an XP algorithm for parameterisation by the tree-width and the distance to cluster. We also give an improved FPT algorithm for parameterisation by the vertex integrity and the first explicit FPT algorithm for the $3$-path vertex cover number. The main ingredient of these algorithms is a formulation of ECP as $N$-fold IP, which clearly indicates that such formulations may, in certain scenarios, significantly outperform existing algorithms based on the famous algorithm of Lenstra.