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

TL;DR

The paper studies the Equitable Connected Partition (ECP) problem, where a graph $G=(V,E)$ and an integer $p$ are given and the goal is to partition $V$ into $p$ connected parts of near-equal size. It maps the parameterized complexity landscape by proving $W[1]$-hardness for the $4$-path vertex cover number and for the feedback-edge set, as well as for the distance to disjoint paths, and shows NP-hardness for shrub-depth and clique-width; it also develops fixed-parameter algorithms for various parameters that tend to be small in dense graphs, including approaches using $N$-fold integer programming. These results place sharp boundaries between tractable and intractable cases and provide practical FPT techniques tailored to dense graph classes. The work thereby advances understanding of how equitability and connectivity constraints interact with structural graph parameters, offering both negative hardness results and positive algorithmic strategies.

Abstract

We study the Equitable Connected Partition (ECP for short) problem, where we are given a graph G=(V,E) together with an integer p, 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. As our main contribution, we resolve a long-standing open question [Enciso et al.; IWPEC '09] regarding the parameterisation by the tree-depth of the underlying graph. In particular, we show that ECP is W[1]-hard with respect to the 4-path vertex cover number, which is an even more restrictive structural parameter than the tree-depth. In addition to that, we show W[1]-hardness of the problem with respect to the feedback-edge set, the distance to disjoint paths, and NP-hardness with respect to the shrub-depth and the clique-width. On a positive note, we propose several novel fixed-parameter algorithms for various parameters that are bounded for dense graphs.

Figures (1)

• Figure 1: An overview of our results. The parameters for which the problem is in FPT are coloured green, the parameters for which ECP is W[1]-hard and in XP have an orange background, and para-NP-hard combinations are highlighted in red. Arrows indicate generalisations; e.g., modular width generalises both neighbourhood diversity and twin-cover number. The solid thick border represents completely new results, and the dashed border represents an improvement of previously known algorithm. All our W[1]-hardness results hold even when the problem is additionally parameterized by the number of parts $p$; however, the results marked with $\star$ becomes fixed-parameter tractable if the size of a larger part $\lceil n/p \rceil$ is an additional parameter.

Theorems & Definitions (11)

• Definition 1: $d$-path vertex cover
• Definition 2: Feedback-edge set
• Definition 3: Vertex integrity
• Definition 4: Distance to $\mathcal{G}$
• Definition 5: Twin-cover Ganian2015
• Definition 6: Neighbourhood-diversity Lampis2012
• Definition 7: Modular-width GajarskyLO2013
• Definition 8: Tree-depth
• Definition 9: Shrub-depth GanianHNOMR2012
• Definition 10: Tree-width