Ing. Jan Pokorný

Stať ve sborníku

10.4230/LIPIcs.MFCS.2024.29

Katedra teoretické informatiky

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.

Stať ve sborníku

10.1609/aaai.v37i5.25771

Katedra teoretické informatiky

Microaggregation is a classical statistical disclosure control technique which requires the input data to be partitioned into clusters while adhering to specified size constraints. We provide novel exact algorithms and lower bounds for the task of microaggregating a given network while considering both unrestricted and connected clusterings, and analyze these from the perspective of the parameterized complexity paradigm. Altogether, our results assemble a complete complexity-theoretic picture for the network microaggregation problem with respect to the most natural parameterizations of the problem, including input-specified parameters capturing the size and homogeneity of the clusters as well as the treewidth and vertex cover number of the network.