Fixed-Parameter Algorithms for Cardinality-Constrained Graph Partitioning Problems on Sparse Graphs

Abstract

For an undirected and edge-weighted graph $G=(V, E)$ and a vertex subset $S\subseteq V$, we define a function $\varphi _{G}(S) := (1-\alpha )\cdot w(S) + \alpha \cdot w(S, V\setminus S)$, where $\alpha \in [0, 1]$ is a real number, w(S) is the sum of weights of edges having two endpoints in S, and $w(S, V\setminus S)$ is the sum of weights of edges having one endpoint in S and the other in $V\setminus S$. Then, given a graph $G=(V, E)$ and a positive integer k, Max (Min) $\alpha$ -Fixed Cardinality Graph Partitioning (Max (Min) $\alpha$ -FCGP) is the problem to find a vertex subset $S\subseteq V$ of size k that maximizes (minimizes) $\varphi _{G}(S)$. In this paper, we first show that Max $\alpha$ -FCGP with $\alpha \in [1/3,1]$ and Min $\alpha$ -FCGP with $\alpha \in [0,1/3]$ can be solved in time $2^{o(kd+k)}(e+ed)^k n^{O(1)}$ where k is the solution size, d is the degeneracy of an input graph, and e is Napier’s constant.Then we consider Max (Min) Connected $\alpha$ -FCGP, which additionally requires the connectivity of a solution. For Max (Min) Connected $\alpha$ -FCGP, we give an $(e (\varDelta -1))^{k-1}n^{O(1)}$-time algorithm on general graphs and a $2^{O(\sqrt{k} \log ^2 k)}n^{O(1)}$-time randomized algorithm on apex-minor-free graphs. Moreover, for Max $\alpha$ -FCGP with $\alpha \in [1/3,1]$ and Min $\alpha$ -FCGP with $\alpha \in [0,1/3]$, we propose an $(1+d)^k 2^{o(kd)+O(k)} n^{O(1)}$-time algorithm. Finally, we show that they admit FPT-ASs when edge weights are constant.

This work is partially supported by JSPS KAKENHI Grant Numbers JP21K17707, JP21H05852, JP22H00513, and JP23H04388.