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 ${\phi }_G\left(S\right)≔(1-\alpha )\cdot w\left(S\right)+\alpha \cdot w(S,V\setminus S)$, where $\alpha \in [0,1]$ is a real number, $w\left(S\right)$ is the sum of the weights of edges having two endpoints in $S$, and $w(S,V\setminus S)$ is the sum of the weights of edges having one endpoint in $S$ and the other in $V\setminus S$. Then, given an undirected and edge-weighted 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) ${\phi }_G\left(S\right)$. 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 $2^{o(kd+k)}{(e+ed)}^k{n}^{O\left(1\right)}$ 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. We show that if Max (Min) $\alpha$-FCGP is W[1]-hard parameterized by $k$, so is Max (Min) Connected $\alpha$-FCGP. Then, we give a $2^{O\left(\sqrt{k}{log}^{2}k\right)}{n}^{O\left(1\right)}$-time randomized algorithm on apex-minor-free graphs. Moreover, for Max Connected $\alpha$-FCGP with $\alpha \in [1/3,1]$ and Min Connected $\alpha$-FCGP with $\alpha \in [0,1/3]$, we propose a ${(1+d)}^k{2}^{o\left(kd\right)+O\left(k\right)}{n}^{O\left(1\right)}$-time algorithm. Finally, we show that they admit FPT-ASs parameterized by $k$ when edge weights are constant.