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

For an undirected and edge-weighted graph G = (V, E) and a vertex subset S subset of V, we define a function phi(G)(S) := (1 - alpha) center dot w(S) + alpha center dot w(S, V \ S), where alpha is an element of [0, 1] is a real number, w(S) is the sum of weights of edges having two endpoints in S, and w(S, V \ S) is the sum of weights of edges having one endpoint in S and the other in V \ 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 subset of V of size k that maximizes (minimizes) (phi G)(S). In this paper, we first show that Max alpha-FCGP with alpha is an element of [1/3, 1] and Min alpha-FCGP with alpha is an element of [0, 1/3] can be solved in time 2 degrees((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(Delta - 1))(k-1) n (O(1))-time algorithm on general graphs and a 2(O(root k log2 k))n(O(1))-time randomized algorithm on apex-minor-free graphs. Moreover, for Max alpha-FCGP with alpha is an element of [1/3, 1] and Min alpha-FCGP with alpha is an element of [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.