TY - JOUR

T1 - On the Parameterized Complexity of Computing Balanced Partitions in Graphs

AU - van Bevern, René

AU - Feldmann, Andreas Emil

AU - Sorge, Manuel

AU - Suchý, Ondřej

PY - 2015/7/8

Y1 - 2015/7/8

N2 - A balanced partition is a clustering of a graph into a given number of equal-sized parts. For instance, the Bisection problem asks to remove at most k edges in order to partition the vertices into two equal-sized parts. We prove that Bisection is FPT for the distance to constant cliquewidth if we are given the deletion set. This implies FPT algorithms for some well-studied parameters such as cluster vertex deletion number and feedback vertex set. However, we show that Bisection does not admit polynomial-size kernels for these parameters. For the VertexBisection problem, vertices need to be removed in order to obtain two equal-sized parts. We show that this problem is FPT for the number of removed vertices k if the solution cuts the graph into a constant number c of connected components. The latter condition is unavoidable, since we also prove that VertexBisection is W[1]-hard w.r.t. (k,c). Our algorithms for finding bisections can easily be adapted to finding partitions into d equal-sized parts, which entails additional running time factors of nO(d). We show that a substantial speed-up is unlikely since the corresponding task is W[1]-hard w.r.t. d, even on forests of maximum degree two. We can, however, show that it is FPT for the vertex cover number.

AB - A balanced partition is a clustering of a graph into a given number of equal-sized parts. For instance, the Bisection problem asks to remove at most k edges in order to partition the vertices into two equal-sized parts. We prove that Bisection is FPT for the distance to constant cliquewidth if we are given the deletion set. This implies FPT algorithms for some well-studied parameters such as cluster vertex deletion number and feedback vertex set. However, we show that Bisection does not admit polynomial-size kernels for these parameters. For the VertexBisection problem, vertices need to be removed in order to obtain two equal-sized parts. We show that this problem is FPT for the number of removed vertices k if the solution cuts the graph into a constant number c of connected components. The latter condition is unavoidable, since we also prove that VertexBisection is W[1]-hard w.r.t. (k,c). Our algorithms for finding bisections can easily be adapted to finding partitions into d equal-sized parts, which entails additional running time factors of nO(d). We show that a substantial speed-up is unlikely since the corresponding task is W[1]-hard w.r.t. d, even on forests of maximum degree two. We can, however, show that it is FPT for the vertex cover number.

KW - Bisection

KW - Cliquewidth

KW - NP-hard problems

KW - Problem kernelization

KW - Treewidth reduction

UR - http://www.scopus.com/inward/record.url?scp=84937191674&partnerID=8YFLogxK

U2 - 10.1007/s00224-014-9557-5

DO - 10.1007/s00224-014-9557-5

M3 - Article

AN - SCOPUS:84937191674

VL - 57

JO - Theory of Computing Systems

JF - Theory of Computing Systems

SN - 1432-4350

IS - 1

ER -