TY - JOUR
T1 - Parameterized Algorithms for Graph Partitioning Problems
AU - Shachnai, Hadas
AU - Zehavi, Meirav
N1 - Publisher Copyright:
© 2016, Springer Science+Business Media New York.
PY - 2017/10/1
Y1 - 2017/10/1
N2 - We study a broad class of graph partitioning problems. Each problem is defined by two constants, α1 and α2. The input is a graph G, an integer k and a number p, and the objective is to find a subset U⊆ V of size k, such that α1m1 + α2m2 is at most (or at least) p, where m1, m2 are the cardinalities of the edge sets having both endpoints, and exactly one endpoint, in U, respectively. This class of fixed-cardinality graph partitioning problems (FGPPs) encompasses Max (k, n − k)-Cut, Mink-Vertex Cover, k-Densest Subgraph, and k-Sparsest Subgraph. Our main result is a 4k + o(k)Δk⋅nO(1) time algorithm for any problem in this class, where Δ ≥ 1 is the maximum degree in the input graph. This resolves an open question posed by Bonnet et al. (Proc. International Symposium on Parameterized and Exact Computation, 2013). We obtain faster algorithms for certain subclasses of FGPPs, parameterized by p, or by (k + p). In particular, we give a 4p + o(p)⋅nO(1) time algorithm for Max (k, n − k)-Cut, thus improving significantly the best known pp⋅nO(1) time algorithm by Bonnet et al.
AB - We study a broad class of graph partitioning problems. Each problem is defined by two constants, α1 and α2. The input is a graph G, an integer k and a number p, and the objective is to find a subset U⊆ V of size k, such that α1m1 + α2m2 is at most (or at least) p, where m1, m2 are the cardinalities of the edge sets having both endpoints, and exactly one endpoint, in U, respectively. This class of fixed-cardinality graph partitioning problems (FGPPs) encompasses Max (k, n − k)-Cut, Mink-Vertex Cover, k-Densest Subgraph, and k-Sparsest Subgraph. Our main result is a 4k + o(k)Δk⋅nO(1) time algorithm for any problem in this class, where Δ ≥ 1 is the maximum degree in the input graph. This resolves an open question posed by Bonnet et al. (Proc. International Symposium on Parameterized and Exact Computation, 2013). We obtain faster algorithms for certain subclasses of FGPPs, parameterized by p, or by (k + p). In particular, we give a 4p + o(p)⋅nO(1) time algorithm for Max (k, n − k)-Cut, thus improving significantly the best known pp⋅nO(1) time algorithm by Bonnet et al.
KW - Graph partitioning
KW - Parameterized algorithm
KW - Random separation
KW - Representative family
UR - http://www.scopus.com/inward/record.url?scp=85027881184&partnerID=8YFLogxK
U2 - 10.1007/s00224-016-9706-0
DO - 10.1007/s00224-016-9706-0
M3 - Article
AN - SCOPUS:85027881184
VL - 61
SP - 721
EP - 738
JO - Theory of Computing Systems
JF - Theory of Computing Systems
SN - 1432-4350
IS - 3
ER -