17 research outputs found
Nonbipartite Dulmage-Mendelsohn Decomposition for Berge Duality
The Dulmage-Mendelsohn decomposition is a classical canonical decomposition
in matching theory applicable for bipartite graphs, and is famous not only for
its application in the field of matrix computation, but also for providing a
prototypal structure in matroidal optimization theory. The Dulmage-Mendelsohn
decomposition is stated and proved using the two color classes, and therefore
generalizing this decomposition for nonbipartite graphs has been a difficult
task. In this paper, we obtain a new canonical decomposition that is a
generalization of the Dulmage-Mendelsohn decomposition for arbitrary graphs,
using a recently introduced tool in matching theory, the basilica
decomposition. Our result enables us to understand all known canonical
decompositions in a unified way. Furthermore, we apply our result to derive a
new theorem regarding barriers. The duality theorem for the maximum matching
problem is the celebrated Berge formula, in which dual optimizers are known as
barriers. Several results regarding maximal barriers have been derived by known
canonical decompositions, however no characterization has been known for
general graphs. In this paper, we provide a characterization of the family of
maximal barriers in general graphs, in which the known results are developed
and unified
Constrained Min-Cut replication for K-Way hypergraph partitioning
Cataloged from PDF version of article.Replication is a widely-used technique in information retrieval and database systems for providing fault tolerance and reducing parallelization and processing costs. Combinatorial models based on hypergraph partitioning are proposed for various problems arising in information retrieval and database systems. We consider the possibility of using vertex replication to improve the quality of hypergraph partitioning. In this study, we focus on the constrained min-cut replication (CMCR) problem, where we are initially given a maximum replication capacity and a K-way hypergraph partition with an initial imbalance ratio. The objective in the CMCR problem is finding the optimal vertex replication sets for each part of the given partition such that the initial cut size of the partition is minimized, where the initial imbalance is either preserved or reduced under the given replication capacity constraint. In this study, we present a complexity analysis of the CMCR problem and propose a model based on a unique blend of coarsening and integer linear programming (ILP) schemes. This coarsening algorithm is derived from a novel utilization of the Dulmage-Mendelsohn decomposition. Experiments show that the ILP formulation coupled with the Dulmage-Mendelsohn decomposition-based coarsening provides high quality results in practical execution times for reducing the cut size of a given K-way hypergraph partition. © 2014 INFORMS