Parameterized algorithms for (r, l)-Partization
We consider the (r, l)-Partization problem of finding a set of at most k vertices whose deletion results in a graph that can be partitioned into r independent sets and l cliques. Restricted to perfect graphs and split graphs, we describe sequacious fixed-parameter tractability results for (r, 0)-Partization, parameterized by k and r. For (r, l)-Partization where r+l = 2, we describe an O*(2k) algorithm for perfect graphs. We then study the parameterized complexity hardness of a generalization of the Above Guarantee Vertex Cover by a reduction from (r, l)-Partization.
Generalized above guarantee vertex cover and r-partization
Vertex cover and odd cycle transversal are minimum cardinality sets of vertices of a graph whose deletion makes the resultant graph 1-colorable and 2-colorable, respectively. As a natural generalization of these well-studied problems, we consider the Graph r-Partization problem of finding a minimum cardinality set of vertices whose deletion makes the graph r-colorable. We explore further connections to Vertex Cover by introducing Generalized Above Guarantee Vertex Cover, a variant of Vertex Cover defined as: Given a graph G, a clique cover K of G and a non-negative integer k, does G have a vertex cover of size at most k +Σ CεK(
Another disjoint compression algorithm for odd cycle transversal
Given a graph G and an odd cycle transversal T, we describe an elegant O* (2|T|) algorithm for determining whether G has a smaller odd cycle transversal that is disjoint from T. We believe that our algorithm, based on a reduction to VERTEX Cover, is conceptually simpler than the known algorithms for the problem and refines the understanding of the relationship between ODD CYCLE TRANSVERSAL and VERTEX COVER. © 2013 Elsevierc B.V. All rights reserved.
Approximability of Clique Transversal in Perfect Graphs
Given an undirected simple graph G, a set of vertices is an r-clique transversal if it has at least one vertex from every r-clique. Such sets generalize vertex covers as a vertex cover is a 2-clique transversal. Perfect graphs are a well-studied class of graphs on which a minimum weight vertex cover can be obtained in polynomial time. Further, an r-clique transversal in a perfect graph is also a set of vertices whose deletion results in an (r- 1) -colorable graph. In this work, we study the problem of finding a minimum weight r-clique transversal in a perfect graph. This problem is known to be NP-hard for r≥ 3 and admits a straightforward r-approximation algorithm. We describe two different r+12-approximation algorithms for the problem. Both the algorithms are based on (different) linear programming relaxations. The first algorithm employs the primal–dual method while the second uses rounding based on a threshold value. We also show that the problem is APX-hard and describe hardness results in the context of parameterized algorithms and kernelization.