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.

N. S. Narayanaswamy

Department of Computer Science and Engineering

Compression algorithms

Graph algorithms

Graph G

ODD CYCLE TRANSVERSALS

PARAMETERIZED COMPLEXITY

Vertex cover

Algorithms

Graph theory

IIT Madras is a public technical and research university located in Chennai, Tamil Nadu. Founded in 1959, it is recognised as an Institute of National Importance.

IIT Madras has been ranked as the top engineering institute in India for four years in a row by the National Institutional Ranking Framework of the MHRD

It currently offers undergraduate, postgraduate and research degrees across 16 disciplines in Engineering, Sciences, Humanities and Management. About 596 faculty belonging to science and engineering departments and centres of the Institute are engaged in teaching, research and industrial consultancy.

IIT Madras

Information Processing Letters

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. © 2017, Springer Science+Business Media New York.

Algorithmica

Approximability of Clique Transversal in Perfect Graphs

Given an undirected simple graph G, a subset T of vertices is an r-clique transversal if it has at least one vertex from every r-clique in G. I.e. T is an r-clique transversal if G-S is K r -free. r-clique transversals generalize vertex covers as a vertex cover is a set of vertices whose deletion results in a graph that is K 2-free. Perfect graphs are a well-studied class of graphs on which a minimum vertex cover can be obtained in polynomial time. However, the problem of finding a minimum r-clique transversal is NP-hard even for r=3. As any induced odd length cycle in a perfect graph is a triangle, a triangle-free perfect graph is bipartite. I.e. in perfect graphs, a 3-clique transversal is an odd cycle transversal. In this work, we describe an(r+1/2) -approximation algorithm for r-clique transversal on weighted perfect graphs improving on the straightforward r-approximation algorithm. We then show that 3-Clique Transversal is APX-hard on perfect graphs and it is NP-hard to approximate it within any constant factor better than 4/3 assuming the unique games conjecture. We also show intractability results in the parameterized complexity framework. © 2014 Springer-Verlag Berlin Heidelberg.

Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)

LP approaches to improved approximation for clique transversal in perfect graphs

Another disjoint compression algorithm for odd cycle transversal

Journal	Information Processing Letters
ISSN	00200190
Open Access	No