A (2 + ε)-approximation scheme for minimum domination on circle graphs

The main result of this paper is a (2 + e)-approximation scheme for the minimum dominating set problem on circle graphs. We first present an O(n 2) time 8-approximation algorithm for the minimum dominating set problem on circle graphs and then extend it to a (2+e)-approximation scheme that solves the minimum dominating set problem in time 6_fi+l~ O(n 3 + -g,~ ,,~]. Here n and m are the number of vertices and the number of edges of the circle graph. We then present simple modifications to this algorithm that yield (3 + ~)approximation schemes for the minimum connected dominating set and the minimum total dominating set problems on circle graphs. Keil (Discrete Applied Mathematics, 42 (1993), 51-63) showed that these problems are NP-complete for circle graphs and left open the problem of devising approximation algorithms for them. These are the first O(1)approximation algorithms for domination problems on circle graphs and involve techniques that may lead to new approximation algorithms for other problems like vertex cover and chromatic number on circle graphs. 1 I n t r o d u c t i o n A graph G = (V, E) is called a circle graph if there is a one-to-one correspondence between vertices in V and a set C of chords in a circle such that two vertices in V are adjacent if and only ff the corresponding chords in C intersect. C is called the chord intersection model for G. Equivalently, the vertices of a circle graph can be placed in one-to-one correspondence with the elements of a set I of intervals such that two vertices are adjacent if and only if the corresponding intervals overlap, but neither contains the other. I is called the interval model of the corresponding circle graph. Representations of a circle graph as a graph or as a set of chords or as a set of intervals are equivalent via linear time transformations. So, without loss of generality, in specifying instances of problems, r ]~a r tmen t of Computer Science, University of Iowa, Iowa City, IA 52242, U.S.A. e-marl: damianjo@cs.uiova.edu tDepartment of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai 400076, India. e-maih sriram~math, iitb .ernet. in we assume the availability of the representation that is most convenient. For a graph G = (V,E), a subset V t of V is a dominating set of G if for all u E V either u E V' or u has a neighbor in W. In addition, if no two vertices in W are adjacent, then V' is called an independent dominating set; if the subgraph of G induced by W, denoted G[V'], is connected, then V' is called a connected dominating see if G[V'] has no isolated nodes, then V' is called a total dominating set; and if G[W] is a clique, then W is called a dominating clique. Garey and Johnson [5] mention that problems of finding a minimum cardinality dominating set (MDS), minimum cardinality independent dominating set (MIDS), minimum cardinality connected dominating set (MCDS), minimum cardinality total dominating set (MTDS), and minimum cardinality dominating clique (MDC) are all NP-complete for general graphs. Johnson [3] seems to be the first to identify MDS as an open problem for circle graphs. Keil [4] resolved the complexity of MDS on circle graphs by showing that it is NP-complete. In the same paper, he also showed that MCDS and MTDS are also NP-complete for circle graphs. In that paper, Keil does not investigate the question of approximation algorithms, however he mentions the construction of approximation algorithms for MDS, MCDS, and MTDS as being open. An a-approximation algorithm for a minimization problem is a polynomial time a lgor i thm that guarantees tha t the ratio of the cost of the solution to the optimal (over all instances of the problem) does not exceed a . The main result of this paper is a (2 + ¢)approximation scheme for MDS. We start by presenting an 8-approximation algorithm for MDS tha t runs in O(n 2) time using O(n) space. We then extend this to a (2 + c)-approximation scheme that solves MDS in time O(n 3 + ~n~+lm). Here n and m are the number of vertices and the number of edges of the circle graph. Finally we explain how to modify this algorithm to solve MCDS and MTDS as well.