New Approximation to the One-sided Radial Crossing Minimization

In this paper, we study a crossing minimization problem in a radial drawing of a graph. Radial drawings have strong application in social network visualization, for example, displaying centrality indices of actors [20]. The main problem of this paper is called the one-sided radial crossing minimization between two concentric circles, named orbits, where the positions of vertices in the outer orbit are fixed. The main task of the problem is to find the vertex ordering of the free orbit and edge routing between two orbits in order to minimize the number of edge crossings. The problem is known to be NP-hard [1], and the first polynomial time 15-approximation algorithm was presented in [9]. In this paper, we present a new 3α-approximation algorithm for the case when the free orbit has no leaf vertex, where α represents the approximation ratio of the one-sided crossing minimization problem in a horizontal drawing. Using the best known result of α =1.4664 [13], our new algorithm can achieve 4.3992-approximation. Submitted: September 2008 Reviewed: February 2009 Revised: March 2009 Accepted: April 2009 Final: May 2009 Published: June 2009 Article type: Regular paper Communicated by: G. Liotta Research supported by a Scientific Grant in Aid from the Ministry of Education, Science, Sports and Culture of Japan. E-mail addresses: [email protected] (Seok-Hee Hong) [email protected] (Hiroshi Nagamochi) 180 Hong and Nagamochi Approximating Radial Crossing Minimization