J. Appl. Math. & Computing Vol. 19(2005), No. 1 - 2, pp. 77 - 92 AN EFFICIENT PRAM ALGORITHM FOR MAXIMUM-WEIGHT INDEPENDENT SET ON PERMUTATION GRAPHS ANITA SAHA, MADHUMANGAL PAL ∗ AND TAPAN K. PAL Abstract. An efficient parallel algorithm is presented to find a maximum weight independent set of a permutation graph which takes O(log n) time using O(n 2 / log n) processors on an EREW PRAM, provided the graph has at most O(n) maximal independent sets. The best known parallel algorithm takes O(log 2 n) time and O(n 3 / log n) processors on a CREW PRAM. AMS Mathematics Subject Classification : 05C69, 05C85. Key words and phrases : Design and analysis of algorithms, parallel algo- rithms, independent set, permutation graph. 1. Introduction An undirected graph G =(V,E) with vertices V = {1, 2,...,n} is called a permutation graph if there exists a permutation π on {1, 2,...,n} such that for all i, j ∈ V , (i − j )(π −1 (i) − π −1 (j )) < 0 if and only if i and j are joined by an edge in G [7]. Graphically, the vertices 1, 2, ... , n are drawn in order on a line and π(1),π(2),... ,π(n) on a parallel line such that for each i ∈ V , i is directly above π(i). Next, for each i ∈ V , a line segment is drawn from i on the upper line to i on the lower line. Then, there is an edge (i, j ) in G if and only if the line segment for i intersects the line segment for j . As an illustration, a permutation graph with its permutation representation is considered in Figure 1. A large amount of works on permutation graphs are done by several scholars [1, 3, 10, 11, 12, 13, 15, 16, 17, 19, 20]. A subset of the vertices of a graph G =(V,E) is an independent set if no two vertices in this subset are adjacent. An Received July 20, 2004. Revised November 28, 2004. * Corresponding author. c  2005 Korean Society for Computational & Applied Mathematics and Korean SIGCAM. 77