Computational Optimization and Applications, 18, 49–62, 2001 c  2001 Kluwer Academic Publishers. Manufactured in The Netherlands. An Efficient Algorithm for Finding a Maximum Weight k-Independent Set on Trapezoid Graphs MRINMOY HOTA Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721 102, India MADHUMANGAL PAL Department of Mathematics, Midnapore College, Midnapore 721 101, India TAPAN K. PAL Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore 721 102, India Received July 22, 1998; Accepted June 7, 1999 Abstract. The maximum weight k -independent set problem has applications in many practical problems like k -machines job scheduling problem, k -colourable subgraph problem, VLSI design layout and routing problem. Based on DAG (Directed Acyclic Graph) approach, an O(kn 2 ) time sequential algorithm is designed in this paper to solve the maximum weight k -independent set problem on weighted trapezoid graphs. The weights considered here are all non-negative and associated with each of the n vertices of the graph. Keywords: trapezoid graph, independent set, path, network flow problem, combinatorial problems, design of algorithms, analysis of algorithms 1. Introduction Trapezoid graphs were first studied in [1, 2]. The graphs of this particular class are perfect and this class is a superclass of both permutation graphs and interval graphs but subclass of co-comparability graphs [3]. Trapezoid graphs can be recognized in O (n 2 ) time by Ma and Spinrad’s algorithm [15]. For the trapezoid representation of a trapezoid graph we consider two parallel lines called the top channel and the bottom channel. For each i = 1, 2,..., n the trapezoid T i is then defined by four corner points [a i , b i , c i , d i ], where a i ≤ b i and c i ≤ d i with a i , b i lying on the top channel and c i , d i lying on the bottom channel. An undirected graph G = (V , E ) with vertex set V ={v 1 ,v 2 ,...,v n } and edge set E ={e 1 , e 2 ,..., e m } is called a trapezoid graph if a trapezoid representation can be obtained such that (i) each vertex v i in V corresponds to a trapezoid T i and (ii) (v i ,v j ) is an edge of G if and only if the trapezoids T i and T j corresponding to the vertices v i and v j intersect each other.