INFORM4 TION SCIENCES 74,1-B 0993) 1 The Summation and Bottleneck Minimization for Single-Step Searching on Weighted Graphs JU YUAN HSIAO CHUAN YI TANG* zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHG Department of Computer Science, National Tsing Hua University, Hsinchu, Taiwan 300, Republic of China and RUAY SHIUNG CHANG Department of Information M anagement, National Taiwan Institute of Technology Taipei, Taiwan, Republic of China Communicated by Benjamin W. Wah ABSTRACT In this paper, three types of problems for single-step searching weighted graphs are defined. They are the summation cost minimization, the bottleneck cost minimization, and a hybrid to minimize the maximum of the summation cost and the bottleneck cost. All three are shown to be NP-hard but polynomially solvable for trees. The bottleneck minimization is shown to be reducible to the summation minimization problem. 1. INTRODUCTION Given a simple undirected graph G = (V, E), where I/ is the vertex set with II vertices and E is the edge set with m edges, the single- step graph searching problem was defined as follows [l]. Assume that every edge of G is of the same distance. A fugitive, that can move at any speed, is hidden in some edge of G. For each edge of G, a searcher was assigned to clear the edge. Assume that every searcher moves at the same speed and each edge in G can be searched by one searcher in a single step. The single-step * To whom correspondence should be addressed. 0 Elsevier Science Publishing Co., Inc. 1993 655 Avenue of the Americas, New York, NY 10010 0020-0255/93/$6.00