JOURNAL OF ALGORITHMS 6, 393-399 (1985) Lower Bounds for Combinatorial Problems on Graphs HIROWKI NAKAYAMA Kyoto Works of Mitsubishi Electric Corp., Nagaokakyo, 617 Japan AND TAKAO NISHIZEKI AND NOBUJI SAITO Department of Electrical Communications, Faculty of Engineering, Tohoku University, Sendai 980. Japan Received October 7.1983; accepted March 1,1984 Nontrivial lower bounds are given for the computation time of various combina- torial problems on graphs under a linear or algebraic decision tree model. An SI( n310g n) bound is obtained for a “travelling salesman problem” on a weighted complete graph of n vertices. Moreover it is shown that the same bound is valid for the “subgraph detection problem” with respect to property P if P is hereditary and determined by components. Thus an Q(n310gn) bound is established in a unified way for a rather large class of problems. 8 1985 Academic PRSS, hc. 1. INTRODUCTION A number of results have been obtained on the upper bounds for combinatorial problems. In contrast, nonlinear lower bounds have been established only for a limited number of problems [l]. An Q(m*) lower bound has been obtained for the m-dimensional knapsack problem under a linear or algebraic decision tree model [2,4,6,8]. Given a set X of m real numbers xi, x2,. . . , x, together with target t, the problem asks whether there is some selection from among the numbers that totals exactly 1. In this case all the subsets of X are feasible solutions of the problem, and there is 393 Ol%-6774/85 $3.00 Copyright Q 1985 by Academic Press. Inc. All rights of reproduction in any form reserved.