LOWER BOUNDS ON TIME COMPLEXITY FOR SOME SUBGRAPH DETECTION PROBLEMS.

Nontrivial lower bounds of the time complexity are given for a class of problems on graphs. Specifically, given a complete graph with weighted edges, the problem of finding a subgraph satisfying a property pi and having edge-weights that sum exactly to unity is considered. An OMEGA (n**3 log n) lower bound is established under the algebraic decision tree model for a property pi that satisfies the degree constraint and is closed with respect to isomorphism, where n is the number of vertices in an input graph. This result is a proper generalization of that of H. Nakayama et al. , in which the same lower bound was obtained for pi that is hereditary on subgraphs and determined by components. Furthermore, an OMEGA (n**3) lower bound is shown for pi equals 'clique' that does not satisfy the degree constraint and hence the above result can not be applied.