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 π and having edge‐weights that sum exactly to unity is considered. An ω(n³ log n) lower bound is established under the algebraic decision tree model for a property π 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 Nakayama et al. [8], in which the same lower bound was obtained for π that is hereditary on subgraphs and determined by components. Furthermore, an ω(n³) lower bound is shown for π = “clique” that does not satisfy the degree constraint and hence the above result can not be applied.