The Homogeneous Set Sandwich Problem

A homogeneous set is a non-trivial module of a graph, i.e. a nonempty, non-unitary, proper subset of a graph’s vertices such that all its elements present exactly the same outer neighborhood. Given two graphs G1(V, E1), G2(V, E2), the Homogeneous Set Sandwich Problem (HSSP) asks whether there exists a sandwich graph GS(V, ES), E1 ⊆ ES ⊆ E2, which has a homogeneous set. In 2001, Tang et al. [15] published an all-fast O(n42) algorithm which was recently proven wrong [5], so that the HSSP’s known upper bound would have been reset thereafter at former O(n) determined by Cerioli et al. [1] in 1998. We present, notwithstanding, new deterministic algorithms which have it established at O(n log mn ). We give as well two even faster O(n ) randomized algorithms, whose simplicity might lend them didactic usefulness. We believe that, besides providing efficient easy-to-implement procedures to solve it, the study of these new approaches allows a fairly thorough understanding of the problem.