Faster Deterministic and Randomized Algorithms on the Homogeneous Set Sandwich Problem

A homogeneous set is a non-trivial, 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), we consider the problem of finding a sandwich graph Gs(V, ES), with E1 ⊆ ES ⊆ E2, which contains a homogeneous set, in case such a graph exists. This is called the Homogeneous Set Sandwich Problem (HSSP). We give an O(n) deterministic algorithm, which updates the known upper bounds for this problem, and an O(n) Monte Carlo algorithm as well. Both algorithms, which share the same underlying idea, are quite easy to be implemented on the computer.