Efficient envelopment algorithms on the Homogeneous Set Sandwich Problem

A homogeneous set is a non-trivial module of a graph, i.e. a non-unitary, proper subset H of a graph’s vertices such that all vertices in H have the same neighbors outside H . Given two graphs G1(V,E1), G2(V,E2), the Homogeneous Set Sandwich Problem asks whether there exists a sandwich graph GS(V,ES), with E1 ⊆ ES ⊆ E2, which contains a homogeneous set. Two are the algorithms which stand for the state-ofthe-art in this problem, allowing it to be solved in O ` Min{n3n,m1m2} ́ time. Besides, there is also an yes-biased O(n) Monte Carlo algorithm, which solves the problem efficiently, albeit from a probabilistic approach. In this paper, we give two easy-to-implement O(n log n) deterministic algorithms, which henceforth constitute the last word on this problem, and an O(n) Las Vegas algorithm, which turns out to be our recommended practical choice.