Algorithmic aspects of homophyly of networks

Abstract

We investigate the algorithmic problems of the homophyly phenomenon in networks. Given an undirected graph G = (V, E) and a vertex coloring c : V → {1, 2, · · · , k} of G, we say that a vertex v ∈ V is happy if v shares the same color with all its neighbors, and unhappy, otherwise, and that an edge e ∈ E is happy, if its two endpoints have the same color, and unhappy, otherwise. Supposing c is a partial vertex coloring of G, we define the Maximum Happy Vertices problem (MHV, for short) as to color all the remaining vertices such that the number of happy vertices is maximized, and the Maximum Happy Edges problem (MHE, for short) as to color all the remaining vertices such that the number of happy edges is maximized.Let k be the number of colors allowed in the problems. We show that both MHV and MHE can be solved in polynomial time if k = 2, and that both MHV and MHE are NP-hard if k ≥ 3. We devise a max{1/k, Ω(∆ −3 )}-approximation algorithm for the MHV problem, where ∆ is the maximum degree of vertices in the input graph, and a 1/2-approximation algorithm for the MHE problem. This is the first theoretical progress of these two natural and fundamental new problems.

On Happy Colorings, Cuts, and Structural Parameterizations

“…Recently, MHV and MHE have attracted a lot of attention and were studied from parameterized [1,2,3,7,19] and approximation [24,25,23,22] points of view as well as from experimantal perspective [18]. Further, dozens of algorithms for the classical Multiway Cut problem have been considered as well, which is the complement of a special case of MHE.…”

Approximation Algorithms for Vertex Happiness

“…For the MHV problem, Zhang and Li [24] proved that it is polynomial time solvable for k = 2 and it becomes NP-hard for k ≥ 3; for k ≥ 3, they presented two approximation algorithms: a greedy algorithm with an approximation ratio of 1 k , and an Ω( 1 ∆ 3 )-approximation based on a subset-growth technique, where ∆ is the maximum vertex degree of the input graph. Recently, Zhang et al [23] presented an improved algorithm with an approximation ratio of 1 ∆+1 based on a combination of randomized LP rounding techniques.…”

“…The multiway uncut problem seems only studied by Langberg et al [15], who presented a 0.8535-approximation based on an LP relaxation. When generalizing the multiway uncut problem to pre-assign multiple terminals in a part of the vertex partition, it becomes the recently studied maximum happy edges (MHE) problem [24]. It is important to note that MHE is not a special case of the Sup-MP problem, but a special case of the Sup-ML problem.…”

“…In a recently studied vertex-coloring problem by Zhang and Li [24], one is given an undirected graph G = (V, E) with a non-negative weight w(v) for each vertex v ∈ V , a color set C = {1, 2, . .…”

A simple and effective algorithm for the maximum happy vertices problem

“…The MHV problem and the concept of "happiness" related to vertices have been proposed in Zhang and Li (2015). A vertex is considered happy if all its neighbors are of the same color.…”