The Unbalanced Connected Subgraph Problem

A large class of decision and optimization problems can be defined as finding a connected sub graph in a graph while satisfying certain requirements. This paper studies the Unbalanced Connected Subgraphs problem, referred to as UBCS problem. Given a graph with nodes belonging to different categories, the objective of UBCS problem is to find the maximum connected subgraph in which the number and proportion of nodes in certain categories can meet given requirements, respectively. This problem has many real-world applications, for example, to find the largest subgraph of a social network in which the proportion of people who likes a specific product is not less than a given value. This paper introduces the formal definition of UBCS problem, studies its computational complexity and proves that it is NP-hard in planar graphs. A mixed integer programming model based on network flow is proposed to formulate this problem. Moreover, a simple and fast heuristic algorithm is designed to solve UBCS problem in large-scale sparse graphs. Through numerical experiments, the advantages and disadvantages of the heuristic are verified and compared with the optimization solver IBM ILOG CPLXE.