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In this paper, we consider the minimum gap partition problem in an undirected connected graph with nonnegative vertex weights. This problem involves in dividing the given graph into a specified number of connected subgraphs such that the total difference between the largest and the smallest vertex weights in each subgraph is minimized. Namely, let G = (V, E, W) be a given undirected connected graph with vertex set V, edge set E, and nonnegative vertex weight function W: V → R+, and let k ≥ 2 be a given positive integer. The minimum gap partition problem consists of constructing a partition P = V1, V2,. . . , Vk of non-empty and pairwise disjoint subsets of V such that each vertex set Vi, i = 1, 2,. . . , k, induces a connected subgraph GVi of G. The objective of the problem is to find such a partition of V that minimizes the total difference 1≤i≤k maxv∈Viw (v) − minv∈Viw (v). This problem, as many other variants of graph partition problems, is known to be NP-hard problem. We designed an efficient metaheuristic algorithm for finding approximate solution of large-scale instances of this problem. The quality of the proposed algorithm is assessed comparing with the previous algorithms proposed for the problem.
Morsy et al. (Mon,) studied this question.