TY - JOUR
T1 - An enhanced MILP-based branch-and-price approach to modularity density maximization on graphs
AU - Sato, Keisuke
AU - Izunaga, Yoichi
N1 - Funding Information:
The authors thank two anonymous reviewers for their valuable comments on the earlier version of this paper. They also thank Alberto Costa, for sharing the instance data files with them, and Rafael de Santiago, for providing them with heuristic solution data of the instances.
Publisher Copyright:
© 2018 Elsevier Ltd
PY - 2019/6
Y1 - 2019/6
N2 - For clustering of an undirected graph, this paper presents an exact algorithm for the maximization of modularity density, a more complicated criterion that overcomes drawbacks of the well-known modularity metric. The problem can be interpreted as the set-partitioning problem, a problem typically solved with an integer linear programming (ILP) formulation. We provide a branch-and-price framework for solving this ILP, i.e., column generation combined with branch-and-bound. Most importantly, we formulate the column generation subproblem to be solved repeatedly as a simpler mixed integer linear programming (MILP) problem. Acceleration techniques called the set-packing relaxation and the multiple-cutting-planes-at-a-time combined with the MILP formulation enable us to optimize the modularity density for famous test instances including ones with over 100 vertices in around four minutes on a PC. Our solution method is deterministic and the computation time is not affected by any stochastic behavior. For one of the instances, column generation at the root node of the branch-and-bound tree provides a fractional upper bound solution and our algorithm finds an integral optimal solution after branching.
AB - For clustering of an undirected graph, this paper presents an exact algorithm for the maximization of modularity density, a more complicated criterion that overcomes drawbacks of the well-known modularity metric. The problem can be interpreted as the set-partitioning problem, a problem typically solved with an integer linear programming (ILP) formulation. We provide a branch-and-price framework for solving this ILP, i.e., column generation combined with branch-and-bound. Most importantly, we formulate the column generation subproblem to be solved repeatedly as a simpler mixed integer linear programming (MILP) problem. Acceleration techniques called the set-packing relaxation and the multiple-cutting-planes-at-a-time combined with the MILP formulation enable us to optimize the modularity density for famous test instances including ones with over 100 vertices in around four minutes on a PC. Our solution method is deterministic and the computation time is not affected by any stochastic behavior. For one of the instances, column generation at the root node of the branch-and-bound tree provides a fractional upper bound solution and our algorithm finds an integral optimal solution after branching.
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U2 - 10.1016/j.cor.2018.01.012
DO - 10.1016/j.cor.2018.01.012
M3 - Article
AN - SCOPUS:85041286159
VL - 106
SP - 236
EP - 245
JO - Computers and Operations Research
JF - Computers and Operations Research
SN - 0305-0548
ER -