Solving Partition Problems

The minimum k-partition (MkP) problem is the problem§of partitioning the set of vertices of a graph into k§disjoint subsets so as to minimize the total weight§of the edges joining vertices in the same partition.§The main contribution is the design and§implementation of a novel iterative clustering§heuristic (ICH) based on semide nite programming to nd feasible solutions for the MkP problem. We§compare ICH to the hyperplane rounding techniques,§and the computational results support the conclusion§that ICH consistently provides better feasible§solutions for the MkP problem. We use ICH in a§branch-and-cut algorithm to provide feasible§solutions at each node of the branch-and-bound tree.§The branch-and-cut algorithm computes globally§optimal solutions for dense graphs with up to 60§vertices, for grid graphs with up to 100 vertices,§and for different values of k, providing the best§exact approach to date for k 2. The minimum k-partition (MkP) problem is the problem§of partitioning the set of vertices of a graph into k§disjoint subsets so as to minimize the total weight§of the edges joining vertices in the same partition.§The main contribution is the design and§implementation of a novel iterative clustering§heuristic (ICH) based on semide nite programming to nd feasible solutions for the MkP problem. We§compare ICH to the hyperplane rounding techniques,§and the computational results support the conclusion§that ICH consistently provides better feasible§solutions for the MkP problem. We use ICH in a§branch-and-cut algorithm to provide feasible§solutions at each node of the branch-and-bound tree.§The branch-and-cut algorithm computes globally§optimal solutions for dense graphs with up to 60§vertices, for grid graphs with up to 100 vertices,§and for different values of k, providing the best§exact approach to date for k 2.