The minimum k-partition (MkP) problem is a well-known
optimization problem encountered in various applications most
notably in telecommunication and physics. Formulated in the early
1990s by Chopra and Rao, the 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 different partitions.
In this thesis, we design and implement a branch-and-cut algorithm
based on semidefinite programming (SBC) for the MkP problem. We
describe and study the properties of two relaxations of the MkP
problem, the linear programming and the semidefinite programming
relaxations. We then derive a new strengthened relaxation based on
semidefinite programming. This new relaxation provides tighter
bounds compared to the other two discussed relaxations but suffers
in term of computational time. We further devise an iterative
clustering heuristic (ICH), a novel heuristic that finds feasible
solution to the MkP problem and we compare it to the hyperplane
rounding techniques of Goemans and Williamson and Frieze and
Jerrum for k=2 and for k=3 respectively. Our computational
results support the conclusion that ICH provides a better feasible
solution for the MkP. Furthermore, unlike the hyperplane
rounding, ICH remains very effective in the presence of negative
edge weights. Next we describe in detail the design and
implementation of a branch-and-cut algorithm based on semidefinite
programming (SBC) to find optimal solution for the MkP problem.
The ICH heuristic is used in our SBC algorithm to provide feasible
solutions at each node of the branch-and-cut tree. Finally, we
present computational results for the SBC algorithm on several
classes of test instances with k=3, 5, and 7. Complete graphs
with up to 60 vertices and sparse graphs with up to 100 vertices
arising from a physics application were considered.
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:OWTU.10012/2766 |
| Date | January 2007 |
| Creators | Ghaddar, Bissan |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Thesis or Dissertation |
| Format | 473235 bytes, application/pdf |
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