Approximate and exact algorithms for the maximum planar subgraph problem
Finding a planar subgraph with maximum weight in a given nonplanar graph is an NP-hard problem which has applications in automatic graph drawing, facility layout, and the design of electronic circuits. We study the polytope PLS(G) of all planar subgraphs of a given graph G. Among others, all the subdivisions of K{sub 5} and K{sub 3,3} give facet-defining inequalities for this polytope. We give an overview about our investigations on the facial structure of PLS(G). Our theoretical results are applied in the implementation of a branch and cut algorithm whose main components are the branch and cut frame developed by Junger, Reineit and Thienel and an efficient implementation of the Hopcroft-Tarjan planarity testing algorithm by Mutzel. Using this method, we were able to find optimal solutions for several problem instances in the literature for the first time, especially instances coming from automatic graph drawing and facility layout applications. During the computation, our algorithm produces solutions with increasing values, as well as upper bounds for the value of a maximum planar subgraph of decreasing values. So we can stop the computation at any time and get a solution together with a quality guarantee, which says that the value of the found solution is less than P% away from the optimum solution value. There are various heuristics for the maximum planar subgraph problem. Such heuristics usually compute (locally) maximal planar subgraphs in the sense that the addition of any edge to the subgraph destroys planarity. Maximal planar subgraphs can be found in time O({vert_bar}E{vert_bar}log {vert_bar}V{vert_bar}) time by an algorithm by Cai, Han, and Tarjan. Many alternative methods have been proposed in the literature. We close with a report on computational experiments with such methods in comparison to our method.
- OSTI ID:
- 36323
- Report Number(s):
- CONF-9408161-; TRN: 94:009753-0661
- Resource Relation:
- Conference: 15. international symposium on mathematical programming, Ann Arbor, MI (United States), 15-19 Aug 1994; Other Information: PBD: 1994; Related Information: Is Part Of Mathematical programming: State of the art 1994; Birge, J.R.; Murty, K.G. [eds.]; PB: 312 p.
- Country of Publication:
- United States
- Language:
- English
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