## An Analysis of Some Heuristics for the Maximum Planar Subgraph Problem (1995)

Venue: | Proc. 6 th Annual ACM-SIAM Symp. on Discrete Algorithms |

Citations: | 4 - 0 self |

### BibTeX

@INPROCEEDINGS{Cimikowski95ananalysis,

author = {Robert Cimikowski},

title = {An Analysis of Some Heuristics for the Maximum Planar Subgraph Problem},

booktitle = {Proc. 6 th Annual ACM-SIAM Symp. on Discrete Algorithms},

year = {1995},

pages = {322--331}

}

### Years of Citing Articles

### OpenURL

### Abstract

Introduction The problem of extracting a maximum planar subgraph from a nonplanar graph, referred to as graph planarization, has important applications in circuit layout, facility layout, and automated graphical display systems [F, TDB]. The problem is NP-hard [LG]; hence, research has focused on heuristics. There are several algorithms for finding maximal planar subgraphs [CHT, CNS, GT, JTS, JM, K, OT]. However, there are graphs (see [CC]) for which the size ratio between two maximal planar subgraphs can be as small as 1=3. Hence, unless some precautions are taken to avoid the extraction of small subgraphs, these heuristics have the potential for poor behavior. In this paper, we analyze the worst-case performance of some heuristics and show that there are graphs which can cause each of them to achieve the 1=3 bound. However, a theoretical analysis of an algorithm's performance is often too pessimistic and somew

### Citations

485 |
Testing for the Consecutive Ones Property, Interval Graphs, and Graph Planarity Using PQ-tree Algorithms
- Booth, Lueker
- 1976
(Show Context)
Citation Context ...anch-and-cut method JM in the empirical analysis. An implementation of the CHT heuristic was unavailable. We describe the other heuristics briefly. 4.1 The Vertex-Addition Heuristic. Booth and Lueker =-=[BL] developed-=- a planarity algorithm based on vertex addition. It embeds one vertex of a graph at each step in an order given by an "st-numbering" of the vertices. After each vertex addition, it may be ne... |

238 | Efficient planarity testing
- Hopcroft, Tarjan
- 1974
(Show Context)
Citation Context ...euristic, thus achieving the desired worst-case size ratio. 3.1 The Path-Embedding Heuristic. The path embedding heuristic [CNS] is based on the linear time planarity algorithm of Hopcroft and Tarjan =-=[HT]-=-. Using depth-first search, an initial cycle is found in a graph G, deleted, and then embedded in the plane. The remainder of G is then decomposed into edge-disjoint paths and an attempt is made to em... |

47 | Graph Theory Applications - Foulds - 1993 |

25 |
O(n 2 ) algorithms for graph planarization
- Jayakumar, Thulasiraman, et al.
- 1989
(Show Context)
Citation Context ...tonian with an optimum solution G 1 \Gamma ff4; 8g; f7; 8gg. G 2 has an optimum solution G 2 \Gamma fu; vg, and G 3 has an optimum solution G 3 \Gamma ff1; 28g; f9; 12gg. G 4 , G 5 , and G 6 are from =-=[JTS]-=-, [K], and [TDB], respectively. 4.5.2 Analysis of Results. The basic performance indicator was solution quality since, for most applications, computational costs are not as critical as optimality. For... |

17 | An O(m log n)-time algorithm for the maximal planar subgraph problem
- Cai, Han, et al.
- 1993
(Show Context)
Citation Context ...f n!1 n+2 3n\Gamma12 = 1=3. 2 We observe that if the edges f1; k + 1g and fk; 2kg are embedded before any other interior edges, the 1=3 ratio is achieved. 3.2 The Edge-Embedding Heuristic. Cai et al. =-=[CHT]-=- proposed a variant of AP which processes an edge rather than a path at a time. The heuristic AE begins by finding a dfs tree with tree edges E T and fronds EF . It then constructs a maximal planar su... |

17 |
An O(n2) maximal planarization algorithm based on PQ-trees
- Kant
- 1992
(Show Context)
Citation Context ...ace HT O(mn) O(mn) CHT O(m log n) O(m) PQ O(n 2 ) O(n 2 ) Inc O(n + mff(m; n)) y O(m) GT O(nm 2 ) O(m) y with incremental planarity testing [L] Table 1: Time and space complexities of the heuristics. =-=[K] where ful-=-l details may be found. 4.2 The Cycle-Packing Heuristic. This method (see [GT]) finds an optimal ordering of vertices along a horizontal "node line" in the plane and tries to embed as many e... |

15 |
An algorithm of maximal planarization of graphs
- Chiba, Nishioka, et al.
(Show Context)
Citation Context ...rtices, edges, or paths are processed, either subgraph is obtainable by the heuristic, thus achieving the desired worst-case size ratio. 3.1 The Path-Embedding Heuristic. The path embedding heuristic =-=[CNS]-=- is based on the linear time planarity algorithm of Hopcroft and Tarjan [HT]. Using depth-first search, an initial cycle is found in a graph G, deleted, and then embedded in the plane. The remainder o... |

13 |
An e#cient graph planarization two-phase heuristic
- Goldschmidt, Takvorian
- 1994
(Show Context)
Citation Context ...T O(nm 2 ) O(m) y with incremental planarity testing [L] Table 1: Time and space complexities of the heuristics. [K] where full details may be found. 4.2 The Cycle-Packing Heuristic. This method (see =-=[GT]) finds an-=- optimal ordering of vertices along a horizontal "node line" in the plane and tries to embed as many edges as possible above and below the line. In the first phase of the heuristic GT , an a... |

8 | Solving the maximum weight planar subgraph problem by branch and cut
- Jünger, Mutzel
- 1993
(Show Context)
Citation Context ...s to a hamiltonian cycle and a 3=4-approximation is guaranteed. Otherwise, a greedy ordering is used, and there is no performance bound. 4.3 The Branch-and-Cut Heuristic. The branchand -cut heuristic =-=[JM]-=- is an exhaustive search algorithm based on linear programming and cutting plane generation, with feasibility bounding performed by a planarity testing algorithm. As planar obstructions are detected d... |

3 |
The sizes of maximal planar, outerplanar, and bipartite planar subgraphs
- Cimikowski, Coppersmith
- 1996
(Show Context)
Citation Context ...he problem is NP-hard [LG]; hence, research has focused on heuristics. There are several algorithms for finding maximal planar subgraphs [CHT, CNS, GT, JTS, JM, K, OT]. However, there are graphs (see =-=[CC]-=-) for which the size ratio between two maximal planar subgraphs can be as small as 1=3. Hence, unless some precautions are taken to avoid the extraction of small subgraphs, these heuristics have the p... |

1 |
Alpha-time algorithms for incremental 100 vertices 200 vertices Size of subgraph obtained Size of subgraph obtained k opt
- Poutr'e
(Show Context)
Citation Context ... with an "empty" graph and adds edges one at a time, discarding an edge if it causes nonplanarity. After each edge addition, a planarity test is performed. Using "incremental" plan=-=arity testing, i.e. [L]-=-, the time complexity is O(n + mff(m; n)), which is roughly O(n +m). 4 (d) 8 7 4 3 2 1 G: (b) 5 6 D: 8 7 6 (1) (1) (1) 5 (1) 4 (1) 3 (1) 2 (1) (1) (4) (3) (2) (1) (a) (2) (1) 1 (c) 8 7 4 3 2 1 5 6 H 1... |