### Abstract

In the paper, we describe a polynomial time algorithm that, for every input graph, either outputs the minimum bisection of the graph or halts without output. More importantly, we show that the algorithm chooses the former course with high probability for many natural classes of graphs. In particular, for every fixed d≧3, all sufficiently large n and all b=o(n^{ 1-1/[(d+1)/2]}), the algorithm finds the minimum bisection for almost all d-regular labelled simple graphs with 2 n nodes and bisection width b. For example, the algorithm succeeds for almost all 5-regular graphs with 2 n nodes and bisection width o(n^{ 2/3}). The algorithm differs from other graph bisection heuristics (as well as from many heuristics for other NP-complete problems) in several respects. Most notably: (i) the algorithm provides exactly the minimum bisection for almost all input graphs with the specified form, instead of only an approximation of the minimum bisection, (ii) whenever the algorithm produces a bisection, it is guaranteed to be optimal (i.e., the algorithm also produces a proof that the bisection it outputs is an optimal bisection), (iii) the algorithm works well both theoretically and experimentally, (iv) the algorithm employs global methods such as network flow instead of local operations such as 2-changes, and (v) the algorithm works well for graphs with small bisections (as opposed to graphs with large bisections, for which arbitrary bisections are nearly optimal).

Original language | English (US) |
---|---|

Pages (from-to) | 171-191 |

Number of pages | 21 |

Journal | Combinatorica |

Volume | 7 |

Issue number | 2 |

DOIs | |

State | Published - Jun 1 1987 |

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### All Science Journal Classification (ASJC) codes

- Discrete Mathematics and Combinatorics
- Computational Mathematics

### Cite this

*Combinatorica*,

*7*(2), 171-191. https://doi.org/10.1007/BF02579448

}

*Combinatorica*, vol. 7, no. 2, pp. 171-191. https://doi.org/10.1007/BF02579448

**Graph bisection algorithms with good average case behavior.** / Bui, Thang Nguyen; Chaudhuri, S.; Leighton, F. T.; Sipser, M.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Graph bisection algorithms with good average case behavior

AU - Bui, Thang Nguyen

AU - Chaudhuri, S.

AU - Leighton, F. T.

AU - Sipser, M.

PY - 1987/6/1

Y1 - 1987/6/1

N2 - In the paper, we describe a polynomial time algorithm that, for every input graph, either outputs the minimum bisection of the graph or halts without output. More importantly, we show that the algorithm chooses the former course with high probability for many natural classes of graphs. In particular, for every fixed d≧3, all sufficiently large n and all b=o(n 1-1/[(d+1)/2]), the algorithm finds the minimum bisection for almost all d-regular labelled simple graphs with 2 n nodes and bisection width b. For example, the algorithm succeeds for almost all 5-regular graphs with 2 n nodes and bisection width o(n 2/3). The algorithm differs from other graph bisection heuristics (as well as from many heuristics for other NP-complete problems) in several respects. Most notably: (i) the algorithm provides exactly the minimum bisection for almost all input graphs with the specified form, instead of only an approximation of the minimum bisection, (ii) whenever the algorithm produces a bisection, it is guaranteed to be optimal (i.e., the algorithm also produces a proof that the bisection it outputs is an optimal bisection), (iii) the algorithm works well both theoretically and experimentally, (iv) the algorithm employs global methods such as network flow instead of local operations such as 2-changes, and (v) the algorithm works well for graphs with small bisections (as opposed to graphs with large bisections, for which arbitrary bisections are nearly optimal).

AB - In the paper, we describe a polynomial time algorithm that, for every input graph, either outputs the minimum bisection of the graph or halts without output. More importantly, we show that the algorithm chooses the former course with high probability for many natural classes of graphs. In particular, for every fixed d≧3, all sufficiently large n and all b=o(n 1-1/[(d+1)/2]), the algorithm finds the minimum bisection for almost all d-regular labelled simple graphs with 2 n nodes and bisection width b. For example, the algorithm succeeds for almost all 5-regular graphs with 2 n nodes and bisection width o(n 2/3). The algorithm differs from other graph bisection heuristics (as well as from many heuristics for other NP-complete problems) in several respects. Most notably: (i) the algorithm provides exactly the minimum bisection for almost all input graphs with the specified form, instead of only an approximation of the minimum bisection, (ii) whenever the algorithm produces a bisection, it is guaranteed to be optimal (i.e., the algorithm also produces a proof that the bisection it outputs is an optimal bisection), (iii) the algorithm works well both theoretically and experimentally, (iv) the algorithm employs global methods such as network flow instead of local operations such as 2-changes, and (v) the algorithm works well for graphs with small bisections (as opposed to graphs with large bisections, for which arbitrary bisections are nearly optimal).

UR - http://www.scopus.com/inward/record.url?scp=51249170866&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=51249170866&partnerID=8YFLogxK

U2 - 10.1007/BF02579448

DO - 10.1007/BF02579448

M3 - Article

VL - 7

SP - 171

EP - 191

JO - Combinatorica

JF - Combinatorica

SN - 0209-9683

IS - 2

ER -