### Abstract

We introduce a new combinatorial formulation of chromosome physical-mapping by the sampling-without-replacement protocol. In this protocol, which is simple, inexpensive, and has been used to successfully map several organisms, equal-length clones are hybridized against a subset of the clones called probes, which are designed to form a maximal nonoverlapping clone-subset. The output of the protocol is the clone-probe hybridization matrix H. The problem of finding a maximum-likelihood reconstruction of the order of the probes along the chromosome in the presence of false positive and negative hybridization error is equivalent to finding the minimum number of entries of H to change to zeros so that the resulting matrix has at most 2 ones per row, and the consecutive-ones property across rows. This combinatorial problem, which we call 2-Consecutive-Ones Mapping, has a concise integer linear-programming formulation, to which we apply techniques from polyhedral combinatorics. The formulation is unique in that it does not explicitly represent the probe permutation, and in contrast to prior linear-programming approaches, the number of variables is small: in practice, linear in the number of clones. We derive a large class of facet-defining inequalities for the 2-consecutive-ones polytope that we call the augmented k-degree inequalities, and we show that the basic k-degree class can be efficiently separated using bipartite matchings. Computational results with an implementation of the resulting branch-and-cut algorithm applied to both simulated and real data from the complete genome of Aspergillus nidulans show that we can solve many problems to provable optimality and find maps of higher quality than previously possible.

Original language | English (US) |
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Title of host publication | Proceedings of the Annual International Conference on Computational Molecular Biology, RECOMB |

Publisher | ACM |

Pages | 115-123 |

Number of pages | 9 |

State | Published - 1999 |

Externally published | Yes |

Event | Proceedings of the 1999 3rd Annual International Conference on Computational Molecular Biology, RECOMB '99 - Lyon Duration: Apr 11 1999 → Apr 14 1999 |

### Other

Other | Proceedings of the 1999 3rd Annual International Conference on Computational Molecular Biology, RECOMB '99 |
---|---|

City | Lyon |

Period | 4/11/99 → 4/14/99 |

### Fingerprint

### ASJC Scopus subject areas

- Biochemistry, Genetics and Molecular Biology(all)
- Computer Science(all)

### Cite this

*Proceedings of the Annual International Conference on Computational Molecular Biology, RECOMB*(pp. 115-123). ACM.

**Computing physical maps of chromosomes with nonoverlapping probes by branch-and-cut.** / Christof, Thomas; Kececioglu, John D.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Proceedings of the Annual International Conference on Computational Molecular Biology, RECOMB.*ACM, pp. 115-123, Proceedings of the 1999 3rd Annual International Conference on Computational Molecular Biology, RECOMB '99, Lyon, 4/11/99.

}

TY - CHAP

T1 - Computing physical maps of chromosomes with nonoverlapping probes by branch-and-cut

AU - Christof, Thomas

AU - Kececioglu, John D

PY - 1999

Y1 - 1999

N2 - We introduce a new combinatorial formulation of chromosome physical-mapping by the sampling-without-replacement protocol. In this protocol, which is simple, inexpensive, and has been used to successfully map several organisms, equal-length clones are hybridized against a subset of the clones called probes, which are designed to form a maximal nonoverlapping clone-subset. The output of the protocol is the clone-probe hybridization matrix H. The problem of finding a maximum-likelihood reconstruction of the order of the probes along the chromosome in the presence of false positive and negative hybridization error is equivalent to finding the minimum number of entries of H to change to zeros so that the resulting matrix has at most 2 ones per row, and the consecutive-ones property across rows. This combinatorial problem, which we call 2-Consecutive-Ones Mapping, has a concise integer linear-programming formulation, to which we apply techniques from polyhedral combinatorics. The formulation is unique in that it does not explicitly represent the probe permutation, and in contrast to prior linear-programming approaches, the number of variables is small: in practice, linear in the number of clones. We derive a large class of facet-defining inequalities for the 2-consecutive-ones polytope that we call the augmented k-degree inequalities, and we show that the basic k-degree class can be efficiently separated using bipartite matchings. Computational results with an implementation of the resulting branch-and-cut algorithm applied to both simulated and real data from the complete genome of Aspergillus nidulans show that we can solve many problems to provable optimality and find maps of higher quality than previously possible.

AB - We introduce a new combinatorial formulation of chromosome physical-mapping by the sampling-without-replacement protocol. In this protocol, which is simple, inexpensive, and has been used to successfully map several organisms, equal-length clones are hybridized against a subset of the clones called probes, which are designed to form a maximal nonoverlapping clone-subset. The output of the protocol is the clone-probe hybridization matrix H. The problem of finding a maximum-likelihood reconstruction of the order of the probes along the chromosome in the presence of false positive and negative hybridization error is equivalent to finding the minimum number of entries of H to change to zeros so that the resulting matrix has at most 2 ones per row, and the consecutive-ones property across rows. This combinatorial problem, which we call 2-Consecutive-Ones Mapping, has a concise integer linear-programming formulation, to which we apply techniques from polyhedral combinatorics. The formulation is unique in that it does not explicitly represent the probe permutation, and in contrast to prior linear-programming approaches, the number of variables is small: in practice, linear in the number of clones. We derive a large class of facet-defining inequalities for the 2-consecutive-ones polytope that we call the augmented k-degree inequalities, and we show that the basic k-degree class can be efficiently separated using bipartite matchings. Computational results with an implementation of the resulting branch-and-cut algorithm applied to both simulated and real data from the complete genome of Aspergillus nidulans show that we can solve many problems to provable optimality and find maps of higher quality than previously possible.

UR - http://www.scopus.com/inward/record.url?scp=0032642168&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0032642168&partnerID=8YFLogxK

M3 - Chapter

SP - 115

EP - 123

BT - Proceedings of the Annual International Conference on Computational Molecular Biology, RECOMB

PB - ACM

ER -