### Abstract

In discrete convex analysis, the scaling and proximity properties for the class of L^{♮}-convex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of integrally convex functions of n variables, we show here that the scaling property only holds when n ≤ 2, while a proximity theorem can be established for any n, but only with an exponential bound. This is, however, sufficient to extend the classical logarithmic complexity result for minimizing a discretely convex function in one dimension to the case of integrally convex functions in two dimensions. Furthermore, we identified a new class of discrete convex functions, called directed integrally convex functions, which is strictly between the classes of L^{♮}-convex and integrally convex functions but enjoys the same scaling and proximity properties that hold for L^{♮}-convex functions.

Original language | English |
---|---|

Title of host publication | 27th International Symposium on Algorithms and Computation, ISAAC 2016 |

Publisher | Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing |

Pages | 57.1-57.13 |

Volume | 64 |

ISBN (Electronic) | 9783959770262 |

DOIs | |

Publication status | Published - 2016 Dec 1 |

Event | 27th International Symposium on Algorithms and Computation, ISAAC 2016 - Sydney, Australia Duration: 2016 Dec 12 → 2016 Dec 14 |

### Other

Other | 27th International Symposium on Algorithms and Computation, ISAAC 2016 |
---|---|

Country | Australia |

City | Sydney |

Period | 16/12/12 → 16/12/14 |

### Keywords

- Discrete convexity
- Discrete optimization
- Proximity theorem
- Scaling algorithm

### ASJC Scopus subject areas

- Software

### Cite this

*27th International Symposium on Algorithms and Computation, ISAAC 2016*(Vol. 64, pp. 57.1-57.13). Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing. https://doi.org/10.4230/LIPIcs.ISAAC.2016.57

**Scaling and proximity properties of integrally convex functions.** / Moriguchi, Satoko; Murota, Kazuo; Tamura, Akihisa; Tardella, Fabio.

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

*27th International Symposium on Algorithms and Computation, ISAAC 2016.*vol. 64, Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing, pp. 57.1-57.13, 27th International Symposium on Algorithms and Computation, ISAAC 2016, Sydney, Australia, 16/12/12. https://doi.org/10.4230/LIPIcs.ISAAC.2016.57

}

TY - GEN

T1 - Scaling and proximity properties of integrally convex functions

AU - Moriguchi, Satoko

AU - Murota, Kazuo

AU - Tamura, Akihisa

AU - Tardella, Fabio

PY - 2016/12/1

Y1 - 2016/12/1

N2 - In discrete convex analysis, the scaling and proximity properties for the class of L♮-convex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of integrally convex functions of n variables, we show here that the scaling property only holds when n ≤ 2, while a proximity theorem can be established for any n, but only with an exponential bound. This is, however, sufficient to extend the classical logarithmic complexity result for minimizing a discretely convex function in one dimension to the case of integrally convex functions in two dimensions. Furthermore, we identified a new class of discrete convex functions, called directed integrally convex functions, which is strictly between the classes of L♮-convex and integrally convex functions but enjoys the same scaling and proximity properties that hold for L♮-convex functions.

AB - In discrete convex analysis, the scaling and proximity properties for the class of L♮-convex functions were established more than a decade ago and have been used to design efficient minimization algorithms. For the larger class of integrally convex functions of n variables, we show here that the scaling property only holds when n ≤ 2, while a proximity theorem can be established for any n, but only with an exponential bound. This is, however, sufficient to extend the classical logarithmic complexity result for minimizing a discretely convex function in one dimension to the case of integrally convex functions in two dimensions. Furthermore, we identified a new class of discrete convex functions, called directed integrally convex functions, which is strictly between the classes of L♮-convex and integrally convex functions but enjoys the same scaling and proximity properties that hold for L♮-convex functions.

KW - Discrete convexity

KW - Discrete optimization

KW - Proximity theorem

KW - Scaling algorithm

UR - http://www.scopus.com/inward/record.url?scp=85010781766&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85010781766&partnerID=8YFLogxK

U2 - 10.4230/LIPIcs.ISAAC.2016.57

DO - 10.4230/LIPIcs.ISAAC.2016.57

M3 - Conference contribution

AN - SCOPUS:85010781766

VL - 64

SP - 57.1-57.13

BT - 27th International Symposium on Algorithms and Computation, ISAAC 2016

PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing

ER -