Universal number partition problem with divisibility

This note examines the likelihood of packing two identical one dimensional shelves of integer length L by items whose individual lengths are divisors of L, given that their combined length sums-up to 2L. We compute the number of packing failures and packing successes for integer shelve lengths L, 1 ≤ L ≤ 1000, by implementing a dynamic programming scheme using a problem specific "boundedness property". The computational results indicate that the likelihood of a packing failure is very rare. We observe that the existence of packing failures is tied to the number of divisors of L and prove that the number of divisors has to be at least 8 for a packing failure to exist.

Download Full-text

The d-Subtree Partition Problem

Chinese Journal of Computers ◽

10.3724/sp.j.1016.2010.00652 ◽

2010 ◽

Vol 33 (4) ◽

pp. 652-665

Author(s):

Yan-Guang CAI ◽

Yun ZHANG ◽

Ji-Xin QIAN

Keyword(s):

Partition Problem

Download Full-text

A Level Set Method for the Dirichlet k-Partition Problem

Journal of Scientific Computing ◽

10.1007/s10915-020-01368-w ◽

2021 ◽

Vol 86 (1) ◽

Author(s):

Kwunlun Chu ◽

Shingyu Leung

Keyword(s):

Level Set Method ◽

Level Set ◽

Partition Problem

Download Full-text

A new multi-level algorithm for balanced partition problem on large scale directed graphs

Advances in Aerodynamics ◽

10.1186/s42774-021-00074-x ◽

2021 ◽

Vol 3 (1) ◽

Author(s):

Xianyue Li ◽

Yufei Pang ◽

Chenxia Zhao ◽

Yang Liu ◽

Qingzhen Dong

Keyword(s):

Large Scale ◽

Directed Graphs ◽

Vlsi Design ◽

Graph Partition ◽

Partition Problem ◽

Multi Level ◽

The Stability ◽

Recursive Partition ◽

Balanced Partition ◽

Partition Method

AbstractGraph partition is a classical combinatorial optimization and graph theory problem, and it has a lot of applications, such as scientific computing, VLSI design and clustering etc. In this paper, we study the partition problem on large scale directed graphs under a new objective function, a new instance of graph partition problem. We firstly propose the modeling of this problem, then design an algorithm based on multi-level strategy and recursive partition method, and finally do a lot of simulation experiments. The experimental results verify the stability of our algorithm and show that our algorithm has the same good performance as METIS. In addition, our algorithm is better than METIS on unbalanced ratio.

Download Full-text