A Polynomial-Time Algorithm to Find the Shortest Cycle Basis of a Graph

1987 ◽  
Vol 16 (2) ◽  
pp. 358-366 ◽  
Author(s):  
J. D. Horton
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Cycle Basis

scholarly journals Interchangeability of Relevant Cycles in Graphs

10.37236/1494 ◽  
2000 ◽  
Vol 7 (1) ◽  
Author(s):  
Petra M. Gleiss ◽  
Josef Leydold ◽  
Peter F. Stadler
Keyword(s):  
Lower Bounds ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Upper And Lower Bounds ◽  
Cycle Basis ◽  
Cycle Bases

The set ${\cal R}$ of relevant cycles of a graph $G$ is the union of its minimum cycle bases. We introduce a partition of ${\cal R}$ such that each cycle in a class ${\cal W}$ can be expressed as a sum of other cycles in ${\cal W}$ and shorter cycles. It is shown that each minimum cycle basis contains the same number of representatives of a given class ${\cal W}$. This result is used to derive upper and lower bounds on the number of distinct minimum cycle bases. Finally, we give a polynomial-time algorithm to compute this partition.

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

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