Closed form of the response function in FDH technologies: Theory, computation and application

2020 ◽  
Vol 54 (1) ◽  
pp. 53-68 ◽  
Author(s):  
Amin Mostafaee ◽  
Majid Soleimani-Damaneh
Keyword(s):  
Performance Analysis ◽  
Closed Form ◽  
Response Function ◽  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Finite Convergence ◽  
Real World Data ◽  
Scale Elasticity ◽  
Elasticity Measurement

Response function (RF), which gives the value of maximum feasible outputs in response to changing the inputs, has a crucial role in performance analysis and scale elasticity measurement. In this paper, a polynomial-time algorithm is provided which is able to obtain the closed form of the RF under (nonconvex) FDH productions technologies. Finite convergence of the presented algorithm is proved; and it is established that the algorithm is polynomial-time from a complexity standpoint. Moreover, an application of the proposed procedure with real-world data accompanying some experiment-based computational discussions are given.

scholarly journals Reconstructing tumor evolutionary histories and clone trees in polynomial-time with SubMARine

2021 ◽  
Vol 17 (1) ◽  
pp. e1008400
Author(s):  
Linda K. Sundermann ◽  
Jeff Wintersinger ◽  
Gunnar Rätsch ◽  
Jens Stoye ◽  
Quaid Morris
Keyword(s):  
Polynomial Time ◽  
Perfect Match ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Ancestral Reconstruction ◽  
Real World Data ◽  
Cancer Dataset ◽  
Reconstruction Methods ◽  
Partial Clone ◽  
Single Clone

Tumors contain multiple subpopulations of genetically distinct cancer cells. Reconstructing their evolutionary history can improve our understanding of how cancers develop and respond to treatment. Subclonal reconstruction methods cluster mutations into groups that co-occur within the same subpopulations, estimate the frequency of cells belonging to each subpopulation, and infer the ancestral relationships among the subpopulations by constructing a clone tree. However, often multiple clone trees are consistent with the data and current methods do not efficiently capture this uncertainty; nor can these methods scale to clone trees with a large number of subclonal populations. Here, we formalize the notion of a partially-defined clone tree (partial clone tree for short) that defines a subset of the pairwise ancestral relationships in a clone tree, thereby implicitly representing the set of all clone trees that have these defined pairwise relationships. Also, we introduce a special partial clone tree, the Maximally-Constrained Ancestral Reconstruction (MAR), which summarizes all clone trees fitting the input data equally well. Finally, we extend commonly used clone tree validity conditions to apply to partial clone trees and describe SubMARine, a polynomial-time algorithm producing the subMAR, which approximates the MAR and guarantees that its defined relationships are a subset of those present in the MAR. We also extend SubMARine to work with subclonal copy number aberrations and define equivalence constraints for this purpose. Further, we extend SubMARine to permit noise in the estimates of the subclonal frequencies while retaining its validity conditions and guarantees. In contrast to other clone tree reconstruction methods, SubMARine runs in time and space that scale polynomially in the number of subclones. We show through extensive noise-free simulation, a large lung cancer dataset and a prostate cancer dataset that the subMAR equals the MAR in all cases where only a single clone tree exists and that it is a perfect match to the MAR in most of the other cases. Notably, SubMARine runs in less than 70 seconds on a single thread with less than one Gb of memory on all datasets presented in this paper, including ones with 50 nodes in a clone tree. On the real-world data, SubMARine almost perfectly recovers the previously reported trees and identifies minor errors made in the expert-driven reconstructions of those trees. The freely-available open-source code implementing SubMARine can be downloaded at https://github.com/morrislab/submarine.

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.

A polynomial-time algorithm for designing FIR filters with power-of-two coefficients

2002 ◽  
Vol 50 (8) ◽  
pp. 1935-1941 ◽  
Author(s):  
Dongning Li ◽  
Yong Ching Lim ◽  
Yong Lian ◽  
Jianjian Song
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Fir Filters
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