Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Scheduling problems are fundamental in combinatorial optimization. Much work has been done on approximation algorithms for NP-hard cases, but relatively little is known about exact solutions when some part of the input is a fixed parameter. In 2014, Mnich and Wiese initiated a systematic study in this direction. #R##N#  continue this study and show that several additional cases of fundamental scheduling problems are fixed parameter tractable for some natural parameters. Our main tool is n-fold integer programming, a recent variable dimension technique which  believe to be highly relevant for the parameterized complexity community.  serves to showcase and highlight this technique. #R##N#Specifically,  show the following four scheduling problems to be fixed-parameter tractable, where p max is the maximum processing time of a job and w max is the maximum ight of a job: #R##N#- Makespan minimization on uniformly related machines $(Q||C_{max} )$ parameterized by $p_{max}$, #R##N#- Makespan minimization on unrelated machines $(R||C_{max} )$ parameterized by $p_{max}$ and the number of kinds of machines, #R##N#- Sum of ighted completion times minimization on unrelated machines $(R|| \sum w_i C_i )$ parameterized by $p_{max} + w_{max}$ and the number of kinds of machines, #R##N#- The same problem, $(R|| \sum w_i C_i),$ parameterized by the number of distinct job times and the number of machines.
['Dušan Knop', 'Martin Koutecký']
I am writing a research paper on a idea Scheduling meets n-fold Integer Programming
[{"from": "human", "value": "I am writing a research paper on a idea Scheduling meets n-fold Integer Programming"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Scheduling problems are fundamental in combinatorial optimization. Much work has been done on approximation algorithms for NP-hard cases, but relatively little is known about exact solutions when some part of the input is a fixed parameter. In 2014, Mnich and Wiese initiated a systematic study in this direction. #R##N#  continue this study and show that several additional cases of fundamental scheduling problems are fixed parameter tractable for some natural parameters. Our main tool is n-fold integer programming, a recent variable dimension technique which  believe to be highly relevant for the parameterized complexity community.  serves to showcase and highlight this technique. #R##N#Specifically,  show the following four scheduling problems to be fixed-parameter tractable, where p max is the maximum processing time of a job and w max is the maximum ight of a job: #R##N#- Makespan minimization on uniformly related machines $(Q||C_{max} )$ parameterized by $p_{max}$, #R##N#- Makespan minimization on unrelated machines $(R||C_{max} )$ parameterized by $p_{max}$ and the number of kinds of machines, #R##N#- Sum of ighted completion times minimization on unrelated machines $(R|| \\sum w_i C_i )$ parameterized by $p_{max} + w_{max}$ and the number of kinds of machines, #R##N#- The same problem, $(R|| \\sum w_i C_i),$ parameterized by the number of distinct job times and the number of machines."}]