High Multiplicity Scheduling on Uniform Machines in FPT-Time
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In high-multiplicity scheduling, jobs of the same size are encoded in an efficient way, that is, for each size the number of jobs of that size is given instead of a list of jobs. Similarly, machines are encoded. We consider scheduling on uniform machines where a job of size p_j takes time p_j/s_i on a machine of speed s_i. Classical (NP-hard) objectives are Makespan minimization (C_max) and Santa Claus (C_min). We show that both objectives can be solved in time ๐ช( p_max^๐ช(d^2)poly |I| ) where p_max is the largest jobs size, d the number of different job sizes and |I| the encoding length of the instance. Our approach incorporates two structural theorems: The first allows us to replace machines of large speed by multiple machines of smaller speed. The second tells us that some fractional assignments can be used to reduce the instance significantly. Using only the first theorem, we show some additional results. For the problem Envy Minimization (C_๐๐๐ฃ๐ฆ), we propose an ๐ช(s_maxยท p_max^๐ช(d^3)poly |I|) time algorithm (where s_max is the largest speed). For C_max and C_min in the Restricted Assignment setting, we give an ๐ช( (d p_max)^๐ช(d^3)poly |I|) time algorithm. As far as we know, those running times are better than the running times of the algorithms known until today.
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