Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: High Multiplicity Scheduling on Uniform Machines in FPT-Time   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 $\mathcal O( p_{\max}^{\mathcal O(d^2)}
\operatorname {poly} |I| )$ 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 ($C_{\mathit{envy}}$), we propose an
$\mathcal O(s_{\max} \cdot p_{\max}^{\mathcal O(d^3)} \operatorname{poly} |I|)$
time algorithm (where $s_{\max}$ is the largest speed). For $C_{\max}$ and
$C_{\min}$ in the Restricted Assignment setting, we give an $\mathcal O( (d
p_{\max})^{\mathcal O(d^3)} \operatorname{poly} |I|)$ 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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The categories of this paper are: quant-ph
37
The categories of this paper are: supr-con
8
The categories of this paper are: hep-ex
9
The categories of this paper are: nlin
32
The categories of this paper are: atom-ph
16
The categories of this paper are: stat
10
The categories of this paper are: q-bio
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The categories of this paper are: hep-lat
35
The categories of this paper are: mtrl-th
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The categories of this paper are: chao-dyn
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The categories of this paper are: solv-int
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The categories of this paper are: plasm-ph
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The categories of this paper are: acc-phys
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The categories of this paper are: alg-geom
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The categories of this paper are: math
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The categories of this paper are: astro-ph