Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Tighter inapproximability for set cover   Set Cover is a classic NP-hard problem; as shown by Slav\'{i}k (1997) the
greedy algorithm gives an approximation ratio of $\ln n - \ln \ln n +
\Theta(1)$. A series of works by Lund \& Yannakakis (1994), Feige (1998),
Moshkovitz (2015) have shown that, under the assumption $P \neq NP$, it is
impossible to obtain a polynomial-time approximation ratio with approximation
ratio $(1 - \alpha) \ln n$, for any constant $\alpha > 0$.
  In this note, we show that under the Exponential Time Hypothesis (a stronger
complexity-theoretic assumptions than $P \neq NP$), there are no
polynomial-time algorithms achieving approximation ratio $\ln n - C \ln \ln n$,
where $C$ is some universal constant. Thus, the greedy algorithm achieves an
essentially optimal approximation ratio (up to the coefficient of $\ln \ln n$).

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The categories of this paper are: q-bio
29
The categories of this paper are: funct-an
35
The categories of this paper are: mtrl-th
0
The categories of this paper are: hep-ph
31
The categories of this paper are: ao-sci
19
The categories of this paper are: eess
34
The categories of this paper are: chem-ph
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The categories of this paper are: math-ph
33
The categories of this paper are: plasm-ph
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The categories of this paper are: stat
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The categories of this paper are: hep-th
8
The categories of this paper are: hep-ex
4
The categories of this paper are: cond-mat
36
The categories of this paper are: cmp-lg
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The categories of this paper are: dg-ga
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The categories of this paper are: bayes-an