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**1 - 2**of**2**### A Note on Set Cover Inapproximability Independent of Universe Size

- ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 105 (2007)
, 2007

"... In the set cover problem we are given a collection of m sets whose union covers [n] = {1,..., n} and must find a minimum-sized subcollection whose union still covers [n]. We investigate the approximability of set cover by an approximation ratio that depends only on m and observe that, for any const ..."

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In the set cover problem we are given a collection of m sets whose union covers [n] = {1,..., n} and must find a minimum-sized subcollection whose union still covers [n]. We investigate the approximability of set cover by an approximation ratio that depends only on m and observe that, for any constant c < 1/2, set cover cannot be approximated to within O(2 log1−1/(log log m)c m) unless SAT can be decided in slightly subexponential time. We conjecture that polynomial time m 1−ɛ-approximation is impossible for any ɛ> 0 unless SAT can be decided in subexponential time.