Let $c\in(0,1/2]$ be a constant. Given a set of positive integers with sum $S$, is there a partition into two subsets such that both subsets have sum at least $cS$?

If $c=1/2$, this is the famous partition problem which is NP-complete. What is known about the complexity for other values of $c$?

  • 2
    $\begingroup$ It is NP-hard for every fixed constant $c$ with $0<c<1/2$. $\endgroup$
    – Gamow
    Mar 24, 2019 at 17:42
  • 3
    $\begingroup$ @Gamow Thanks. Could you please give a reference? $\endgroup$
    – user52378
    Mar 24, 2019 at 18:10
  • $\begingroup$ @Gamow This is incorrect, according to Tom van der Zanden's answer on CS SE. $\endgroup$ Jun 19, 2019 at 15:07


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