Title: A Simple PTAS for the Dual Bin Packing Problem and Advice Complexity of Its Online Version,
Abstract: Recently, Renault (2016) studied the dual bin packing problem in the
per-request advice model of online algorithms. He showed that given
$O(1/\epsilon)$ advice bits for each input item allows approximating the dual
bin packing problem online to within a factor of $1+\epsilon$. Renault asked
about the advice complexity of dual bin packing in the tape-advice model of
online algorithms. We make progress on this question. Let $s$ be the maximum
bit size of an input item weight. We present a conceptually simple online
algorithm that with total advice $O\left(\frac{s + \log n}{\epsilon^2}\right)$
approximates the dual bin packing to within a $1+\epsilon$ factor. To this end,
we describe and analyze a simple offline PTAS for the dual bin packing problem.
Although a PTAS for a more general problem was known prior to our work
(Kellerer 1999, Chekuri and Khanna 2006), our PTAS is arguably simpler to state
and analyze. As a result, we could easily adapt our PTAS to obtain the
advice-complexity result.
We also consider whether the dependence on $s$ is necessary in our algorithm.
We show that if $s$ is unrestricted then for small enough $\epsilon > 0$
obtaining a $1+\epsilon$ approximation to the dual bin packing requires
$\Omega_\epsilon(n)$ bits of advice. To establish this lower bound we analyze
an online reduction that preserves the advice complexity and approximation
ratio from the binary separation problem due to Boyar et al. (2016). We define
two natural advice complexity classes that capture the distinction similar to
the Turing machine world distinction between pseudo polynomial time algorithms
and polynomial time algorithms. Our results on the dual bin packing problem
imply the separation of the two classes in the advice complexity world.