Theory of Combinatorial Algorithms
Prof. Emo Welzl

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Mittagsseminar (in cooperation with A. Steger)

 Mittagsseminar Talk Information

Date and Time: Thursday, September 27, 2012, 12:15 pm

Duration: 30 minutes

Location: CAB G51

Speaker: Alexander Souza (Apixxo AG)

## Approximation Algorithms for Generalized and Variable Sized Bin Covering

We consider the Generalized Bin Covering problem: We are given $m$ bin types, where each bin of type $i$ has profit $p_i$ and demand $d_i$. Furthermore, there are $n$ items, where item $j$ has size $s_j$. A bin of type $i$ is covered if the set of items assigned to it has total size at least the demand $d_i$. Then, the profit of $p_i$ is earned and the objective is to maximize the total profit. To the best of our knowledge, only the cases $p_i = d_i = 1$ (Bin Covering) and $p_i = d_i$ (Variable-Sized Bin Covering) have been treated before. We study two models of bin supply: In the unit supply model, we have exactly one bin of each type, i.e., we have individual bins. By contrast, in the infinite supply model, we have arbitrarily many bins of each type. Both versions of the problem are $\NP$-hard and can not be approximated better than $2$, unless $\P = \NP$.

Our results in the unit supply model hold not only asymptotically, but for all instances. This contrasts most of the previous work on Bin Covering, which has been asymptotic. We prove that there is a combinatorial $5$-approximation algorithm for Generalized Bin Covering with unit supply, which has running time $O(nm\sqrt{m+n})$. This also transfers to the infinite supply model. Furthermore, for Variable-Sized Bin Covering, in which we have $p_i = d_i$, we show that the natural and fast Next Fit Decreasing (NFD) algorithm is a $9/4$-approximation in the unit supply model. The bound is tight for the algorithm and close to being best-possible. Our analysis gives detailed insight into the limited extent to which the optimum can significantly outperform NFD.

Then we discuss the difficulty of defining asymptotics in the unit supply model. For two natural definitions, the negative result holds that Variable-Sized Bin Covering in the \emph{unit} supply model does not allow an APTAS. Clearly, this also excludes an APTAS for Generalized Bin Covering in that model. Nonetheless, we show that there is an AFPTAS for Variable-Sized Bin Covering in the \emph{infinite} supply model.

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