An EPTAS for Budgeted Matroid Independent Set

Doron-Arad, Ilan and Kulik, Ariel and Shachnai, Hadas
(2022) An EPTAS for Budgeted Matroid Independent Set.
In: SIAM Symposium on Simplicity in Algorithms (SOSA23).
(In Press)

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%each new sentence starts on a new line We study the $2$-Dimensional Vector Bin Packing Problem (2VBP), a generalization of classic Bin Packing that is widely applicable in resource allocation and scheduling. In 2VBP we are given a set of items, where each item is associated with a two-dimensional {\em volume} vector. The objective is to partition the items into a minimal number of subsets (bins), such that the total volume of items in each subset is at most $1$ in each dimension. We give an asymptotic $\left(\frac{4}{3}+\eps\right)$-approximation for the problem, thus improving upon the best known asymptotic ratio of $\left(1+\ln \frac{3}{2}+\eps\right)\approx 1.406$ due to Bansal, Eli{\'a}{\v{s}} and Khan (SODA 2016). Our algorithm applies a novel {\em Round\&Round} approach which iteratively solves a configuration LP relaxation for the residual\mmcom{residual?}\akcom{modified.} instance (from the previous iteration) and samples a small number of configurations based on the solution for the configuration LP. For the analysis we derive an iteration-dependent upper bound on the solution size for the configuration LP, which holds with high probability. To facilitate the analysis, we discover key structural properties of 2VBP instances, leveraging the recent {\em fractional grouping} technique of Fairstein et al. (ESA 2021).


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