Kolloquiumsvortrag, Prof. Grandoni, IDSIA USI-SUPSI in Lugano / am 07.07.2017

07.07.2017 von 14:15 bis 15:45

Institut für Informatik, Ludewig-Meyn-Str. 2, Raum Ü2/K (LMS2, R. Ü2/K), 24114 Kiel

Titel: Approximating Geometric Knapsack via L-packings

Abstract: Joint work with: Waldo Galvez, Sandy Heydrich, Salvatore Ingala,
Arindam Khan, Andreas Wiese

In the 2-dimensional geometric knapsack problem (2DK) we are given a
set of n axis-aligned rectangular items, each one with an associated
profit, and an axis-aligned square knapsack. The goal is to find a
(non-overlapping) packing of a maximum profit subset of items inside
the knapsack (without rotating items). The best-known polynomial-time
approximation factor for this problem (even just in the cardinality
case) is 2 + ε [Jansen and Zhang, SODA 2004]. In this work we break
the 2 approximation barrier, achieving a polynomial-time 17/9 + ε <
1.89 approximation, which improves to 558/325+ ε < 1.72 in the
cardinality case.

Essentially all prior work on 2DK approximation packs items inside a
constant number of rectangular containers, where items inside each
container are packed using a simple greedy strategy. We deviate for
the first time from this setting: we show that there exists a large
profit solution where items are packed inside a constant number of
containers plus one L-shaped region at the boundary of the knapsack
which contains items that are high and narrow and items that are wide
and thin. The items of these two types possibly interact in a complex
manner at the corner of the L.

The above structural result is not enough however: the best-known
approximation ratio for the sub-problem in the L-shaped region is 2 +
ε (obtained via a trivial reduction to 1-dimensional knapsack by
considering tall or wide items only). Indeed this is one of the
simplest special settings of the problem for which this is the best
known approximation factor. As a second major, and the main
algorithmic contribution of this work, we present a PTAS for this
case. We believe that this will turn out to be useful in future work
in geometric packing problems.

We also consider the variant of the problem with rotations (2DKR),
where items can be rotated by 90 degrees. Also in this case the
best-known polynomial-time approximation factor (even for the
cardinality case) is 2 + ε [Jansen and Zhang, SODA 2004]. Exploiting
part of the machinery developed for 2DK plus a few additional ideas,
we obtain a polynomial-time 3/2 + ε-approximation for 2DKR, which
improves to 4/3 + ε in the cardinality case.

Prof. Jansen

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