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- W4200631013 abstract "We investigate various online packing problems in which convex polygons arrive one by one and have to be placed irrevocably into a container before the next piece is revealed; the pieces must not be rotated, but only translated. The aim is to minimize the used space depending on the specific problem at hand, e.g., the strip length in strip packing, the number of bins in bin packing, etc. We draw interesting connections to the following online sorting problem OnlineSorting{}$[gamma,n]$: We receive a stream of real numbers $s_1,ldots,s_n$, $s_iin[0,1]$, one by one. Each real must be placed in an array~$A$ with $gamma n$ initially empty cells without knowing the subsequent reals. The goal is to minimize the sum of differences of consecutive reals in $A$. The offline optimum is to place the reals in sorted order so the cost is at most $1$. We show that for any $Delta$-competitive online algorithm of OnlineSorting{}$[gamma,n]$, it holds that $gamma Delta inOmega(log n/log log n)$. We use this lower bound to prove the non-existence of competitive algorithms for various online translational packing problems of convex polygons, among them strip packing, bin packing and perimeter packing. This also implies that there exists no online algorithm that can pack all streams of pieces of diameter and total area at most $delta$ into the unit square. These results are in contrast to the case when the pieces are restricted to rectangles, for which competitive algorithms are known. Likewise, the offline versions of packing convex polygons have constant factor approximation algorithms. On the positive side, we present an algorithm with competitive ratio $O(n^{0.59})$ for online translational strip packing of convex polygons. In the case of OnlineSorting{}$[C,n]$ for any constant $C>1$, we present an $O(2^{O(sqrt{log nloglog n})})$-competitive algorithm." @default.
- W4200631013 created "2021-12-31" @default.
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- W4200631013 date "2021-12-07" @default.
- W4200631013 modified "2023-09-28" @default.
- W4200631013 title "Online Sorting and Translational Packing of Convex Polygons" @default.
- W4200631013 doi "https://doi.org/10.48550/arxiv.2112.03791" @default.
- W4200631013 hasPublicationYear "2021" @default.
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