Minimum Star Partitions of Simple Polygons in Polynomial Time.
CoRR(2023)
摘要
We devise a polynomial-time algorithm for partitioning a simple polygon $P$
into a minimum number of star-shaped polygons. The question of whether such an
algorithm exists has been open for more than four decades [Avis and Toussaint,
Pattern Recognit., 1981] and it has been repeated frequently, for example in
O'Rourke's famous book [Art Gallery Theorems and Algorithms, 1987]. In addition
to its strong theoretical motivation, the problem is also motivated by
practical domains such as CNC pocket milling, motion planning, and shape
parameterization.
The only previously known algorithm for a non-trivial special case is for $P$
being both monotone and rectilinear [Liu and Ntafos, Algorithmica, 1991]. For
general polygons, an algorithm was only known for the restricted version in
which Steiner points are disallowed [Keil, SIAM J. Comput., 1985], meaning that
each corner of a piece in the partition must also be a corner of $P$.
Interestingly, the solution size for the restricted version may be linear for
instances where the unrestricted solution has constant size. The covering
variant in which the pieces are star-shaped but allowed to overlap--known as
the Art Gallery Problem--was recently shown to be $\exists\mathbb R$-complete
and is thus likely not in NP [Abrahamsen, Adamaszek and Miltzow, STOC 2018 & J.
ACM 2022]; this is in stark contrast to our result. Arguably the most related
work to ours is the polynomial-time algorithm to partition a simple polygon
into a minimum number of convex pieces by Chazelle and Dobkin~[STOC, 1979 &
Comp. Geom., 1985].
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