On a Class of 3SUM-HARD problems in Computational Geometry

Given a simple polygon P and an integer K 1, we
want to compute the set of
straight lines in the Cartesian plane that cut this
polygon into exactly K simple
polygons. We call this set of lines a K-separator and
call this problem the K-separator
problem.
We present an algorithm that finds the K-separators
of an n-vertex simple polygon,
for all K 0, in O(n2) total time.
We prove that the decision problem given an integer K 2 and an edge of the
polygon, is there a line through this edge that cuts
the polygon in exactly K pieces?, is
3SUM-HARD. For the special case when K = 2, we show
that the decision problem
can be solved in O(n log(n)) time.
Several other complexity results may be obtained. We
suspect that the problem
of finding the cell of maximum depth is also
3SUM-hard, and as a corollary the
problem of identifying the line that cuts the polygon
in the maximum possible number
of pieces is also 3SUM-hard.

Autorentext

Born in Vlore, Albania where he lived until the age of 18, went
abroad to study (Applied Math, Mount Allison University;
Computational Geometry, Carleton University) and work (MTA,
CIRA). Is now back in Vlore, teaching at the University of Vlora.

Klappentext

Given a simple polygon P and an integer K > 1, we want to compute the set of straight lines in the Cartesian plane that cut this polygon into exactly K simple polygons. We call this set of lines a K-separator and call this problem the K-separator problem. We present an algorithm that finds the K-separators of an n-vertex simple polygon, for all K > 0, in O(n2) total time. We prove that the decision problem given an integer K > 2 and an edge of the polygon, is there a line through this edge that cuts the polygon in exactly K pieces?, is 3SUM-HARD. For the special case when K = 2, we show that the decision problem can be solved in O(n log(n)) time. Several other complexity results may be obtained. We suspect that the problem of finding the cell of maximum depth is also 3SUM-hard, and as a corollary the problem of identifying the line that cuts the polygon in the maximum possible number of pieces is also 3SUM-hard.