We study the computational complexity of computing solutions for the  square-cut pizza sharing problem. In this problem, we have n mass distributions in the plane, and the task is to find a path that uses horizontal and vertical segments that splits each of the masses in half while making at most n-1 turns. We show that finding an approximate solution to this problem is PPA-complete  , while finding an exact solution  is FIXP-hard and in BU. Our PPA-hardness result applies even when all mass distributions are unions of non-overlapping squares, and our FIXP-hardness result applies even when all mass distributions are unions of weighted squares and right-angled triangles. When the path is restricted to have at most n-2 turns, we show that the approximate problem becomes NP-complete, and the exact problem becomes ETR-complete.
We study the computational complexity of computing solutions for the straight-cut and square-cut pizza sharing problems. We show that finding an approximate solution is PPA-hard for the straight-cut problem, and PPA-complete for the square-cut problem , while finding an exact solution for the square-cut problem is FIXP-hard and in BU. Our PPA-hardness results apply even when all mass distributions are unions of non-overlapping squares, and our FIXP-hardness result applies even when all mass distributions are unions of weighted squares and right-angled triangles. We also show that decision variants of the square-cut problem are hard: we show that the approximate problem is NP-complete, and the exact problem is ETR-complete.
A
straight-cut and
R
problem. In this problem, we have n mass distributions in the plane, and the task is to find a path that uses horizontal and vertical segments that splits each of the masses in half while making at most n-1 turns.
problems.
R
to this problem is
is PPA-hard for the straight-cut problem, and
A
for the square-cut problem
A
for the square-cut problem
R
result applies
results apply
R
When the path is restricted to have at most n-2 turns,
We also show that decision variants of the square-cut problem are hard:
R
becomes
is
R
becomes
is