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Both volume computation and lattice point enumeration of convex polyhedron are $\#P$ hard. However there is a randomized polytime algorithm for constant factor approximation for volume computation.

1. Is there a randomized polytime algorithm for constant factor approximation for lattice point enumeration as well?

2. Is it $\oplus P$ complete to decide if a convex body has odd number of integer points?

If the polytope is convex and also centrally symmetric then what is the situation for 1., 2. and approximate volume computation?

Both volume computation and lattice point enumeration of convex polyhedron are $\#P$ hard. However there is a randomized polytime algorithm for constant factor approximation for volume computation.

1. Is there a randomized polytime algorithm for constant factor approximation for lattice point enumeration as well?

2. Is it $\oplus P$ complete to decide if a convex body has odd number of integer points?

If the polytope is convex and also centrally symmetric then what is the situation for 1., 2. and approximate volume computation?

Update If you know the number of lattice points approximately then we can guess volume approximately.

The converse is not true. What additional assumptions could give a healthy converse?

Both volume computation and lattice point enumeration of convex polyhedron is $\#P$ hard. However there is a randomized polytime algorithm for constant factor approximation for volume computation.

1. Is there a randomized polytime algorithm for constant factor approximation for lattice point enumeration as well?

2. Is it $\oplus P$ complete to decide if a convex body has odd number of integer points?

If the polytope is convex and also centrally symmetric then what is the situation for 1., 2. and approximate volume computation?

Both volume computation and lattice point enumeration of convex polyhedron are $\#P$ hard. However there is a randomized polytime algorithm for constant factor approximation for volume computation.

1. Is there a randomized polytime algorithm for constant factor approximation for lattice point enumeration as well?

2. Is it $\oplus P$ complete to decide if a convex body has odd number of integer points?

If the polytope is convex and also centrally symmetric then what is the situation for 1., 2. and approximate volume computation?

We know that both volume computation and lattice point enumeration of convex polyhedron is $\#P$ hard. However there is a randomized polytime algorithm for constant factor approximation for volume computation.

1. Is there a randomized polytime algorithm for constant factor approximation for lattice point enumeration as well?

2. Is it $\oplus P$ complete to decide if a convex body has odd number of integer points?

If the polytope is convex and also centrally symmetric then what is the situation for 1., 2. and approximate volume computation?

Both volume computation and lattice point enumeration of convex polyhedron is $\#P$ hard. However there is a randomized polytime algorithm for constant factor approximation for volume computation.

1. Is there a randomized polytime algorithm for constant factor approximation for lattice point enumeration as well?

2. Is it $\oplus P$ complete to decide if a convex body has odd number of integer points?

If the polytope is convex and also centrally symmetric then what is the situation for 1., 2. and approximate volume computation?