Polyhedral Circuits and Their Applications

Abstract

To better compute the volume and count the lattice points in geometric objects, we propose polyhedral circuits. Each polyhedral circuit characterizes a geometric region in $\mathbb {R}^d$. They can be applied to represent a rich class of geometric objects, which include all polyhedra and the union of a finite number of polyhedron. They can be also used to approximate a large class of d-dimensional manifolds in $\mathbb {R}^d$. Barvinok  [3] developed polynomial time algorithms to compute the volume of a rational polyhedron, and to count the number of lattice points in a rational polyhedron in $\mathbb {R}^d$ with a fixed dimensional number d. Let d be a fixed dimensional number, $T_V(d,\, n)$ be polynomial time in n to compute the volume of a rational polyhedron, $T_L(d,\, n)$ be polynomial time in n to count the number of lattice points in a rational polyhedron, where n is the total number of linear inequalities from input polyhedra, and $T_I(d,\, n)$ be polynomial time in n to solve integer linear programming problem with n be the total number of input linear inequalities. We develop algorithms to count the number of lattice points in geometric region determined by a polyhedral circuit in $O\,\left( nd\cdot r_d(n)\cdot T_V(d,\, n)\right)$ time and to compute the volume of geometric region determined by a polyhedral circuit in $O\,\left( n\cdot r_d(n)\cdot T_I(d,\, n)+r_d(n)T_L(d,\, n)\right)$ time, where $r_d(n)$ is the maximum number of atomic regions that n hyperplanes partition $\mathbb {R}^d$. The applications to continuous polyhedra maximum coverage problem, polyhedra maximum lattice coverage problem, polyhedra $\left( 1-\beta \right)$-lattice set cover problem, and $\left( 1-\beta \right)$-continuous polyhedra set cover problem are discussed. We also show the NP-hardness of the geometric version of maximum coverage problem and set cover problem when each set is represented as union of polyhedra.