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.