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 Rd . 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 Rd . Barvinok  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 Rd with a fixed dimensional number d. Let d be a fixed dimensional number, TV(d,n) be polynomial time in n to compute the volume of a rational polyhedron, TL(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 TI(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(nd⋅rd(n)⋅TV(d,n)) time and to compute the volume of geometric region determined by a polyhedral circuit in O(n⋅rd(n)⋅TI(d,n)+rd(n)TL(d,n)) time, where rd(n) is the maximum number of atomic regions that n hyperplanes partition Rd . The applications to continuous polyhedra maximum coverage problem, polyhedra maximum lattice coverage problem, polyhedra (1−β) -lattice set cover problem, and (1−β) -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.
Fu B., Gu P., Zhao Y. (2020) Polyhedral Circuits and Their Applications. In: Zhang Z., Li W., Du DZ. (eds) Algorithmic Aspects in Information and Management. AAIM 2020. Lecture Notes in Computer Science, vol 12290. Springer, Cham. https://doi.org/10.1007/978-3-030-57602-8_2
International Conference on Algorithmic Applications in Management