Polyhedra Circuits and Their Applications

TL;DR

This paper introduces polyhedra circuits as a new way to represent complex geometric regions, providing algorithms for counting lattice points and computing volumes efficiently in fixed dimensions.

Contribution

It develops algorithms for lattice point counting and volume computation for regions defined by polyhedra circuits, extending existing methods to more complex geometric objects.

Findings

Algorithms run in polynomial time relative to input size and dimension.

Polyhedra circuits can represent a broad class of geometric objects.

Efficient computation of volume and lattice points in complex regions.

Abstract

We introduce polyhedra circuits. Each polyhedra 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 polyhedra. They can be used to approximate a large class of $d$-dimensional manifolds in $\mathbb{R}^d$. Barvinok developed polynomial time algorithms to compute the volume of a rational polyhedra, and to count the number of lattice points in a rational polyhedra in a fixed dimensional space $\mathbb{R}^d$ with a fix $d$. Define $T_V(d,\, n)$ be the polynomial time in $n$ to compute the volume of one rational polyhedra, $T_L(d,\, n)$ be the polynomial time in $n$ to count the number of lattice points in one rational polyhedra with $d$ be a fixed dimensional number, $T_I(d,\, n)$ be the polynomial time in $n$ to solve integer linear programming time with $d$ be the fixed dimensional number, where $n$ is the total number of linear inequalities from input polyhedra. We develop algorithms to count the number of lattice points in the geometric region determined by a polyhedra circuit in $O\left(nd\cdot r_d(n)\cdot T_V(d,\, n)\right)$ time and to compute the volume of the geometric region determined by a polyhedra circuit in $O\left(n\cdot r_d(n)\cdot T_I(d,\, n)+r_d(n)T_L(d,\, n)\right)$ time, where $n$ is the number of input linear inequalities, $d$ is number of variables and $r_d(n)$ be the maximal number of regions that $n$ linear inequalities with $d$ variables partition $\mathbb{R}^d$.