Approximating the volume of unions and intersections of high-dimensional geometric objects (0809.0835v2)

Abstract: We consider the computation of the volume of the union of high-dimensional geometric objects. While showing that this problem is #P-hard already for very simple bodies (i.e., axis-parallel boxes), we give a fast FPRAS for all objects where one can: (1) test whether a given point lies inside the object, (2) sample a point uniformly, (3) calculate the volume of the object in polynomial time. All three oracles can be weak, that is, just approximate. This implies that Klee's measure problem and the hypervolume indicator can be approximated efficiently even though they are #P-hard and hence cannot be solved exactly in time polynomial in the number of dimensions unless P=NP. Our algorithm also allows to approximate efficiently the volume of the union of convex bodies given by weak membership oracles. For the analogous problem of the intersection of high-dimensional geometric objects we prove #P-hardness for boxes and show that there is no multiplicative polynomial-time $2^{d^{1-\epsilon}}$-approximation for certain boxes unless NP=BPP, but give a simple additive polynomial-time $\epsilon$-approximation.

• The paper addresses the high computational complexity of computing volumes of unions and intersections of high-dimensional geometric objects, showing they are #P-hard.
• It presents a Fully Polynomial-Time Randomized Approximation Scheme (FPRAS) for approximating the volume of unions, relying on point, sample, and volume oracle access.
• The paper demonstrates strong computational barriers against simple polynomial-time approximation for intersection volumes, even for simple geometric objects like co-boxes.

Approximating the Volume of Unions and Intersections of High-Dimensional Geometric Objects

The paper by Karl Bringmann and Tobias Friedrich addresses the computational challenge of estimating the volume of unions and intersections of high-dimensional geometric objects. This investigation touches upon the fundamental complexity issues inherent in these geometric problems and introduces approximation methodologies that overcome these challenges under certain conditions.

Complexity of Volume Computation

The authors exhibit that computing the volume of the union and intersection of high-dimensional geometric objects poses significant complexity challenges. The problem is identified as #P-hard, even for relatively simple geometric structures such as axis-parallel boxes and general boxes when considering intersections. It is particularly underscored that while these problems are computationally prohibitive due to their #P-hard nature, they can nonetheless be approximately solved under specific conditions through sophisticated approximation algorithms.

For union calculations, the paper presents a Fully Polynomial-Time Randomized Approximation Scheme (FPRAS). This is contingent on three types of oracle operations: determining if a point lies within a geometric body (POINTQUERY), obtaining a uniformly random sample from the body (SAMPLEQUERY), and estimating the volume of the body (VOLUMEQUERY). These oracles need only be approximately accurate, enabling the application of the authors' algorithm across a variety of geometric structures such as convex bodies, polytopes, and even simpler structures like boxes and spheres.

Volume of Intersections and Approximation Barriers

In the domain of intersections, the paper reveals that, unlike the case for unions, there is no simple polynomial-time approximation possible. The volume of intersections is not only #P-hard to compute exactly but also presented as inappropriately complex for approximation. The authors emphasize that even for seemingly straightforward geometric objects like co-boxes, the intractability extends to preclude the possibility of achieving a multiplicative polynomial-time approximation unless NP equals BPP, establishing a strong computational barrier.

Implications and Application of FPRAS

The introduction of the FPRAS represents a significant step forward for approximating the volume of unions, providing efficiency and feasibility in instances where these problems are otherwise intractable. As notably evidenced for Klee's Measure Problem, which remains #P-hard, this paper's contribution in algorithmic development illustrates a valuable pathway for practical applications where exact solutions are infeasible due to the combinatorial explosion in high dimensions.

The implications of this research extend beyond mere theoretical interest; they offer practical solutions in areas where multidimensional geometric calculations are pivotal—such as in computational geometry, operations research, and optimization problems associated with multi-objective evolutionary algorithms. Additionally, the findings undergird further explorations into related computational problems, potentially fueling advancements in deterministic approximation methodologies for other complex geometric challenges.

Future Directions

While the authors successfully provide an FPRAS for the union volume approximation problem, the limitations regarding intersection volumes beg further inquiry. Future research might explore deterministic or other innovative theoretical frameworks for intersection volume approximations, particularly for complex and higher-dimensional geometric configurations. Moreover, exploring the boundaries and extensions of the FPRAS might reveal further classes of problems that can benefit from similar approximation techniques.

In summary, the work of Bringmann and Friedrich robustly advances our understanding of geometric volume approximations in high-dimensional spaces, combining both intricate theoretical insights and pragmatic solutions to tackle complexity barriers in computational geometry.