Approximating the volume of unions and intersections of high-dimensional geometric objects

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.

💡 Research Summary

The paper tackles two fundamental computational geometry problems in high dimensions: estimating the volume of the union and the volume of the intersection of a collection of geometric objects. It begins by establishing hardness results. By a parsimonious reduction from #SAT, the authors prove that even for the simplest class—axis‑parallel boxes (hyper‑rectangles)—computing the exact union volume is #P‑hard. This immediately implies that classic problems such as Klee’s measure problem and the hypervolume indicator, which are central in multi‑objective optimization and database range queries, cannot be solved exactly in polynomial time unless P = NP.

Despite this intractability, the authors present a positive result for the union case: a Fully Polynomial‑Randomized Approximation Scheme (FPRAS). The scheme assumes three oracles for each object: (1) a (possibly weak) membership test, (2) a (possibly weak) uniform sampler, and (3) a (possibly weak) volume estimator that runs in polynomial time. Using these oracles, the algorithm embeds the union into a known bounding convex body and runs a Markov‑chain Monte Carlo (MCMC) walk inside this container. At each step the algorithm queries which objects contain the current point, weights the contribution of each object by its (approximate) volume, and accumulates an unbiased estimator of the total union volume. By carefully controlling the number of samples—on the order of O((n·d)/ε²)—the estimator achieves a relative error of (1 ± ε) with probability at least 3/4. The total running time is polynomial in the number of objects n, the dimension d, and 1/ε, satisfying the definition of an FPRAS. Importantly, the framework extends to convex bodies that are given only by a weak membership oracle; the same sampling‑based approach yields an efficient union‑volume approximation even when exact volume computation for individual bodies is unavailable.

The second part of the paper addresses the intersection problem. The authors again show #P‑hardness for axis‑parallel boxes, and they strengthen the negative result by proving that, unless NP = BPP, no polynomial‑time algorithm can achieve a multiplicative approximation factor better than 2^{d^{1‑ε}} for any fixed ε > 0. This hardness stems from a reduction that embeds a SAT instance into the geometry of the boxes so that any sufficiently tight multiplicative approximation would solve the underlying decision problem. Consequently, the powerful FPRAS technique that works for unions cannot be transferred to intersections.

Nevertheless, the authors provide a simple additive approximation algorithm for intersections. The method discretizes the ambient space into a fine grid whose cell size is chosen proportional to ε. For each grid cell the algorithm checks (via the membership oracle) whether the cell’s representative point lies in the intersection; if so, the cell’s volume contributes to the total estimate. Because each cell has a known volume, the sum of accepted cells yields an estimate with absolute error at most ε·V_max, where V_max is an upper bound on the true intersection volume. The algorithm runs in time polynomial in n, d, and 1/ε, offering a practical way to obtain ε‑additive guarantees when multiplicative guarantees are provably impossible.

The paper concludes by situating its contributions within the broader literature. Prior work on Klee’s measure problem offered exact algorithms with exponential dependence on d and heuristic approximations limited to low dimensions. This work is the first to deliver a rigorous FPRAS for union volumes in arbitrary dimensions under realistic oracle assumptions, and it delineates the precise complexity boundary for intersection volumes: multiplicative approximation is essentially out of reach, while additive approximation remains feasible. The authors suggest future directions, including extending the framework to non‑convex or implicitly defined objects, improving the dependence on ε, and conducting empirical studies to validate the theoretical performance on real‑world high‑dimensional datasets.