[Paper Review] Hardness of discrepancy computation and epsilon-net verification in high dimension

Panos Giannopoulos, Christian Knauer+2 more|arXiv (Cornell University)|Mar 23, 2011

Mathematical Approximation and Integration19 references4 citations

TL;DR

This paper proves that computing discrepancy and verifying epsilon-nets in high-dimensional geometric range spaces are W[1]-hard with respect to dimension d, implying that no fixed-parameter tractable algorithm exists under standard complexity assumptions. The results show that these problems require n^{Ω(d)} time unless 3-SAT can be solved in 2^{o(n)} time, establishing fundamental limits on efficient computation in high dimensions.

ABSTRACT

Discrepancy measures how uniformly distributed a point set is with respect to a given set of ranges. There are two notions of discrepancy, namely continuous discrepancy and combinatorial discrepancy. Depending on the ranges, several possible variants arise, for example star discrepancy, box discrepancy, and discrepancy of half-spaces. In this paper, we investigate the hardness of these problems with respect to the dimension d of the underlying space. All these problems are solvable in time {n^O(d)}, but such a time dependency quickly becomes intractable for high-dimensional data. Thus it is interesting to ask whether the dependency on d can be moderated. We answer this question negatively by proving that the canonical decision problems are W[1]-hard with respect to the dimension. This is done via a parameterized reduction from the Clique problem. As the parameter stays linear in the input parameter, the results moreover imply that these problems require {n^Ω(d)} time, unless 3-Sat can be solved in {2^o(n)} time. Further, we derive that testing whether a given set is an ε-net with respect to half-spaces takes {n^Ω(d)} time under the same assumption. As intermediate results, we discover the W[1]-hardness of other well known problems, such as determining the largest empty star inside the unit cube. For this, we show that it is even hard to approximate within a factor of {2^n}.

Motivation & Objective

• To determine whether discrepancy computation and epsilon-net verification in high-dimensional geometric range spaces can be solved efficiently with respect to dimension d.
• To investigate whether these problems admit fixed-parameter tractable (FPT) algorithms parameterized by dimension d.
• To establish computational lower bounds for these problems under standard complexity-theoretic assumptions.
• To explore the inapproximability of related geometric problems such as the largest empty star.
• To demonstrate that the dependency on dimension d is inherently necessary, not just a limitation of current algorithms.

Proposed method

• Uses parameterized reductions from the Clique problem to prove W[1]-hardness of discrepancy computation and epsilon-net verification in high dimensions.
• Constructs geometric point sets in the d-dimensional unit cube to encode instances of the Clique problem via range space properties.
• Leverages the Koksma–Hlawka inequality and the structure of half-spaces, boxes, and convex sets to simulate clique existence in geometric terms.
• Applies a transformation to lift point sets and ensure that large empty boxes or ranges correspond to cliques in the original graph.
• Adapts the construction to different range spaces, including cubes, convex sets, and half-spaces, to generalize the hardness results.
• Uses the fact that a set S is not an ε-net if there exists a range containing ε|P| points but no points from S, linking this to the existence of a large empty range.

Experimental results

Research questions

• RQ1 Can discrepancy computation in high-dimensional geometric range spaces be solved in FPT time with respect to dimension d?
• RQ2 Is there a fixed-parameter tractable algorithm for verifying whether a given set is an ε-net with respect to half-spaces in high dimensions?
• RQ3 What is the computational complexity of finding the largest empty star in the d-dimensional unit cube?
• RQ4 Can the star discrepancy of a point set be approximated within a constant factor in FPT time when parameterized by dimension?
• RQ5 What are the implications of W[1]-hardness for the existence of efficient algorithms in high-dimensional numerical integration and geometric optimization?

Key findings

• The decision problems for star discrepancy, box discrepancy, and half-space discrepancy are W[1]-hard with respect to dimension d.
• These problems require n^{Ω(d)} time unless 3-Sat can be solved in 2^{o(n)} time, under standard complexity assumptions.
• The problem of finding the largest empty star in the d-dimensional unit cube is W[1]-hard and inapproximable within a factor of 2^n.
• Verifying whether a set is an ε-net with respect to half-spaces is W[1]-hard with respect to dimension d.
• The hardness results extend to other geometric range spaces, including cubes, convex sets, and unrestricted boxes.
• The results imply that no FPT algorithm exists for these problems under the assumption that FPT ≠ W[1], establishing fundamental limits on efficient computation in high dimensions.