[Paper Review] Bringing Order to Special Cases of Klee's Measure Problem

Karl Bringmann|arXiv (Cornell University)|Jan 30, 2013

Computational Geometry and Mesh Generation15 references4 citations

TL;DR

This paper establishes a comprehensive hierarchy among special cases of Klee's Measure Problem (KMP) through novel reductions, proving that Hypervolume is the easiest variant and that Cube-KMP and Unitcube-KMP have polynomially related runtimes. Crucially, it shows that any algorithm for KMP with runtime $ T(n,d) $ implies an algorithm for the general case with runtime $ T(n,2d) $, transferring $\textup{W}[1]$-hardness and tight lower bounds to all special cases under the Exponential Time Hypothesis.

ABSTRACT

Klee's Measure Problem (KMP) asks for the volume of the union of n axis-aligned boxes in d-space. Omitting logarithmic factors, the best algorithm has runtime O*(n^{d/2}) [Overmars,Yap'91]. There are faster algorithms known for several special cases: Cube-KMP (where all boxes are cubes), Unitcube-KMP (where all boxes are cubes of equal side length), Hypervolume (where all boxes share a vertex), and k-Grounded (where the projection onto the first k dimensions is a Hypervolume instance). In this paper we bring some order to these special cases by providing reductions among them. In addition to the trivial inclusions, we establish Hypervolume as the easiest of these special cases, and show that the runtimes of Unitcube-KMP and Cube-KMP are polynomially related. More importantly, we show that any algorithm for one of the special cases with runtime T(n,d) implies an algorithm for the general case with runtime T(n,2d), yielding the first non-trivial relation between KMP and its special cases. This allows to transfer W[1]-hardness of KMP to all special cases, proving that no n^{o(d)} algorithm exists for any of the special cases under reasonable complexity theoretic assumptions. Furthermore, assuming that there is no improved algorithm for the general case of KMP (no algorithm with runtime O(n^{d/2 - eps})) this reduction shows that there is no algorithm with runtime O(n^{floor(d/2)/2 - eps}) for any of the special cases. Under the same assumption we show a tight lower bound for a recent algorithm for 2-Grounded [Yildiz,Suri'12].

Motivation & Objective

• To clarify the relative complexity of prominent special cases of Klee's Measure Problem: Cube-KMP, Unitcube-KMP, Hypervolume, and $k$-Grounded.
• To establish a formal hierarchy among these special cases through algorithmic reductions.
• To transfer hardness results from the general KMP to its special cases, proving that no $ n^{o(d)} $ algorithm exists under the Exponential Time Hypothesis.
• To provide tight lower bounds for specific algorithms, including the $ \mathcal{O}(n^{(d-1)/2}\log^2 n) $ algorithm for 2-Grounded.
• To identify open problems in the complexity of KMP and its variants, particularly regarding optimal runtime exponents and generalizations.

Proposed method

• Reduction from general KMP to $ 2k $-Grounded instances, showing that an algorithm for $ k $-Grounded with runtime $ T(n,d) $ implies an algorithm for general KMP with runtime $ T(n,2d) $.
• Iterative coordinate-wise splitting in each dimension to decompose a KMP instance into $ 2d+1 $ smaller $ k $-Grounded instances, each with at most $ (1 - \alpha)n $ boxes.
• Use of the fact that boxes trivial in all dimensions (i.e., equal to a fixed box $[a_1,b_1] \times \cdots \times [a_d,b_d]$) contribute volume only if they are identical, allowing simplification of the remaining instance.
• Recursive application of the reduction to amplify the size parameter, enabling the derivation of asymptotic runtime bounds via recurrence relations.
• Establishment of a recurrence $ T_{\textsc{$k$-Grounded}}'(n,d) \leqslant \mathcal{O}(T_{\textsc{$k$-Grounded}}((1-\alpha)n,d)) $, which allows bounding the runtime of the general problem in terms of $ k $-Grounded.
• Use of the $ \textup{W}[1] $-hardness of KMP (via reduction from $ k $-Clique) to infer that no $ n^{o(d)} $ algorithm exists for any special case under the Exponential Time Hypothesis.

Experimental results

Research questions

• RQ1 Is there a polynomial relationship between the runtimes of Hypervolume and Unitcube-KMP, similar to the one known between Cube-KMP and Unitcube-KMP?
• RQ2 Can the $ \mathcal{O}(n^{(d-1)/2}\log^2 n) $ algorithm for 2-Grounded be generalized to $ \mathcal{O}(n^{(d-k)/2+o(1)}) $ for $ 2k $-Grounded?
• RQ3 What is the exact value of the exponent $ c_d $ in the optimal runtime $ n^{c_d \cdot d \pm \mathcal{O}(1)} $ for Hypervolume and Cube-KMP/Unitcube-KMP?
• RQ4 Is there a reverse reduction from $ k $-Grounded to general KMP with $ d' < d $, establishing a tighter correspondence between the two?
• RQ5 Does the absence of an improved algorithm for general KMP imply a tight lower bound of $ \Omega(n^{\lfloor d/2\rfloor/2 - \varepsilon}) $ for all special cases?

Key findings

• The paper establishes that Hypervolume is the easiest among the studied special cases, as it admits the fastest known algorithms and is reducible to from other cases.
• Cube-KMP and Unitcube-KMP have polynomially related runtimes, meaning that an improvement in one implies an improvement in the other.
• Any algorithm for $ k $-Grounded with runtime $ T(n,d) $ implies an algorithm for general KMP with runtime $ T(n,2d) $, establishing a fundamental connection between the general problem and its special cases.
• Under the Exponential Time Hypothesis, no $ n^{o(d)} $ algorithm exists for any of the special cases, due to the transfer of $ \textup{W}[1] $-hardness from KMP.
• Assuming no improved algorithm exists for general KMP (i.e., no $ \mathcal{O}(n^{d/2 - \varepsilon}) $ algorithm), the paper proves a tight lower bound of $ \Omega(n^{\lfloor d/2\rfloor/2 - \varepsilon}) $ for all special cases.
• The paper provides a tight lower bound of approximately $ n^{(d-1)/4} $ for all special cases under the same assumption, and confirms the optimality of the $ \mathcal{O}(n^{(d-1)/2}\log^2 n) $ algorithm for 2-Grounded.