Karl Bringmann

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, 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, and (3) calculate the volume of the object in polynomial time. It suffices to be able to answer all three questions approximately. We show that this holds for a large class of objects. It implies that Klees measure problem can be approximated efficiently even though it is #P-hard and hence cannot be solved exactly in polynomial time in the number of dimensions unless P=NP. Our algorithm also allows to efficiently approximate 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^^^-^^^@e-approximation for certain boxes unless NP=BPP, but give a simple additive polynomial-time @e-approximation.

Approximating the least hypervolume contributor: NP-hard in general, but fast in practice

The hypervolume indicator is an increasingly popular set measure to compare the quality of two Pareto sets. The basic ingredient of most hypervolume indicator based optimization algorithms is the calculation of the hypervolume contribution of single solutions regarding a Pareto set. We show that exact calculation of the hypervolume contribution is #P-hard while its approximation is NP-hard. The same holds for the calculation of the minimal contribution. We also prove that it is NP-hard to decide whether a solution has the least hypervolume contribution. Even deciding whether the contribution of a solution is at most (1+@e) times the minimal contribution is NP-hard. This implies that it is neither possible to efficiently find the least contributing solution (unless P=NP) nor to approximate it (unless NP=BPP). Nevertheless, in the second part of the paper we present a fast approximation algorithm for this problem. We prove that for arbitrarily given @e,@d>0 it calculates a solution with contribution at most (1+@e) times the minimal contribution with probability at least (1-@d). Though it cannot run in polynomial time for all instances, it performs extremely fast on various benchmark datasets. The algorithm solves very large problem instances which are intractable for exact algorithms (e.g., 10,000 solutions in 100 dimensions) within a few seconds.

Why Walking the Dog Takes Time: Frechet Distance Has No Strongly Subquadratic Algorithms Unless SETH Fails

The Fréchet distance is a well-studied and very popular measure of similarity of two curves. Many variants and extensions have been studied since Alt and Godau introduced this measure to computational geometry in 1991. Their original algorithm to compute the Fréchet distance of two polygonal curves with n vertices has a runtime of O(n^2 log n). More than 20 years later, the state of the art algorithms for most variants still take time more than O(n2 / log n), but no matching lower bounds are known, not even under reasonable complexity theoretic assumptions. To obtain a conditional lower bound, in this paper we assume the Strong Exponential Time Hypothesis or, more precisely, that there is no O*((2-δ)N) algorithm for CNF-SAT for any delta > 0. Under this assumption we show that the Fréchet distance cannot be computed in strongly subquadratic time, i.e., in time O(n2-δ) for any delta > 0. This means that finding faster algorithms for the Fréchet distance is as hard as finding faster CNF-SAT algorithms, and the existence of a strongly subquadratic algorithm can be considered unlikely. Our result holds for both the continuous and the discrete Fréchet distance. We extend the main result in various directions. Based on the same assumption we (1) show non-existence of a strongly subquadratic 1.001-approximation, (2) present tight lower bounds in case the numbers of vertices of the two curves are imbalanced, and (3) examine realistic input assumptions (c-packed curves).

Approximation-guided evolutionary multi-objective optimization

Multi-objective optimization problems arise frequently in applications but can often only be solved approximately by heuristic approaches. Evolutionary algorithms have been widely used to tackle multi-objective problems. These algorithms use different measures to ensure diversity in the objective space but are not guided by a formal notion of approximation. We present a new framework of an evolutionary algorithm for multi-objective optimization that allows to work with a formal notion of approximation. Our experimental results show that our approach outperforms state-of-the-art evolutionary algorithms in terms of the quality of the approximation that is obtained in particular for problems with many objectives.

Approximating the Least Hypervolume Contributor: NP-Hard in General, But Fast in Practice

The hypervolume indicator is an increasingly popular set measure to compare the quality of two Pareto sets. The basic ingredient of most hypervolume indicator based optimization algorithms is the calculation of the hypervolume contribution of single solutions regarding a Pareto set. We show that exact calculation of the hypervolume contribution is #P-hard while its approximation is NP-hard. The same holds for the calculation of the minimal contribution. We also prove that it is NP-hard to decide whether a solution has the least hypervolume contribution. Even deciding whether the contribution of a solution is at most (1 + *** ) times the minimal contribution is NP-hard. This implies that it is neither possible to efficiently find the least contributing solution (unless P = NP) nor to approximate it (unless NP = BPP). Nevertheless, in the second part of the paper we present a very fast approximation algorithm for this problem. We prove that for arbitrarily given *** ,*** > 0 it calculates a solution with contribution at most (1 + *** ) times the minimal contribution with probability at least (1 *** *** ). Though it cannot run in polynomial time for all instances, it performs extremely fast on various benchmark datasets. The algorithm solves very large problem instances which are intractable for exact algorithms (e.g., 10000 solutions in 100 dimensions) within a few seconds.

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 Klees 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 2d1-z-approximation for certain boxes unless NP=BPP, but give a simple additive polynomial-time e-approximation.

An efficient algorithm for computing hypervolume contributions

The hypervolume indicator serves as a sorting criterion in many recent multi-objective evolutionary algorithms (MOEAs). Typical algorithms remove the solution with the smallest loss with respect to the dominated hypervolume from the population. We present a new algorithm which determines for a population of size n with d objectives, a solution with minimal hypervolume contribution in time (nd2 log n) for d > 2. This improves all previously published algorithms by a factor of n for all d > 3 and by a factor of for d 3. We also analyze hypervolume indicator based optimization algorithms which remove > 1 solutions from a population of size n . We show that there are populations such that the hypervolume contribution of iteratively chosen solutions is much larger than the hypervolume contribution of an optimal set of solutions. Selecting the optimal set of solutions implies calculating conventional hypervolume contributions, which is considered to be computationally too expensive. We present the first hypervolume algorithm which directly calculates the contribution of every set of solutions. This gives an additive term of in the runtime of the calculation instead of a multiplicative factor of . More precisely, for a population of size n with d objectives, our algorithm can calculate a set of solutions with minimal hypervolume contribution in time (nd2 log n n) for d > 2. This improves all previously published algorithms by a factor of nmin{,d2} for d > 3 and by a factor of n for d 3.

Two-dimensional subset selection for hypervolume and epsilon-indicator

The goal of bi-objective optimization is to find a small set of good compromise solutions. A common problem for bi-objective evolutionary algorithms is the following subset selection problem (SSP): Given n solutions P ⊂ R2 in the objective space, select k solutions P* from P that optimize an indicator function. In the hypervolume SSP we want to select k points P* that maximize the hypervolume indicator IHYP(P*, r) for some reference point r ∈ R2. Similarly, the ε-indicator SSP aims at selecting k~points P* that minimize the ε-indicator Iε(P*,R) for some reference set R ⊂ R2 of size m (which can be R=P). We first present a new algorithm for the hypervolume SSP with runtime O(n (k + log n)). Our second main result is a new algorithm for the ε-indicator SSP with runtime O(n log n + m log m). Both results improve the current state of the art runtimes by a factor of (nearly)

The maximum hypervolume set yields near-optimal approximation

n

Approximation quality of the hypervolume indicator

and make the problems tractable for new applications. Preliminary experiments confirm that the theoretical results translate into substantial empirical runtime improvements.