On the complexity of computing the volume of a polyhedron
SIAM Journal on Computing
Monte-Carlo approximation algorithms for enumeration problems
Journal of Algorithms
A random polynomial-time algorithm for approximating the volume of convex bodies
Journal of the ACM (JACM)
Sampling and integration of near log-concave functions
STOC '91 Proceedings of the twenty-third annual ACM symposium on Theory of computing
New upper bounds in Klee's measure problem
SIAM Journal on Computing
On the hardness of approximate reasoning
Artificial Intelligence
Random walks and an O*(n5) volume algorithm for convex bodies
Random Structures & Algorithms
On the complexity of computing the measure of ∪[ai,bi]
Communications of the ACM
A note on a method for generating points uniformly on n-dimensional spheres
Communications of the ACM
Semi-Online Maintenance of Geometric Optima and Measures
SIAM Journal on Computing
Counting without sampling: new algorithms for enumeration problems using statistical physics
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Counting independent sets up to the tree threshold
Proceedings of the thirty-eighth annual ACM symposium on Theory of computing
Simulated annealing in convex bodies and an O*(n4) volume algorithm
Journal of Computer and System Sciences - Special issue on FOCS 2003
Computing the volume of the union of cubes
SCG '07 Proceedings of the twenty-third annual symposium on Computational geometry
Simple deterministic approximation algorithms for counting matchings
Proceedings of the thirty-ninth annual ACM symposium on Theory of computing
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Correlation decay and deterministic FPTAS for counting list-colorings of a graph
SODA '07 Proceedings of the eighteenth annual ACM-SIAM symposium on Discrete algorithms
Fully polynomial time approximation schemes for stochastic dynamic programs
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
The mixing rate of Markov chains, an isoperimetric inequality, and computing the volume
SFCS '90 Proceedings of the 31st Annual Symposium on Foundations of Computer Science
Approximating the Volume of Unions and Intersections of High-Dimensional Geometric Objects
ISAAC '08 Proceedings of the 19th International Symposium on Algorithms and Computation
A (slightly) faster algorithm for Klee's measure problem
Computational Geometry: Theory and Applications
Thin partitions: isoperimetric inequalities and a sampling algorithm for star shaped bodies
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
Multiobjective evolutionary algorithms: a comparative case studyand the strength Pareto approach
IEEE Transactions on Evolutionary Computation
Convergence of hypervolume-based archiving algorithms I: effectiveness
Proceedings of the 13th annual conference on Genetic and evolutionary computation
Approximating the least hypervolume contributor: NP-hard in general, but fast in practice
Theoretical Computer Science
LION'05 Proceedings of the 5th international conference on Learning and Intelligent Optimization
Weighted preferences in evolutionary multi-objective optimization
AI'11 Proceedings of the 24th international conference on Advances in Artificial Intelligence
Approximation-guided evolutionary multi-objective optimization
IJCAI'11 Proceedings of the Twenty-Second international joint conference on Artificial Intelligence - Volume Volume Two
Convergence of hypervolume-based archiving algorithms ii: competitiveness
Proceedings of the 14th annual conference on Genetic and evolutionary computation
Convergence of set-based multi-objective optimization, indicators and deteriorative cycles
Theoretical Computer Science
On set-based local search for multiobjective combinatorial optimization
Proceedings of the 15th annual conference on Genetic and evolutionary computation
A fast approximation-guided evolutionary multi-objective algorithm
Proceedings of the 15th annual conference on Genetic and evolutionary computation
Parameterized average-case complexity of the hypervolume indicator
Proceedings of the 15th annual conference on Genetic and evolutionary computation
Population size matters: rigorous runtime results for maximizing the hypervolume indicator
Proceedings of the 15th annual conference on Genetic and evolutionary computation
Speeding up many-objective optimization by Monte Carlo approximations
Artificial Intelligence
Annals of Mathematics and Artificial Intelligence
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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 Klee's 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.