## A subexponential algorithm for abstract optimization problems (1995)

Venue: | SIAM J. Comput |

Citations: | 46 - 4 self |

### BibTeX

@ARTICLE{Gärtner95asubexponential,

author = {Bernd Gärtner},

title = {A subexponential algorithm for abstract optimization problems},

journal = {SIAM J. Comput},

year = {1995},

volume = {24},

pages = {464--472}

}

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### OpenURL

### Abstract

An Abstract Optimization Problem (AOP) is a triple (H, <, Φ) where H is a finite set, < a total order on 2 H and Φ an oracle that, for given F ⊆ G ⊆ H, either reports that F = min<{F ′ | F ′ ⊆ G} or returns a set F ′ ⊆ G with F ′ < F. To solve the problem means to find the minimum set in H. We present a randomized algorithm that solves any AOP with an expected number of at most e 2 √ n+O ( 4 √ n ln n) oracle calls, n = |H|. In contrast, any deterministic algorithm needs to make 2 n − 1 oracle calls in the worst case. The algorithm is applied to the problem of finding the distance between two n-vertex (or nfacet) convex polyhedra in d-space, and to the computation of the smallest ball containing n points in d-space; for both problems we give the first subexponential bounds in the arithmetic model of computation.