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# Peeling Potatoes Near-optimally in Near-linear Time

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SYSNO ASEP 0478998 J - Journal Article Journal Article Článek ve WOS Peeling Potatoes Near-optimally in Near-linear Time Cabello, S. (SI) Cibulka, J. (CZ) Kynčl, J. (CZ) Saumell, Maria (UIVT-O) RID, SAI, ORCID Valtr, P. (CZ) Siam Journal on Computing - ISSN 0097-5397 Roč. 46, č. 5 (2017), s. 1574-1602 29 s. eng - English US - United States geometric optimization ; potato peeling ; visibility graph ; geometric probability ; approximation algorithm BA - General Mathematics Pure mathematics GBP202/12/G061 GA ČR - Czech Science Foundation (CSF) UIVT-O - RVO:67985807 000416763900004 85032943193 10.1137/16M1079695 We consider the following geometric optimization problem: find a convex polygon of maximum area contained in a given simple polygon $P$ with $n$ vertices. We give a randomized near-linear-time $(1-\varepsilon)$-approximation algorithm for this problem: in $O(n( \log^2 n + (1/\varepsilon^3) \log n + 1/\varepsilon^4))$ time we find a convex polygon contained in $P$ that, with probability at least $2/3$, has area at least $(1-\varepsilon)$ times the area of an optimal solution. We also obtain similar results for the variant of computing a convex polygon inside $P$ with maximum perimeter. To achieve these results we provide new results in geometric probability. The first result is a bound relating the area of the largest convex body inside $P$ to the probability that two points chosen uniformly at random inside $P$ are mutually visible. The second result is a bound on the expected value of the difference between the perimeter of any planar convex body $K$ and the perimeter of the convex hull of a uniform random sample inside $K$. Institute of Computer Science Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 2018
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