{"product_id":"convex-optimization-techniques-for-geometric-covering-problems-9783754346754","title":"Convex Optimization Techniques for Geometric Covering Problems","description":"\u003cp\u003e • Author(s): Jan Hendrik Rolfes\u003cbr\u003e • Publisher: Bod - Books on Demand\u003cbr\u003e • Publisher Imprint: Bod - Books on Demand\u003cbr\u003e • BISAC: Optimization\u003c\/p\u003e\u003cp\u003eThe present thesis is a commencement of a generalization of covering results in specific settings, such as the Euclidean space or the sphere, to arbitrary compact metric spaces. In particular we consider coverings of compact metric spaces $(X, d)$ by balls of radius $r$. We are interested in the minimum number of such balls needed to cover $X$, denoted by $\\Ncal(X, r)$. For finite $X$ this problem coincides with an instance of the combinatorial \\textsc{set cover} problem, which is $\\mathrm{NP}$-complete. We illustrate approximation techniques based on the moment method of Lasserre for finite graphs and generalize these techniques to compact metric spaces $X$ to obtain upper and lower bounds for $\\Ncal(X, r)$. \\\\ The upper bounds in this thesis follow from the application of a greedy algorithm on the space $X$. Its approximation quality is obtained by a generalization of the analysis of Chv\\'atal's algorithm for the weighted case of \\textsc{set cover}. We apply this greedy algorithm to the spherical case $X=S n$ and retrieve the best non-asymptotic bound of B\\\"or\\\"oczky and Wintsche. Additionally, the algorithm can be used to determine coverings of Euclidean space with arbitrary measurable objects having non-empty interior. The quality of these coverings slightly improves a bound of Nasz\\'odi. \\\\ For the lower bounds we develop a sequence of bounds $\\Ncal t(X, r)$ that converge after finitely (say $\\alpha\\in\\N$) many steps: $$\\Ncal 1(X, r)\\leq \\ldots \\leq \\Ncal \\alpha(X, r)=\\Ncal(X, r).$$ The drawback of this sequence is that the bounds $\\Ncal t(X, r)$ are increasingly difficult to compute, since they are the objective values of infinite-dimensional conic programs whose number of constraints and dimension of underlying cones grow accordingly to $t$. We show that these programs satisfy strong duality and derive a finite dimensional semidefinite program to approximate $\\Ncal 2(S 2, r)$ to arbitrary precision. Our results rely in part on the moment methods developed by de Laat a\u003c\/p\u003e","brand":"Atlantic Books","offers":[{"title":"Paperback","offer_id":46342022758551,"sku":"9783754346754","price":1088.0,"currency_code":"INR","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0666\/3471\/1191\/files\/9783754346754.webp?v=1768689483","url":"https:\/\/atlanticbooks.com\/products\/convex-optimization-techniques-for-geometric-covering-problems-9783754346754","provider":"Atlantic Books","version":"1.0","type":"link"}