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Person# Klaus Jansen

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Problem solving is the process of achieving a goal by overcoming obstacles, a frequent part of most activities. Problems in need of solutions range from simple personal tasks (e.g. how to turn on an a

In mathematics, particularly in order theory, an upper bound or majorant of a subset S of some preordered set (K, ≤) is an element of K that is greater

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Klaus Jansen, Kim-Manuel Klein, Alexandra Anna Lassota

We consider fundamental algorithmic number theoretic problems and their relation to a class of block structured Integer Linear Programs (ILPs) called 2-stage stochastic. A 2-stage stochastic ILP is an integer program of the form min{c(T)x vertical bar Ax = b, l

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We consider the fully dynamic bin packing problem, where items arrive and depart in an online fashion and repacking of previously packed items is allowed. The goal is, of course, to minimize both the number of bins used as well as the amount of repacking. A recently introduced way of measuring the repacking costs at each timestep is the migration factor, defined as the total size of repacked items divided by the size of an arriving or departing item. Concerning the trade-off between number of bins and migration factor, if we wish to achieve an asymptotic competitive ratio of 1+epsilon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1 + \epsilon$\end{document} for the number of bins, a relatively simple argument proves a lower bound of omega(1/epsilon)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Omega ({1}/{\epsilon })$\end{document} for the migration factor. We establish a nearly matching upper bound of O(1/epsilon 4log1/epsilon)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O({1}/{\epsilon }4 \log {1}/{\epsilon })$\end{document} using a new dynamic rounding technique and new ideas to handle small items in a dynamic setting such that no amortization is needed. The running time of our algorithm is polynomial in the number of items nand in 1/epsilon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${1}/{\epsilon }$\end{document}. The previous best trade-off was for an asymptotic competitive ratio of 5/4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${5}/{4}$\end{document} for the bins (rather than 1+epsilon\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$1+\epsilon$\end{document}) and needed an amortized number of O(logn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$O(\log n)$\end{document} repackings (while in our scheme the number of repackings is independent of n and non-amortized).

In the restricted Santa Claus problem we are given resources R and players P. Every resource j is an element of R. has a value v(j) and every player i is an element of P desires a set of resources R(i). We are interested in distributing the resources to players that desire them. The quality of a solution is measured by the least happy player, i.e., the lowest sum of resource values. This value should be maximized. The local search algorithm by Asadpour et al. [1] and its connection to the configuration LP has proved itself to be a very influential technique for this and related problems. Asadpour et al. used a local search to obtain a bound of 4 for the ratio of the fractional to the integral optimum of the configuration LP (the integrality gap). This bound is non-constructive, since the local search has not been shown to terminate in polynomial time. On the negative side, the worst instance known has an integrality gap of 2. Although much progress was made in this area, neither bound has been improved since. We present a better analysis that shows the integrality gap is not worse than 3 + 5/6 approximate to 3.8333. (C) 2020 Elsevier B.V. All rights reserved.