@misc{KrumkePoensgen, author = {Krumke, Sven and Poensgen, Diana}, title = {Online Call Admission in Optical Networks with Larger Wavelength Demands}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6890}, number = {02-22}, abstract = {In the problem of \emph{Online Call Admission in Optical Networks}, briefly called \textsc{oca}, we are given a graph \$G=(V,E)\$ together with a set of wavelengths~\$W\$ and a finite sequence \$\sigma=r_1,r_2,\dots\$ of calls which arrive in an online fashion. Each call~\$r_j\$ specifies a pair of nodes to be connected and an integral demand indicating the number of required lightpaths. A lightpath is a path in~\$G\$ together with a wavelength~\$\lambda \in W\$. Upon arrival of a call, an online algorithm must decide immediately and irrevocably whether to accept or to reject the call without any knowledge of calls which appear later in the sequence. If the call is accepted, the algorithm must provide the requested number of lightpaths to connect the specified nodes. The essential restriction is the wavelength conflict constraint: each wavelength is available only once per edge, which implies that two lightpaths sharing an edge must have different wavelengths. Each accepted call contributes a benefit equal to its demand to the overall profit. The objective in \textsc{oca} is to maximize the overall profit. Competitive algorithms for \textsc{oca} have been known for the special case where every call requests just a single lightpath. In this paper we present the first competitive online algorithms for the general case of larger demands.}, language = {en} } @misc{KrumkeMarathePoensgenetal., author = {Krumke, Sven and Marathe, Madhav and Poensgen, Diana and Ravi, Sekharipuram S. and Wirth, Hans-Christoph}, title = {Budgeted Maximal Graph Coverage}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6918}, number = {02-24}, abstract = {An instance of the \emph{maximum coverage} problem is given by a set of weighted ground elements and a cost weighted family of subsets of the ground element set. The goal is to select a subfamily of total cost of at most that of a given budget maximizing the weight of the covered elements. We formulate the problem on graphs: In this situation the set of ground elements is specified by the nodes of a graph, while the family of covering sets is restricted to connected subgraphs. We show that on general graphs the problem is polynomial time solvable if restricted to sets of size at most~\$2\$, but becomes NP-hard if sets of size~\$3\$ are permitted. On trees, we prove polynomial time solvability if each node appears in a fixed number of sets. In contrast, if vertices are allowed to appear an unbounded number of times, the problem is NP-hard even on stars. We finally give polynomial time algorithms for special cases where the subgraphs form paths and the host graph is a line, a cycle or a star.}, language = {en} } @phdthesis{Krumke, author = {Krumke, Sven}, title = {Online Optimization: Competitive Analysis and Beyond}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6925}, number = {02-25}, abstract = {Traditional optimization techniques assume, in general, knowledge of all data of a problem instance. There are many cases in practice, however, where decisions have to be made before complete information about the data is available. In fact, it may be necessary to produce a part of the problem solution as soon as a new piece of information becomes known. This is called an \emph{online situation}, and an algorithm is termed \emph{online}, if it makes a decision (computes a partial solution) whenever a new piece of data requests an action. \emph{Competitive analysis} has become a standard yardstick to measure the quality of online algorithms. One compares the solution produced by an online algorithm to that of an optimal (clairvoyant) offline algorithm. An online algorithm is called \$c\$-competitive if on every input the solution it produces has cost'' at most \$c\$~times that of the optimal offline algorithm. This situation can be imagined as a game between an online player and a malicious adversary. Although competitive analysis is a worst-case analysis and henceforth pessimistic, it often allows important insights into the problem structure. One can obtain an idea about what kind of strategies are promising for real-world systems and why. On the other hand there are also cases where the offline adversary is simply too powerful and allows only trivial competitiveness results. This phenomenon is called hitting the triviality barrier''. We investigate several online problems by means of competitive analysis. We also introduce new concepts to overcome the weaknesses of the standard approach and to go beyond the triviality barrier.}, language = {en} } @misc{Groetschel, author = {Gr{\"o}tschel, Martin}, title = {P=NP?}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-6939}, number = {02-26}, abstract = {Hinter der f{\"u}r den Uneingeweihten etwas kryptischen Frage "P = NP?" verbirgt sich das derzeit wichtigste Problem der Komplexit{\"a}tstheorie. Dieser Artikel erl{\"a}utert einige Aspekte der Theorie und erkl{\"a}rt informell, was "P = NP?" bedeutet. Es geht nicht nur um komplizierte algorithmische Mathematik und Informatik, sondern um grunds{\"a}tzliche Fragen unserer Lebensumwelt. Kann man vielleicht beweisen, dass es f{\"u}r viele Probleme unseres Alltags keine effizienten L{\"o}sungsmethoden gibt?}, language = {de} }