TY - GEN
A1 - Hauptmeier, Dietrich
A1 - Krumke, Sven O.
A1 - Rambau, Jörg
A1 - Wirth., Hans-Christoph
T1 - Euler is Standing in Line
N2 - In this paper we study algorithms for ``Dial-a-Ride'' transportation problems. In the basic version of the problem we are given transportation jobs between the vertices of a graph and the goal is to find a shortest transportation that serves all the jobs. This problem is known to be NP-hard even on trees. We consider the extension when precedence relations between the jobs with the same source are given. Our results include a polynomial time algorithm on paths and an approximation algorithm on general graphs with a performance of~$9/4$. For trees we improve the performance to~$5/3$.
T3 - ZIB-Report - SC-99-06
KW - NP-completeness
KW - polynomial-time approximation algorithms
KW - stacker-crane problem
KW - vehicle routing
KW - elevator system
KW - Eulerian Cycle
Y1 - 1999
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3947
ER -
TY - GEN
A1 - Konjevod, Goran
A1 - Krumke, Sven O.
A1 - Marathe, Madhav
T1 - Budget Constrained Minimum Cost Connected Medians
N2 - Several practical instances of network design problems require the network to satisfy multiple constraints. In this paper, we address the \emph{Budget Constrained Connected Median Problem}: We are given an undirected graph $G = (V,E)$ with two different edge-weight functions $c$ (modeling the construction or communication cost) and $d$ (modeling the service distance), and a bound~$B$ on the total service distance. The goal is to find a subtree~$T$ of $G$ with minimum $c$-cost $c(T)$ subject to the constraint that the sum of the service distances of all the remaining nodes $v \in V\setminus T$ to their closest neighbor in~$T$ does not exceed the specified budget~$B$. This problem has applications in optical network design and the efficient maintenance of distributed databases. We formulate this problem as bicriteria network design problem, and present bicriteria approximation algorithms. We also prove lower bounds on the approximability of the problem that demonstrate that our performance ratios are close to best possible
T3 - ZIB-Report - 00-10
KW - NP-hardness
KW - Approximation Algorithms
KW - Network
Y1 - 2000
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-5783
ER -
TY - GEN
A1 - Bley, Andreas
T1 - On the Complexity of Vertex-Disjoint Length-Restricted Path Problems
N2 - Let $G=(V,E)$ be a simple graph and $s$ and $t$ be two distinct vertices of $G$. A path in $G$ is called $\ell$-bounded for some $\ell\in\mathbb{N}$, if it does not contain more than $\ell$ edges. We study the computational complexity of approximating the optimum value for two optimization problems of finding sets of vertex-disjoint $\ell$-bounded $s,t$-paths in $G$. First, we show that computing the maximum number of vertex-disjoint $\ell$-bounded $s,t$-paths is $\mathcal{AP\kern-1pt X}$--complete for any fixed length bound $\ell\geq 5$. Second, for a given number $k\in\mathbb{N}$, $1\leq k \leq |V|-1$, and non-negative weights on the edges of $G$, the problem of finding $k$ vertex-disjoint $\ell$-bounded $s,t$-paths with minimal total weight is proven to be $\mathcal{NPO}$--complete for any length bound $\ell\geq 5$. Furthermore, we show that, even if $G$ is complete, it is $\mathcal{NP}$--complete to approximate the optimal solution value of this problem within a factor of $2^{\langle\phi\rangle^\epsilon}$ for any constant $0<\epsilon<1$, where $\langle\phi\rangle$ denotes the encoding size of the given problem instance $\phi$. We prove that these results are tight in the sense that for lengths $\ell\leq 4$ both problems are polynomially solvable, assuming that the weights satisfy a generalized triangle inequality in the weighted problem. All results presented also hold for directed and non-simple graphs. For the analogous problems where the path length restriction is replaced by the condition that all paths must have length equal to $\ell$ or where vertex-disjointness is replaced by edge-disjointness we obtain similar results.
T3 - ZIB-Report - SC-98-20
KW - disjoint paths
KW - length bounded paths
KW - approximation
KW - reducibility
KW - completeness
Y1 - 1998
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-3639
ER -
TY - GEN
A1 - Krumke, Sven O.
A1 - Rambau, Jörg
A1 - Weider, Steffen
T1 - An Approximation Algorithm for the Non-Preemptive Capacitated Dial-a-Ride Problem
N2 - In the Capacitated Dial-a-Ride Problem (CDARP) we are given a transportation network and a finite set of transportation jobs. Each job specifies the source and target location which are both part of the network. A server which can carry at most $C$~objects at a time can move on the transportation network in order to process the transportation requests. The problem CDARP consists of finding a shortest transportation for the jobs starting and ending at a designated start location. In this paper we are concerned with the restriction of CDARP to graphs which are simple paths. This setting arises for instance when modelling applications in elevator transportation systems. It is known that even for this restricted class of graphs CDARP is NP-hard to solve. We provide a polynomial time approximation algorithm that finds a transportion of length at most thrice the length of the optimal transportation.
T3 - ZIB-Report - 00-53
KW - NP-completeness
KW - polynomial-time approximation algorithms
KW - stacker-crane problem
KW - vehicle
Y1 - 2000
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6217
ER -
TY - GEN
A1 - Bley, Andreas
T1 - On the Hardness of Finding Small Shortest Path Routing Conflicts
N2 - Nowadays most data networks use shortest path protocols such as OSPF or IS-IS to route traffic. Given administrative routing lengths for the links of a network, all data packets are sent along shortest paths with respect to these lengths from their source to their destination. One of the most fundamental problems in planning shortest path networks is to decide whether a given set of routing paths forms a valid routing and, if this is not the case, to find a small subset of the given paths that cannot be shortest paths simultaneously for any routing lengths. In this paper we show that it is NP-hard to approximate the size of the smallest shortest path conflict by a factor less than 7/6.
T3 - ZIB-Report - 09-15
KW - shortest path routing
KW - computational complexity
Y1 - 2009
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11276
SN - 1438-0064
ER -
TY - GEN
A1 - Orlowski, Sebastian
A1 - Pióro, Michal
T1 - On the complexity of column generation in survivable network design with path-based survivability mechanisms
N2 - This survey concerns optimization problems arising in the design of survivable communication networks. It turns out that such problems can be modeled in a natural way as non-compact linear programming formulations based on multicommodity flow network models. These non-compact formulations involve an exponential number of path flow variables, and therefore require column generation to be solved to optimality. We consider several path-based survivability mechanisms and present results, both known and new, on the complexity of the corresponding column generation problems (called the pricing problems). We discuss results for the case of the single link (or node) failures scenarios, and extend the considerations to multiple link failures. Further, we classify the design problems corresponding to different survivability mechanisms according to the structure of their pricing problem. Finally, we show that almost all encountered pricing problems are hard to solve for scenarios admitting multiple failures.
T3 - ZIB-Report - 08-51
KW - Netzplanung
KW - Dimensionierung
KW - Spaltengenerierung
KW - Mehrfachausfälle
KW - survivable network design
KW - routing
KW - column generation
KW - pricing
KW - complexity
Y1 - 2008
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-11058
SN - 1438-0064
ER -
TY - GEN
A1 - Krumke, Sven O.
A1 - Marathe, Madhav V.
A1 - Poensgen, Diana
A1 - Ravi, Sekharipuram S.
A1 - Wirth, Hans-Christoph
T1 - Budgeted Maximal Graph Coverage
N2 - 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.
T3 - ZIB-Report - 02-24
KW - budgeted maximum coverage
KW - approximation algorithm
KW - dynamic programming
Y1 - 2002
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6918
ER -
TY - THES
A1 - Krumke, Sven O.
T1 - Online Optimization: Competitive Analysis and Beyond
N2 - 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.
T3 - ZIB-Report - 02-25
KW - competitive analysis
KW - online optimization
KW - online algorithm
KW - approximation algorithm
Y1 - 2002
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6925
ER -
TY - GEN
A1 - Grötschel, Martin
T1 - P=NP?
N2 - Hinter der für den Uneingeweihten etwas kryptischen Frage "P = NP?" verbirgt sich das derzeit wichtigste Problem der Komplexitätstheorie. Dieser Artikel erläutert einige Aspekte der Theorie und erklärt informell, was "P = NP?" bedeutet. Es geht nicht nur um komplizierte algorithmische Mathematik und Informatik, sondern um grundsätzliche Fragen unserer Lebensumwelt. Kann man vielleicht beweisen, dass es für viele Probleme unseres Alltags keine effizienten Lösungsmethoden gibt?
T3 - ZIB-Report - 02-26
KW - computational complexity
KW - P=NP
Y1 - 2002
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6939
ER -
TY - GEN
A1 - Krumke, Sven O.
A1 - Poensgen, Diana
T1 - Online Call Admission in Optical Networks with Larger Wavelength Demands
N2 - 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.
T3 - ZIB-Report - 02-22
KW - Call Admission
KW - Routing and Wavelength Allocation
KW - Optical Networks
KW - Competitive Analysis
KW - Colorability
Y1 - 2002
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-6890
ER -
TY - GEN
A1 - Wolle, Thomas
A1 - Koster, Arie M.C.A.
A1 - Bodlaender, Hans L.
T1 - A Note on Contraction Degeneracy
N2 - The parameter contraction degeneracy -- the maximum minimum degree over all minors of a graph -- is a treewidth lower bound and was first defined in (Bodlaender, Koster, Wolle, 2004). In experiments it was shown that this lower bound improves upon other treewidth lower bounds. In this note, we examine some relationships between the contraction degeneracy and connected components of a graph, block s of a graph and the genus of a graph. We also look at chordal graphs, and we study an upper bound on the contraction degeneracy and another lower bound for treewidth. A data structure that can be used for algorithms computing the degeneracy and similar parameters, is also described.
T3 - ZIB-Report - 04-43
KW - contraction degeneracy
KW - genus of a graph
KW - treewidth lower bounds
Y1 - 2004
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8180
ER -
TY - GEN
A1 - Koster, Arie M.C.A.
A1 - Wolle, Thomas
A1 - Bodlaender, Hans L.
T1 - Degree-Based Treewidth Lower Bounds
N2 - Every lower bound for treewidth can be extended by taking the maximum of the lower bound over all subgraphs or minors. This extension is shown to be a very vital idea for improving treewidth lower bounds. In this paper, we investigate a total of nine graph parameters, providing lower bounds for treewidth. The parameters have in common that they all are the vertex-degree of some vertex in a subgra ph or minor of the input graph. We show relations between these graph parameters and study their computational complexity. To allow a practical comparison of the bounds, we developed heuristic algorithms for those parameters that are NP-hard to compute. Computational experiments show that combining the treewidth lower bounds with minors can considerably improve the lower bounds.
T3 - ZIB-Report - 04-44
KW - contraction degeneracy
KW - Ramachandramurthi parameter
KW - treewidth lower bounds
Y1 - 2004
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8193
ER -
TY - GEN
A1 - Bodlaender, Hans L.
A1 - Koster, Arie M.C.A.
T1 - On the Maximum Cardinality Search Lower Bound for Treewidth
N2 - The Maximum Cardinality Search algorithm visits the vertices of a graph in some order, such that at each step, an unvisited vertex that has the largest number of visited neighbors becomes visited. An MCS-ordering of a graph is an ordering of the vertices that can be generated by the Maximum Cardinality Search algorithm. The visited degree of a vertex $v$ in an MCS-ordering is the number of neighbors of $v$ that are before $v$ in the ordering. The visited degree of an MCS-ordering $\psi$ of $G$ is the maximum visited degree over all vertices $v$ in $\psi$. The maximum visited degree over all MCS-orderings of graph $G$ is called its {\em maximum visited degree}. Lucena (2003) showed that the treewidth of a graph $G$ is at least its maximum visited degree. We show that the maximum visited degree is of size $O(\log n)$ for planar graphs, and give examples of planar graphs $G$ with maximum visited degree $k$ with $O(k!)$ vertices, for all $k\in \Bbb{N}$. Given a graph $G$, it is NP-complete to determine if its maximum visited degree is at least $k$, for any fixed $k\geq 7$. Also, this problem does not have a polynomial time approximation algorithm with constant ratio, unless P=NP. Variants of the problem are also shown to be NP-complete. We also propose and experimentally analyses some heuristics for the problem. Several tiebreakers for the MCS algorithm are proposed and evaluated. We also give heuristics that give upper bounds on the value of the maximum visited degree of a graph, which appear to give results close to optimal on many graphs from real life applications.
T3 - ZIB-Report - 04-45
KW - maximum cardinality search
KW - treewidth
KW - lower bounds
KW - planar graphs
KW - graph algorithms
Y1 - 2004
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8201
ER -
TY - GEN
A1 - Grötschel, Martin
T1 - Das Problem mit der Komplexität: P = NP?
N2 - Was Komplexität ist, weiß niemand so richtig. In vielen Wissenschaftsgebieten wird der Begriff Komplexität verwendet, überall mit etwas anderer Bedeutung. Mathematik und Informatik hab en eine eigene Theorie hierzu entwickelt: die Komplexitätstheorie. Sie stellt zwar grundlegende Begriffe bereit, aber leider sind die meisten wichtigen Fragestellungen noch ungelöst. Diese kurze Einführung konzentriert sich auf einen speziellen, aber bedeutenden Aspekt der Theorie: Lösbarkeit von Problemen in deterministischer und nichtdeterministischer polynomialer Zeit. Hinter der für Uneingeweihte etwas kryptischen Frage "P = NP?" verbirgt sich das derzeit wichtigste Problem der Komplexitätstheorie. Anhand dieser Fragestellung werden einige Aspekte der Theorie erläutert und formell erklärt, was "P = NP?" bedeutet. Es geht nicht nur um komplizierte algorithmische Mathematik und Informatik, sondern um grundsätzliche Fragen unserer Lebensumwelt. Kann man vielleicht beweisen, dass es für viele Probleme unseres Alltags keine effizienten Lösungsmethoden gibt?
T3 - ZIB-Report - 05-58
KW - Komplexität
KW - Komplexitätstheorie
Y1 - 2005
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8916
ER -
TY - GEN
A1 - Bley, Andreas
T1 - Approximability of Unsplittable Shortest Path Routing Problems
N2 - In this paper, we discuss the relation of unsplittable shortest path routing (USPR) to other routing schemes and study the approximability of three USPR network planning problems. Given a digraph $D=(V,A)$ and a set $K$ of directed commodities, an USPR is a set of flow paths $\Phi_{(s,t)}$, $(s,t)\in K$, such that there exists a metric $\lambda=(\lambda_a)\in \mathbb{Z}^A_+$ with respect to which each $\Phi_{(s,t)}$ is the unique shortest $(s,t)$-path. In the \textsc{Min-Con-USPR} problem, we seek for an USPR that minimizes the maximum congestion over all arcs. We show that this problem is hard to approximate within a factor of $\mathcal{O}(|V|^{1-\epsilon})$, but easily approximable within min$(|A|,|K|)$ in general and within $\mathcal{O}(1)$ if the underlying graph is an undirected cycle or a bidirected ring. We also construct examples where the minimum congestion that can be obtained by USPR is a factor of $\Omega(|V|^2)$ larger than that achievable by unsplittable flow routing or by shortest multi-path routing, and a factor of $\Omega(|V|)$ larger than by unsplittable source-invariant routing. In the CAP-USPR problem, we seek for a minimum cost installation of integer arc capacities that admit an USPR of the given commodities. We prove that this problem is $\mathcal{NP}$-hard to approximate within $2-\epsilon$ (even in the undirected case), and we devise approximation algorithms for various special cases. The fixed charge network design problem \textsc{Cap-USPR}, where the task is to find a minimum cost subgraph of $D$ whose fixed arc capacities admit an USPR of the commodities, is shown to be $\mathcal{NPO}$-complete. All three problems are of great practical interest in the planning of telecommunication networks that are based on shortest path routing protocols. Our results indicate that they are harder than the corresponding unsplittable flow or shortest multi-path routing problems.
T3 - ZIB-Report - 06-02
KW - Shortest path routing
KW - unsplittable flow
KW - computational complexity
KW - approximation
Y1 - 2006
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8968
ER -
TY - GEN
A1 - Hiller, Benjamin
T1 - Probabilistic Competitive Analysis of a Dial-a-Ride Problem on Trees Under High Load
N2 - In this paper we consider a simple variant of the Online Dial-a-Ride Problem from a probabilistic point of view. To this end, we look at a probabilistic version of this online Dial-a-Ride problem and introduce a probabilistic notion of the competitive ratio which states that an algorithm performs well on the vast majority of the instances. Our main result is that under the assumption of high load a certain online algorithm is probabilistically $(1+o(1))$-competitive if the underlying graph is a tree. This result can be extended to general graphs by using well-known approximation techniques at the expense of a distortion factor~$O(\log\|V\|)$.
T3 - ZIB-Report - 05-56
KW - probabilistic competitive analysis
KW - Dial-a-Ride problem
KW - online algorithms
KW - IGNORE strategy
Y1 - 2005
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8896
ER -
TY - THES
A1 - Hiller, Benjamin
T1 - Bad Guys are Rare: Probabilistic Analysis of an Elementary Dial-a-Ride Problem
N2 - This thesis deals with a Dial-a-Ride problem on trees and considers both offline and online versions of this problem. We study the behavior of certain algorithms on random instances, i.e. we do probabilistic analysis. The focus is on results describing the typical behavior of the algorithms, i.e. results holding with (asymptotically) high probability. For the offline version, we present a simplified proof of a result of Coja-Oghlan, Krumke und Nierhoff. The results states that some heuristic using a minimum spanning tree to approximate a Steiner tree gives optimal results with high probability. This explains why this heuristic produces optimal solutions quite often. In the second part, probabilistic online versions of the problem are introduced. We study the online strategies REPLAN and IGNORE. Regarding the IGNORE strategy we can show that it works almost optimal under high load with high probability.
KW - Dial-a-Ride problem on trees
KW - probabilistic analysis
KW - online algorithms
Y1 - 2004
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-10114
ER -
TY - GEN
A1 - Bley, Andreas
T1 - Inapproximability Results for the Inverse Shortest Paths Problem with Integer Length and Unique Shortest Paths
N2 - We study the complexity of two Inverse Shortest Paths (ISP) problems with integer arc lengths and the requirement for uniquely determined shortest paths. Given a collection of paths in a directed graph, the task is to find positive integer arc lengths such that the given paths are uniquely determined shortest paths between their respective terminals. The first problem seeks for arc lengths that minimize the length of the longest of the prescribed paths. In the second problem, the length of the longest arc is to be minimized. We show that it is $np-hard$ to approximate the minimal longest path length within a factor less than $8/7$ or the minimal longest arc length within a factor less than $9/8$. This answers the (previously) open question whether these problems are $np-hard$ or not. We also present a simple algorithm that achieves an $\mathcal{O}(|V|)$-approximation guarantee for both variants. Both ISP problems arise in the planning of telecommunication networks with shortest path routing protocols. Our results imply that it is $\mathcal{NP}$-hard to decide whether a given path set can be realized with a real shortest path routing protocol such as OSPF, IS-IS, or RIP.
T3 - ZIB-Report - 05-04
KW - Inverse Shortest Paths
KW - Computational Complexity
KW - Approximation
KW - Shortest Path Routing
Y1 - 2005
U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:0297-zib-8388
ER -