68Q25 Analysis of algorithms and problem complexity [See also 68W40]
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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.
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.
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.
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.
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.
P=NP?
(2002)
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?
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.
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.
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.
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.