## 68Q25 Analysis of algorithms and problem complexity [See also 68W40]

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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.

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.

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.

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.

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

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.

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?

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.

Euler is Standing in Line
(1999)

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~$5/3$.

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.