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

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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.

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?

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~$O(\log\|V\|)$.

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.

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.

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.