Approximating set multi-covers

Johnson and Lov\'asz and Stein proved independently that any hypergraph satisfies $\tau\leq (1+\ln \Delta)\tau^{\ast}$, where $\tau$ is the transversal number, $\tau^{\ast}$ is its fractional version, and $\Delta$ denotes the maximum degree. We prove $\tau_f\leq c \tau^{\ast}\max\{\ln \Delta, f\}$ for the $f$-fold transversal number $\tau_f$. Similarly to Johnson, Lov\'asz and Stein, we also show that this bound can be achieved non-probabilistically, using a greedy algorithm. As a combinatorial application, we prove an estimate on how fast $\tau_f/f$ converges to $\tau^{\ast}$. As a geometric application, we obtain an upper bound on the minimal density of an $f$-fold covering of the $d$-dimensional Euclidean space by translates of any convex body.Comment: THE TITLE CHANGED! This is the final version. 7 page

On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint

In the problem of minimum connected dominating set with routing cost constraint, we are given a graph $G=(V,E)$, and the goal is to find the smallest connected dominating set $D$ of $G$ such that, for any two non-adjacent vertices $u$ and $v$ in $G$, the number of internal nodes on the shortest path between $u$ and $v$ in the subgraph of $G$ induced by $D \cup \{u,v\}$ is at most $\alpha$ times that in $G$. For general graphs, the only known previous approximability result is an $O(\log n)$-approximation algorithm ($n=|V|$) for $\alpha = 1$ by Ding et al. For any constant $\alpha > 1$, we give an $O(n^{1-\frac{1}{\alpha}}(\log n)^{\frac{1}{\alpha}})$-approximation algorithm. When $\alpha \geq 5$, we give an $O(\sqrt{n}\log n)$-approximation algorithm. Finally, we prove that, when $\alpha =2$, unless $NP \subseteq DTIME(n^{poly\log n})$, for any constant $\epsilon > 0$, the problem admits no polynomial-time $2^{\log^{1-\epsilon}n}$-approximation algorithm, improving upon the $\Omega(\log n)$ bound by Du et al. (albeit under a stronger hardness assumption)

Approximate Clustering via Metric Partitioning

In this paper we consider two metric covering/clustering problems - \textit{Minimum Cost Covering Problem} (MCC) and $k$-clustering. In the MCC problem, we are given two point sets $X$ (clients) and $Y$ (servers), and a metric on $X \cup Y$. We would like to cover the clients by balls centered at the servers. The objective function to minimize is the sum of the $\alpha$-th power of the radii of the balls. Here $\alpha \geq 1$ is a parameter of the problem (but not of a problem instance). MCC is closely related to the $k$-clustering problem. The main difference between $k$-clustering and MCC is that in $k$-clustering one needs to select $k$ balls to cover the clients. For any \eps > 0, we describe quasi-polynomial time (1 + \eps) approximation algorithms for both of the problems. However, in case of $k$-clustering the algorithm uses (1 + \eps)k balls. Prior to our work, a $3^{\alpha}$ and a ${c}^{\alpha}$ approximation were achieved by polynomial-time algorithms for MCC and $k$-clustering, respectively, where $c > 1$ is an absolute constant. These two problems are thus interesting examples of metric covering/clustering problems that admit (1 + \eps)-approximation (using (1+\eps)k balls in case of $k$-clustering), if one is willing to settle for quasi-polynomial time. In contrast, for the variant of MCC where $\alpha$ is part of the input, we show under standard assumptions that no polynomial time algorithm can achieve an approximation factor better than $O(\log |X|)$ for $\alpha \geq \log |X|$.Comment: 19 page

Fault Tolerant Max-Cut

In this work, we initiate the study of fault tolerant Max-Cut, where given an edge-weighted undirected graph G = (V,E), the goal is to find a cut S ? V that maximizes the total weight of edges that cross S even after an adversary removes k vertices from G. We consider two types of adversaries: an adaptive adversary that sees the outcome of the random coin tosses used by the algorithm, and an oblivious adversary that does not. For any constant number of failures k we present an approximation of (0.878-?) against an adaptive adversary and of ?_{GW}? 0.8786 against an oblivious adversary (here ?_{GW} is the approximation achieved by the random hyperplane algorithm of [Goemans-Williamson J. ACM `95]). Additionally, we present a hardness of approximation of ?_{GW} against both types of adversaries, rendering our results (virtually) tight. The non-linear nature of the fault tolerant objective makes the design and analysis of algorithms harder when compared to the classic Max-Cut. Hence, we employ approaches ranging from multi-objective optimization to LP duality and the ellipsoid algorithm to obtain our results

EPTAS and Subexponential Algorithm for Maximum Clique on Disk and Unit Ball Graphs

A (unit) disk graph is the intersection graph of closed (unit) disks in the plane. Almost three decades ago, an elegant polynomial-time algorithm was found for Maximum Cliqe on unit disk graphs [Clark, Colbourn, Johnson; Discrete Mathematics ’90]. Since then, it has been an intriguing open question whether or not tractability can be extended to general disk graphs. We show that the disjoint union of two odd cycles is never the complement of a disk graph nor of a unit (3-dimensional) ball graph. From that fact and existing results, we derive a simple QPTAS and a subexponential algorithm running in time 2O˜(n2/3) for Maximum Cliqe on disk and unit ball graphs. We then obtain a randomized EPTAS for computing the independence number on graphs having no disjoint union of two odd cycles as an induced subgraph, bounded VC-dimension, and linear independence number. This, in combination with our structural results, yields a randomized EPTAS for Max Cliqe on disk and unit ball graphs. Max Cliqe on unit ball graphs is equivalent to finding, given a collection of points in R3, a maximum subset of points with diameter at most some fixed value. In stark contrast, Maximum Cliqe on ball graphs and unit 4-dimensional ball graphs, as well as intersection graphs of filled ellipses (even close to unit disks) or filled triangles is unlikely to have such algorithms. Indeed, we show that, for all those problems, there is a constant ratio of approximation which cannot be attained even in time 2n1−ε, unless the Exponential Time Hypothesis fails

Ambulance routing problems with rich constraints and multiple objectives

Humanitäre non-profit Organisationen im Bereich des Patiententransports sehen sich dazu verpflichtet alle möglichen Einsparungs- und Optimierungspotentiale auszuloten um ihre Ausgaben zu reduzieren. Im Gegensatz zu Notfalleinsatzfahrten, bei denen ein Zusammenlegen mehrerer Transportaufträge normalerweise nicht möglich ist, besteht bei regulären Patiententransporten durchaus Einsparungspotential. Diese Tatsache gibt Anlass zur wissenschaftlichen Analyse jener Problemstellung, welche die täglich notwendige Planung regulärer Patiententransportaufträge umfasst. Solche Aufgabenstellungen werden als Dial-A-Ride-Probleme modelliert. Eine angemessene Service-Qualität kann entweder durch entsprechende Nebenbedingungen gewährleistet oder durch eine zusätzliche Zielfunktion minimiert werden. Beide Herangehensweisen werden hier untersucht. Zuerst wird eine vereinfachte Problemstellung aus der Literatur behandelt und ein kompetitives heuristisches Lösungsverfahren entwickelt. Diese vereinfachte Problemstellung wird in zwei Richtungen erweitert. Einerseits wird, zusätzlich zur Minimierung der Gesamtkosten, eine zweite benutzerorientierte Zielfunktion eingeführt. Andererseits werden eine heterogene Fahrzeugflotte und unterschiedliche Patiententypen in die Standardproblemstellung integriert. Letztendlich wird das reale Patiententransportproblem, basierend auf Informationen des Roten Kreuzes, definiert und gelöst. Neben heterogenen Fahrzeugen und unterschiedlichen Patienten, werden nun auch die Zuordnung von Fahrern und sonstigem Personal zu den verschiedenen Fahrzeugen, Mittagspausen und weitere Aufenthalte am Depot berücksichtigt. Alle eingesetzten exakten Methoden, obwohl sie auf neuesten Erkenntnissen aus der Literatur aufbauen, können Instanzen von realistischer Größe nicht lösen. Dieser Umstand macht die Entwicklung von passenden heuristischen Verfahren nach wie vor unumgänglich. In der vorliegenden Arbeit wird ein relativ generisches System basierend auf der Variable Neighborhood Search Idee entwickelt, das auf alle behandelten Einzielproblemversionen angewandt werden kann; auch für die bi-kriterielle Problemstellung, in Kombination mit Path Relinking, werden gute Ergebnisse erzielt.Humanitarian non-profit ambulance dispatching organizations are committed to look at cost reduction potentials in order to decrease their expenses. While in the context of emergency transportation cost reduction cannot be achieved by means of combined passenger routes, this can be done when dealing with regular patients. This research work is motivated by the problem situation faced by ambulance dispatchers in the field of patient transportation. Problems of this kind are modeled as dial-a-ride problems. In the field of patient transportation, the provision of a certain quality of service is necessary; the term “user inconvenience” is used in this context. User inconvenience can either be considered in terms of additional constraints or in terms of additional objectives. Both approaches are investigated in this book. The aim is to model and solve the real world problem based on available information from the Austrian Red Cross. In a first step, a competitive heuristic solution method for a simplified problem version is developed. This problem version is extended in two ways. On the one hand, besides routing costs, a user-oriented objective, minimizing user inconvenience, in terms of mean user ride time, is introduced. On the other hand, heterogeneous patient types and a heterogeneous vehicle fleet are integrated into the standard dial-a-ride model. In a final step, in addition to heterogeneous patients and vehicles, the assignment of drivers and other staff members to vehicles, the scheduling of lunch breaks, and additional stops at the depot are considered. All exact methods employed, although based on state of the art concepts, are not capable of solving instances of realistic size. This fact makes the development of according heuristic solution methods necessary. In this book a rather generic variable neighborhood search framework is proposed. It is able to accommodate all single objective problem versions and also proves to work well when applied to the bi-objective problem in combination with path relinking