Approximation Algorithms for Network Design in Non-Uniform Fault Models

Classical network design models, such as the Survivable Network Design problem (SNDP), are (partly) motivated by robustness to faults under the assumption that any subset of edges upto a specific number can fail. We consider non-uniform fault models where the subset of edges that fail can be specified in different ways. Our primary interest is in the flexible graph connectivity model [Adjiashvili, 2013; Adjiashvili et al., 2020; Adjiashvili et al., 2022; Boyd et al., 2023], in which the edge set is partitioned into safe and unsafe edges. Given parameters p,q ? 1, the goal is to find a cheap subgraph that remains p-connected even after the failure of q unsafe edges. We also discuss the bulk-robust model [Adjiashvili et al., 2015; Adjiashvili, 2015] and the relative survivable network design model [Dinitz et al., 2022]. While SNDP admits a 2-approximation [K. Jain, 2001], the approximability of problems in these more complex models is much less understood even in special cases. We make two contributions. Our first set of results are in the flexible graph connectivity model. Motivated by a conjecture that a constant factor approximation is feasible when p and q are fixed, we consider two special cases. For the s-t case we obtain an approximation ratio that depends only on p,q whenever p+q > pq/2 which includes (p,2) and (2,q) for all p,q ? 1. For the global connectivity case we obtain an O(q) approximation for (2,q), and an O(p) approximation for (p,2) and (p,3) for any p ? 1, and for (p,4) when p is even. These are based on an augmentation framework and decomposing the families of cuts that need to be covered into a small number of uncrossable families. Our second result is a poly-logarithmic approximation for a generalization of the bulk-robust model when the "width" of the given instance (the maximum number of edges that can fail in any particular scenario) is fixed. Via this, we derive corresponding approximations for the flexible graph connectivity model and the relative survivable network design model. We utilize a recent framework due to Chen et al. [Chen et al., 2022] that was designed for handling group connectivity

Approximation algorithms for network design and cut problems in bounded-treewidth

This thesis explores two optimization problems, the group Steiner tree and firefighter problems, which are known to be NP-hard even on trees. We study the approximability of these problems on trees and bounded-treewidth graphs. In the group Steiner tree, the input is a graph and sets of vertices called groups; the goal is to choose one representative from each group and connect all the representatives with minimum cost. We show an O(log^2 n)-approximation algorithm for bounded-treewidth graphs, matching the known lower bound for trees, and improving the best possible result using previous techniques. We also show improved approximation results for group Steiner forest, directed Steiner forest, and a fault-tolerant version of group Steiner tree. In the firefighter problem, we are given a graph and a vertex which is burning. At each time step, we can protect one vertex that is not burning; fire then spreads to all unprotected neighbors of burning vertices. The goal is to maximize the number of vertices that the fire does not reach. On trees, a classic (1-1/e)-approximation algorithm is known via LP rounding. We prove that the integrality gap of the LP matches this approximation, and show significant evidence that additional constraints may improve its integrality gap. On bounded-treewidth graphs, we show that it is NP-hard to find a subpolynomial approximation even on graphs of treewidth 5. We complement this result with an O(1)-approximation on outerplanar graphs.Diese Arbeit untersucht zwei Optimierungsprobleme, von welchen wir wissen, dass sie selbst in Bäumen NP-schwer sind. Wir analysieren Approximationen für diese Probleme in Bäumen und Graphen mit begrenzter Baumweite. Im Gruppensteinerbaumproblem, sind ein Graph und Mengen von Knoten (Gruppen) gegeben; das Ziel ist es, einen Knoten von jeder Gruppe mit minimalen Kosten zu verbinden. Wir beschreiben einen O(log^2 n)-Approximationsalgorithmus für Graphen mit beschränkter Baumweite, dies entspricht der zuvor bekannten unteren Schranke für Bäume und ist zudem eine Verbesserung über die bestmöglichen Resultate die auf anderen Techniken beruhen. Darüber hinaus zeigen wir verbesserte Approximationsresultate für andere Gruppensteinerprobleme. Im Feuerwehrproblem sind ein Graph zusammen mit einem brennenden Knoten gegeben. In jedem Zeitschritt können wir einen Knoten der noch nicht brennt auswählen und diesen vor dem Feuer beschützen. Das Feuer breitet sich anschließend zu allen Nachbarn aus. Das Ziel ist es die Anzahl der Knoten die vom Feuer unberührt bleiben zu maximieren. In Bäumen existiert ein lang bekannter (1-1/e)-Approximationsalgorithmus der auf LP Rundung basiert. Wir zeigen, dass die Ganzzahligkeitslücke des LP tatsächlich dieser Approximation entspricht, und dass weitere Einschränkungen die Ganzzahligkeitslücke möglicherweise verbessern könnten. Für Graphen mit beschränkter Baumweite zeigen wir, dass es NP-schwer ist, eine sub-polynomielle Approximation zu finden

06481 Abstracts Collection -- Geometric Networks and Metric Space Embeddings

The Dagstuhl Seminar 06481 ``Geometric Networks and Metric Space Embeddings\u27\u27 was held from November~26 to December~1, 2006 in the International Conference and Research Center (IBFI), Schloss Dagstuhl. During the seminar, several participants presented their current research, and ongoing work and open problems were discussed. In this paper we describe the seminar topics, we have compiled a list of open questions that were posed during the seminar, there is a list of all talks and there are abstracts of the presentations given during the seminar. Links to extended abstracts or full papers are provided where available

Algorithms and complexity analyses for some combinational optimization problems

The main focus of this dissertation is on classical combinatorial optimization problems in two important areas: scheduling and network design. In the area of scheduling, the main interest is in problems in the master-slave model. In this model, each machine is either a master machine or a slave machine. Each job is associated with a preprocessing task, a slave task and a postprocessing task that must be executed in this order. Each slave task has a dedicated slave machine. All the preprocessing and postprocessing tasks share a single master machine or the same set of master machines. A job may also have an arbitrary release time before which the preprocessing task is not available to be processed. The main objective in this dissertation is to minimize the total completion time or the makespan. Both the complexity and algorithmic issues of these problems are considered. It is shown that the problem of minimizing the total completion time is strongly NP-hard even under severe constraints. Various efficient algorithms are designed to minimize the total completion time under various scenarios. In the area of network design, the survivable network design problems are studied first. The input for this problem is an undirected graph G = (V, E), a non-negative cost for each edge, and a nonnegative connectivity requirement ruv for every (unordered) pair of vertices &ruv. The goal is to find a minimum-cost subgraph in which each pair of vertices u,v is joined by at least ruv edge (vertex)-disjoint paths. A Polynomial Time Approximation Scheme (PTAS) is designed for the problem when the graph is Euclidean and the connectivity requirement of any point is at most 2. PTASs or Quasi-PTASs are also designed for 2-edge-connectivity problem and biconnectivity problem and their variations in unweighted or weighted planar graphs. Next, the problem of constructing geometric fault-tolerant spanners with low cost and bounded maximum degree is considered. The first result shows that there is a greedy algorithm which constructs fault-tolerant spanners having asymptotically optimal bounds for both the maximum degree and the total cost at the same time. Then an efficient algorithm is developed which finds fault-tolerant spanners with asymptotically optimal bound for the maximum degree and almost optimal bound for the total cost