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The <i>Steiner tree</i> problem is one of the most fundamental NP-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum-cost tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from 2 to the current best 1.55 [Robins,Zelikovsky-SIDMA'05]. All these algorithms… (More)

Davis-Putnam-style exponential-time backtracking algorithms are the most common algorithms used for finding exact solutions of NP-hard problems. The analysis of such recursive algorithms is based on the bounded search tree technique: a measure of the size of the sub-problems is defined; this measure is used to lower bound the progress made by the algorithm… (More)

We provide an algorithm listing all minimal dominating sets of a graph on <i>n</i> vertices in time <i>O</i>(1.7159<sup><i>n</i></sup>). This result can be seen as an algorithmic proof of the fact that the number of minimal dominating sets in a graph on <i>n</i> vertices is at most 1.7159<sup><i>n</i></sup>, thus improving on the trivial… (More)

For more than 30 years Davis-Putnam-style exponential-time backtracking algorithms have been the most common tools used for finding exact solutions of NP-hard problems. Despite of that, the way to analyze such recursive algorithms is still far from producing tight worst case running time bounds.The "Measure and Conquer" approach is one of the recent… (More)

For more than 40 years, Branch & Reduce exponential-time backtracking algorithms have been among the most common tools used for finding exact solutions of NP-hard problems. Despite that, the way to analyze such recursive algorithms is still far from producing tight worst-case running time bounds. Motivated by this, we use an approach, that we call… (More)

The Steiner tree problem is one of the most fundamental <b>NP</b>-hard problems: given a weighted undirected graph and a subset of terminal nodes, find a minimum-cost tree spanning the terminals. In a sequence of papers, the approximation ratio for this problem was improved from 2 to 1.55 [Robins and Zelikovsky 2005]. All these algorithms are purely… (More)

- Daniel Lokshtanov, Maria Chudnovsky, Rolf Niedermeier, Sverre Storøy, Noga Alon, Oren Ben-Zwi +28 others
- 2009

i Acknowledgements First of all, I would like to thank my supervisor Pinar Heggernes for her help and guidance, both with my reasearch and with all other aspects of being a graduate student. This thesis would not have existed without her. Thomassen and Yngve Villanger. A special thanks goes to Saket Saurabh, with whom I have shared many sleepless nights and… (More)

We present a simple randomized algorithmic framework for connected facility location problems. The basic idea is as follows: We run a black-box approximation algorithm for the unconnected facility location problem, randomly sample the clients, and open the facilities serving sampled clients in the approximate solution. Via a novel analytical tool, which we… (More)

This survey concerns techniques in design and analysis of algorithms that can be used to solve NP hard problems faster than exhaustive search algorithms (but still in exponential time). We discuss several of such techniques: Measure & Conquer, Exponential Lower Bounds, Bounded Tree-width, and Memorization. We also consider some extensions of the mentioned… (More)