This comprehensive textbook presents a clean and coherent account of most fundamental tools and techniques in Parameterized Algorithms and is a self-contained guide to the area. The book covers many of the recent developments of the field, including application of important separators, branching based on linear programming, Cut & Count to obtain faster algorithms on tree decompositions, algorithms based on representative families of matroids, and use of the Strong Exponential Time Hypothesis. A number of older results are revisited and explained in a modern and didactic way. The book provides a toolbox of algorithmic techniques. Part I is an overview of basic techniques, each chapter discussing a certain algorithmic paradigm. The material covered in this part can be used for an introductory course on fixed-parameter tractability. Part II discusses more advanced and specialized algorithmic ideas, bringing the reader to the cutting edge of current research. Part III presents complexity results and lower bounds, giving negative evidence by way of W[1]-hardness, the Exponential Time Hypothesis, and kernelization lower bounds. All the results and concepts are introduced at a level accessible to graduate students and advanced undergraduate students. Every chapter is accompanied by exercises, many with hints, while the bibliographic notes point to original publications and related work.

Parameterized Algorithms

http://www.cs.uu.nl/research/techreps/repo/CS-2007/2007-046.pdf

On problems without polynomial kernels

Denial-of-Service (DoS) attacks pose a significant threat to the Internet today especially if they are distributed, i.e., launched simultaneously at a large number of systems. Reactive techniques that try to detect such an attack and throttle down malicious traffic prevail today but usually require an additional infrastructure to be really effective. In this paper we show that preventive mechanisms can be as effective with much less effort: We present an approach to (distributed) DoS attack prevention that is based on the observation that coordinated automated activity by many hosts needs a mechanism to remotely control them. To prevent such attacks, it is therefore possible to identify, infiltrate and analyze this remote control mechanism and to stop it in an automated fashion. We show that this method can be realized in the Internet by describing how we infiltrated and tracked IRC-based botnets which are the main DoS technology used by attackers today.

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Botnet tracking: exploring a root-cause methodology to prevent distributed denial-of-service attacks

The Steiner tree 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 are purely combinatorial. A long-standing open problem is whether there is an LP-relaxation for Steiner tree with integrality gap smaller than 2 [Vazirani,Rajagopalan-SODA'99]. In this paper we improve the approximation factor for Steiner tree, developing an LP-based approximation algorithm. Our algorithm is based on a, seemingly novel, iterative randomized rounding technique. We consider a directed-component cut relaxation for the k-restricted Steiner tree problem. We sample one of these components with probability proportional to the value of the associated variable in the optimal fractional solution and contract it. We iterate this process for a proper number of times and finally output the sampled components together with a minimum-cost terminal spanning tree in the remaining graph. Our algorithm delivers a solution of cost at most ln(4) times the cost of an optimal k-restricted Steiner tree. This directly implies a ln(4)+e

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An improved LP-based approximation for steiner tree

For the vast majority of local graph problems standard dynamic programming techniques give c^tw V^O(1) algorithms, where tw is the treewidth of the input graph. On the other hand, for problems with a global requirement (usually connectivity) the best-known algorithms were naive dynamic programming schemes running in tw^O(tw) V^O(1) time. 
We breach this gap by introducing a technique we dubbed Cut&Count that allows to produce c^tw V^O(1) Monte Carlo algorithms for most connectivity-type problems, including Hamiltonian Path, Feedback Vertex Set and Connected Dominating Set, consequently answering the question raised by Lokshtanov, Marx and Saurabh [SODA'11] in a surprising way. We also show that (under reasonable complexity assumptions) the gap cannot be breached for some problems for which Cut&Count does not work, like CYCLE PACKING. 
The constant c we obtain is in all cases small (at most 4 for undirected problems and at most 6 for directed ones), and in several cases we are able to show that improving those constants would cause the Strong Exponential Time Hypothesis to fail. 
Our results have numerous consequences in various fields, like FPT algorithms, exact and approximate algorithms on planar and H-minor-free graphs and algorithms on graphs of bounded degree. In all these fields we are able to improve the best-known results for some problems.

Solving connectivity problems parameterized by treewidth in single exponential time

For decades, the algorithm providing the smallest proven worst-case running time (with respect to the number of terminals) for the Steiner tree problem has been the one by Dreyfus and Wagner. In this paper, a new algorithm is developed, which improves the running time from O(3kn+2kn2+n3) to (2+δ)k ·poly(n) for arbitrary but fixed δ > 0. Like its predecessor, this algorithm follows the dynamic programming paradigm. Whereas in effect the Dreyfus–Wagner recursion splits the optimal Steiner tree in two parts of arbitrary sizes, our approach looks for a set of nodes that separate the tree into parts containing only few terminals. It is then possible to solve an instance of the Steiner tree problem more efficiently by combining partial solutions.

/pdf/a-faster-algorithm-for-the-steiner-tree-problem-76vsmyjnmo.pdf

A faster algorithm for the steiner tree problem

Parameterized power domination complexity

We present a new dynamic programming algorithm that solves the minimum Steiner tree problem on graphs with k terminals in time O*(ck) for any c > 2. This improves the running time of the previously fastest parameterized algorithm by Dreyfus-Wagner of order O*(3k) and the so-called "full set dynamic programming" algorithm solving rectilinear instances in time O*(2.38k).

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Dynamic Programming for Minimum Steiner Trees

We propose a randomized divide-and-conquer technique that leads to improved randomized and deterministic algorithms for NP-hard path, matching, and packing problems. For the parameterized max-path problem, our randomized algorithm runs in time $O(4^{k}k^{2.7}m)$ and polynomial space (where m is the number of edges in the input graph), improving the previous best randomized algorithm for the problem that runs in time $O(5.44^{k}km)$ and exponential space. Our randomized algorithms for the parameterized max r-d matching and max r-set packing problems run in time $4^{(r-1)k}n^{O(1)}$ and polynomial space, improving the previous best algorithms for the problems that run in time $10.88^{rk}n^{O(1)}$ and exponential space. Moreover, our randomized algorithms can be derandomized to result in significantly improved deterministic algorithms for the problems, and they can be extended to solve other matching and packing problems.

/pdf/randomized-divide-and-conquer-improved-path-matching-and-3559s7icrq.pdf

Randomized Divide-and-Conquer: Improved Path, Matching, and Packing Algorithms

We present a new method of solving graph problems related to Vertex Cover by enumerating and expanding appropriate sets of nodes. As an application, we obtain dramatically improved runtime bounds for two variants of the Vertex Cover problem. In the case of Connected Vertex Cover, we take the upper bound from O *(6 k ) to O *(2.7606 k ) without large hidden factors. For Tree Cover, we show a complexity of O *(3.2361 k ), improving over the previous bound of O *((2k) k ). In the process, faster algorithms for solving subclasses of the Steiner tree problem on graphs are investigated.

Stefan Richter

Papers

A faster algorithm for the steiner tree problem

Parameterized power domination complexity

Dynamic Programming for Minimum Steiner Trees

Randomized Divide-and-Conquer: Improved Path, Matching, and Packing Algorithms

Enumerate and Expand: Improved Algorithms for Connected Vertex Cover and Tree Cover