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

We deal with the problem of a center sending a message to a group of users such that some subset of the users is considered revoked and should not be able to obtain the content of the message. We concentrate on the stateless receiver case, where the users do not (necessarily) update their state from session to session. We present a framework called the Subset-Cover framework, which abstracts a variety of revocation schemes including some previously known ones. We provide sufficient conditions that guarantees the security of a revocation algorithm in this class.

We describe two explicit Subset-Cover revocation algorithms; these algorithms are very flexible and work for any number of revoked users. The schemes require storage at the receiver of log N and 1/2 log2 N keys respectively (N is the total number of users), and in order to revoke r users the required message lengths are of r log N and 2r keys respectively. We also provide a general traitor tracing mechanism that can be integrated with any Subset-Cover revocation scheme that satisfies a "bifurcation property". This mechanism does not need an a priori bound on the number of traitors and does not expand the message length by much compared to the revocation of the same set of traitors.

The main improvements of these methods over previously suggested methods, when adopted to the stateless scenario, are: (1) reducing the message length to O(r) regardless of the coalition size while maintaining a single decryption at the user's end (2) provide a seamless integration between the revocation and tracing so that the tracing mechanisms does not require any change to the revocation algorithm.

/pdf/revocation-and-tracing-schemes-for-stateless-receivers-15vjpzqhub.pdf

Revocation and Tracing Schemes for Stateless Receivers

We prove the Minimum Vertex Cover problem to be NP-hard to approximate to within a factor of 1.3606, extending on previous PCP and hardness of approximation technique. To that end, one needs to develop a new proof framework, and to borrow and extend ideas from several fields.

/pdf/on-the-hardness-of-approximating-minimum-vertex-cover-3op116c3qn.pdf

On the hardness of approximating minimum vertex cover

Old and new problems and results in combinatorial number theory

Recently, Razborov obtained superpolynomial lower bounds for monotone circuits that cliques in graphs. In particular, Razborov showed that detecting cliques of sizes in a graphm vertices requires monotone circuits of size Ω(m
 s
 /(logm)2s
 ) for fixeds, and sizem
 Ω(logm) form/4]. In this paper we modify the arguments of Razborov to obtain exponential lower bounds for circuits. In particular, detecting cliques of size (1/4) (m/logm)2/3 requires monotone circuits exp (Ω((m/logm)1/3)). For fixeds, any monotone circuit that detects cliques of sizes requiresm)
 s
 ) AND gates. We show that even a very rough approximation of the maximum clique of a graph requires superpolynomial size monotone circuits, and give lower bounds for some Boolean functions. Our best lower bound for an NP function ofn variables is exp (Ω(n
 1/4 · (logn)1/2)), improving a recent result of exp (Ω(n
 1/8-e)) due to Andreev.

http://www.thi.informatik.uni-frankfurt.de/~jukna/boolean/Alon-Boppana.pdf

Denis Hanson

Papers

Intersection Theorems for Systems of Sets

Lower bounds for certain types of van der Waerden numbers