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

Communication Complexity: Basics

We present a fast algorithm for the subset convolution problem:given functions f and g defined on the lattice of subsets of ann-element set n, compute their subset convolution f*g, defined for S⊆ N by [ (f * g)(S) = [T ⊆ S] f(T) g(S/T),,]where addition and multiplication is carried out in an arbitrary ring. Via Mobius transform and inversion, our algorithm evaluates the subset convolution in O(n2 2n) additions and multiplications, substanti y improving upon the straightforward O(3n) algorithm. Specifically, if the input functions have aninteger range [-M,-M+1,...,M], their subset convolution over the ordinary sum--product ring can be computed in O(2n log M) time; the notation O suppresses polylogarithmic factors.Furthermore, using a standard embedding technique we can compute the subset convolution over the max--sum or min--sum semiring in O(2n M) time.To demonstrate the applicability of fast subset convolution, wepresent the first O(2k n2 + n m) algorithm for the Steiner tree problem in graphs with n vertices, k terminals, and m edges with bounded integer weights, improving upon the O(3kn + 2k n2 + n m) time bound of the classical Dreyfus-Wagner algorithm. We also discuss extensions to recent O(2n)-time algorithms for covering and partitioning problems (Bjorklund and Husfeldt, FOCS 2006; Koivisto, FOCS 2006).

Fourier meets Möbius: fast subset convolution

https://hal.univ-angers.fr/hal-02709508/document

A review on algorithms for maximum clique problems

Many large decentralized systems rely on information propagation to ensure their proper function. We examine a common scenario in which only participants that are aware of the information can compete for some reward, and thus informed participants have an incentive not to propagate information to others. One recent example in which such tension arises is the 2009 DARPA Network Challenge (finding red balloons). We focus on another prominent example: Bitcoin, a decentralized electronic currency system.Bitcoin represents a radical new approach to monetary systems. It has been getting a large amount of public attention over the last year, both in policy discussions and in the popular press. Its cryptographic fundamentals have largely held up even as its usage has become increasingly widespread. We find, however, that it exhibits a fundamental problem of a different nature, based on how its incentives are structured. We propose a modification to the protocol that can eliminate this problem.Bitcoin relies on a peer-to-peer network to track transactions that are performed with the currency. For this purpose, every transaction a node learns about should be transmitted to its neighbors in the network. As the protocol is currently defined and implemented, it does not provide an incentive for nodes to broadcast transactions they are aware of. In fact, it provides an incentive not to do so. Our solution is to augment the protocol with a scheme that rewards information propagation. Since clones are easy to create in the Bitcoin system, an important feature of our scheme is Sybil-proofness.We show that our proposed scheme succeeds in setting the correct incentives, that it is Sybil-proof, and that it requires only a small payment overhead, all this is achieved with iterated elimination of dominated strategies. We complement this result by showing that there are no reward schemes in which information propagation and no self-cloning is a dominant strategy.

/pdf/on-bitcoin-and-red-balloons-3r9qolagic.pdf

On bitcoin and red balloons

In this paper we present a simple but powerful subgraph sampling primitive that is applicable in a variety of computational models including dynamic graph streams (where the input graph is defined by a sequence of edge/hyperedge insertions and deletions) and distributed systems such as MapReduce. In the case of dynamic graph streams, we use this primitive to prove the following results:• Matching: Our main result for matchings is that there exists an O(k2) space algorithm that returns the edges of a maximum matching on the assumption the cardinality is at most k. The best previous algorithm used O(kn) space where n is the number of vertices in the graph and we prove our result is optimal up to logarithmic factors. Our algorithm has O(1) update time. We also show that there exists an O(n2/α3) space algorithm that returns an α-approximation for matchings of arbitrary size. In independent work, Assadi et al. (SODA 2016) proved this approximation algorithm is optimal and provided an alternative algorithm. We generalize our exact and approximate algorithms to weighted matching. For graphs with low arboricity such as planar graphs, the space required for constant approximation can be further reduced. While there has been a substantial amount of work on approximate matching in insert-only graph streams, these are the first non-trivial results in the dynamic setting.• Vertex Cover and Hitting Set: There exists an O(kd) space algorithm that solves the minimum hitting set problem where d is the cardinality of the input sets and k is an upper bound on the size of the minimum hitting set. We prove this is optimal up to logarithmic factors. Our algorithm has O(1) update time. The case d = 2 corresponds to minimum vertex cover.Finally, we consider a larger family of parameterized problems (including b-matching, disjoint paths, vertex coloring among others) for which our subgraph sampling primitive yields fast, small-space dynamic graph stream algorithms. We then show lower bounds for natural problems outside this family.

/pdf/kernelization-via-sampling-with-applications-to-finding-57kfka4vdx.pdf

Kernelization via sampling with applications to finding matchings and related problems in dynamic graph streams

We introduce a new technique for designing fixed-parameter algorithms for cut problems, namely randomized contractions. With our framework: * We obtain the first FPT algorithm for the parameterized version of the UNIQUE LABEL COVER problem, with single exponential dependency on the size of the cutset and the size of the alphabet. As a consequence, we extend the set of the polynomial time solvable instances of UNIQUE GAMES to those with at most O(sqrt{log n}) violated constraints. * We obtain a new FPT algorithm for the STEINER CUT problem with exponential speed-up over the recent work of Kawarabayashi and Thorup (FOCS'11). * We show how to combine considering 'cut' and 'uncut' constraints at the same time. We define a robust problem NODE MULTIWAY CUT-UNCUT that can serve as an abstraction of introducing uncut constraints, and show that it admits an FPT algorithm with single exponential dependency on the size of the cutset. To the best of our knowledge, the only known way of tackling uncut constraints was via the approach of Marx, O'Sullivan and Razgon (STACS'10), which yields algorithms with double exponential running time. An interesting aspect of our algorithms is that they can handle real weights, to the best of our knowledge, the technique of important separators does not work in the weighted version.

/pdf/designing-fpt-algorithms-for-cut-problems-using-randomized-40r9kspblg.pdf

Designing FPT Algorithms for Cut Problems Using Randomized Contractions

We introduce a new technique for designing fixed-parameter algorithms for cut problems, called randomized contractions. We apply our framework to obtain the first fixed-parameter algorithms (FPT algorithms) with exponential speed up for the Steiner Cut and Node Multiway Cut-Uncut problems. We prove that the parameterized version of the Unique Label Cover problem, which is the base of the Unique Games Conjecture, can be solved in $2^{O(k^2\log |\Sigma|)}n^4\log n$ deterministic time (even in the stronger, vertex-deletion variant), where $k$ is the number of unsatisfied edges and $|\Sigma|$ is the size of the alphabet. As a consequence, we show that one can in polynomial time solve instances of Unique Games where the number of edges allowed not to be satisfied is upper bounded by $O(\sqrt{\log n})$ to optimality, which improves over the trivial $O(1)$ upper bound. We prove that the Steiner Cut problem can be solved in $2^{O(k^2\log k)}n^4\log n$ deterministic time and $\tilde{O}(2^{O(k^2\log k)}n^2)$ random...

/pdf/designing-fpt-algorithms-for-cut-problems-using-randomized-1u3mijpmwt.pdf

Given a directed graph $G$, a set of $k$ terminals and an integer $p$, the \textsc{Directed Vertex Multiway Cut} problem asks if there is a set $S$ of at most $p$ (nonterminal) vertices whose removal disconnects each terminal from all other terminals. \textsc{Directed Edge Multiway Cut} is the analogous problem where $S$ is a set of at most $p$ edges. These two problems indeed are known to be equivalent. A natural generalization of the multiway cut is the \emph{multicut} problem, in which we want to disconnect only a set of $k$ given pairs instead of all pairs. Marx (Theor. Comp. Sci. 2006) showed that in undirected graphs multiway cut is fixed-parameter tractable (FPT) parameterized by $p$. Marx and Razgon (STOC 2011) showed that undirected multicut is FPT and directed multicut is W[1]-hard parameterized by $p$. We complete the picture here by our main result which is that both \textsc{Directed Vertex Multiway Cut} and \textsc{Directed Edge Multiway Cut} can be solved in time $2^{2^{O(p)}}n^{O(1)}$, i.e., FPT parameterized by size $p$ of the cutset of the solution. This answers an open question raised by Marx (Theor. Comp. Sci. 2006) and Marx and Razgon (STOC 2011). It follows from our result that \textsc{Directed Multicut} is FPT for the case of $k=2$ terminal pairs, which answers another open problem raised in Marx and Razgon (STOC 2011).

Fixed-Parameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset

Given a directed graph $G$, a set of $k$ terminals, and an integer $p$, the Directed Vertex Multiway Cut problem asks whether there is a set $S$ of at most $p$ (nonterminal) vertices whose removal disconnects each terminal from all other terminals. Directed Edge Multiway Cut is the analogous problem where $S$ is a set of at most $p$ edges. These two problems are indeed known to be equivalent. A natural generalization of the multiway cut is the Multicut problem, in which we want to disconnect only a set of $k$ given pairs instead of all pairs. Marx [Theoret. Comput. Sci., 351 (2006), pp. 394--406] showed that in undirected graphs Vertex/Edge Multiway cut is fixed-parameter tractable (FPT) parameterized by $p$. Marx and Razgon [Proceedings of the 43rd ACM Symposium on Theory of Computing, 2011, pp. 469--478] showed that undirected Multicut is FPT and Directed Multicut is W[1]-hard parameterized by $p$. We complete the picture here by our main result, which is that both Directed Vertex Multiway Cut and Direc...

Rajesh Chitnis

Papers

Kernelization via sampling with applications to finding matchings and related problems in dynamic graph streams

Designing FPT Algorithms for Cut Problems Using Randomized Contractions

Designing FPT Algorithms for Cut Problems Using Randomized Contractions

Fixed-Parameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset

Fixed-Parameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset