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

Convex bodies: the brunn–minkowski theory

Let M = (E, I) be a matroid and let S = {S1, ...,St} be a family of subsets of E of size p. A subfamily S ⊆ S is q-representative for S if for every set Y ⊆ E of size at most q, if there is a set X e S disjoint from Y with X ∪ Y e I, then there is a set X e S disjoint from Y with X ∪ Y e I. By the classical result of Bollobas, in a uniform matroid, every family of sets of size p has a q-representative family with at most (p+qp) sets. In his famous "two families theorem" from 1977, Lovasz proved that the same bound also holds for any matroid representable over a field F. As observed by Marx, Lovasz's proof is constructive. In this paper we show how Lovasz's proof can be turned into an algorithm constructing a q-representative family of size at most (p+qp) in time bounded by a polynomial in (p+qp), t, and the time required for field operations.We demonstrate how the efficient construction of representative families can be a powerful tool for designing single-exponential parameterized (2O(k)) and exact exponential time (2O(k)) algorithms. The applications of our approach include the following.• In the Long Directed Cycle problem the input is a directed n-vertex graph G and the positive integer k. The task is to find a directed cycle of length at least k in G, if such a cycle exists. As a consequence of our 8k+onO(1) time algorithm, we have that a directed cycle of length at least log n, if such cycle exists, can be found in polynomial time. As it was shown by Bjorklund, Husfeldt, and Khanna [ICALP 2004], under an appropriate complexity assumption, it is impossible to improve this guarantee by more than a constant factor. Thus our algorithm not only improves over the best previous log n/log log n bound of Gabow and Nie [SODA 2004] but also closes the gap between known lower and upper bounds for this problem.• In the Minimum Equivalent Graph (MEG) problem we are seeking a spanning subdigraph D' of a given n-vertex digraph D with as few arcs as possible in which the reachability relation is the same as in the original digraph D. The existence of a single-exponential cn-time algorithm for some constant c > 1 for MEG was open since the work of Moyles and Thompson [JACM 1969].• To demonstrate the diversity of applications of the approach, we provide an alternative proof of the results recently obtained by Bodlaender, Cygan, Kratsch and Nederlof for algorithms on graphs of bounded treewidth, who showed that many "connectivity" problems such as Hamiltonian Cycle or Steiner Tree can be solved in time 2O(t)n on n-vertex graphs of treewidth at most t. We believe that expressing graph problems in "matroid language" shed light on what makes it possible to solve connectivity problems single-exponential time parameterized by treewidth.For the special case of uniform matroids on n elements, we give a faster algorithm computing a representative family in time O((p+q/q)q · 2o(p+q) · t · log n). We use this algorithm to provide the fastest known deterministic parameterized algorithms for k-Path, k-Tree, and more generally, for k-Subgraph Isomorphism, where the k-vertex pattern graph is of constant treewidth. For example, our k-Path algorithm runs in time O(2.851kn log2n log W) on weighted graphs with maximum edge weight W.

https://epubs.siam.org/doi/pdf/10.1137/1.9781611973402.10

Efficient computation of representative sets with applications in parameterized and exact algorithms

Combinatorial Mathematics: THE PRINCIPLE OF INCLUSION AND EXCLUSION

Let M=(E, I) be a matroid and let S=lS1, ċ , Str be a family of subsets of E of size p. A subfamily S ⊆ S is q-representative for S if for every set Y⊆E of size at most q, if there is a set X ∈ S disjoint from Y with X∪ Y ∈ I, then there is a set Xˆ ∈ S disjoint from Y with Xˆ ∪ Y ∈ I. By the classic result of Bollobas, in a uniform matroid, every family of sets of size p has a q-representative family with at most (p+qp) sets. In his famous “two families theorem” from 1977, Lovasz proved that the same bound also holds for any matroid representable over a field F. We give an efficient construction of a q-representative family of size at most (p+qp) in time bounded by a polynomial in (p+qp), t, and the time required for field operations.We demonstrate how the efficient construction of representative families can be a powerful tool for designing single-exponential parameterized and exact exponential time algorithms. The applications of our approach include the following:—In the Long Directed Cycle problem, the input is a directed n-vertex graph G and the positive integer k. The task is to find a directed cycle of length at least k in G, if such a cycle exists. As a consequence of our 6.75k+o(k)nO(1) time algorithm, we have that a directed cycle of length at least log n, if such a cycle exists, can be found in polynomial time.—In the Minimum Equivalent Graph (MEG) problem, we are seeking a spanning subdigraph D′ of a given n-vertex digraph D with as few arcs as possible in which the reachability relation is the same as in the original digraph D.—We provide an alternative proof of the recent results for algorithms on graphs of bounded treewidth showing that many “connectivity” problems such as Hamiltonian Cycle or Steiner Tree can be solved in time 2O(t)n on n-vertex graphs of treewidth at most t.For the special case of uniform matroids on n elements, we give a faster algorithm to compute a representative family. We use this algorithm to provide the fastest known deterministic parameterized algorithms for k-Path, k-Tree, and, more generally, k-Subgraph Isomorphism, where the k-vertex pattern graph is of constant treewidth.

Efficient Computation of Representative Families with Applications in Parameterized and Exact Algorithms

http://www.ii.uib.no/~daniello/papers/CountingSubgraphsJCSS.pdf

Faster algorithms for finding and counting subgraphs

In this paper we study a natural generalization of both {\sc $k$-Path} and {\sc $k$-Tree} problems, namely, the {\sc Subgraph Isomorphism} problem. 
In the {\sc Subgraph Isomorphism} problem we are given two graphs $F$ and $G$ on $k$ and $n$ vertices respectively as an input, and the question is whether there exists a subgraph of $G$ isomorphic to $F$. We show that if the treewidth of $F$ is at most $t$, then there is a randomized algorithm for the {\sc Subgraph Isomorphism} problem running in time $\cO^*(2^k n^{2t})$. To do so, we associate a new multivariate {Homomorphism polynomial} of degree at most $k$ with the {\sc Subgraph Isomorphism} problem and construct an arithmetic circuit of size at most $n^{\cO(t)}$ for this polynomial. Using this polynomial, we also give a deterministic algorithm to count the number of homomorphisms from $F$ to $G$ that takes $n^{\cO(t)}$ time and uses polynomial space. For the counting version of the {\sc Subgraph Isomorphism} problem, where the objective is to count the number of distinct subgraphs of $G$ that are isomorphic to $F$, we give a deterministic algorithm running in time and space $\cO^*({n \choose k/2}n^{2p})$ or ${n\choose k/2}n^{\cO(t \log k)}$. We also give an algorithm running in time $\cO^{*}(2^{k}{n \choose k/2}n^{5p})$ and taking space polynomial in $n$. Here $p$ and $t$ denote the pathwidth and the treewidth of $F$, respectively. Thus our work not only improves on known results on {\sc Subgraph Isomorphism} but it also extends and generalize most of the known results on {\sc $k$-Path} and {\sc $k$-Tree}.

Faster Algorithms for Finding and Counting Subgraphs

In the uniform circuit model of computation, the width of a boolean circuit exactly characterises the "space" complexity of the computed function. Looking for a similar relationship in Valiant's algebraic model of computation, we propose width of an arithmetic circuit as a possible measure of space. We introduce the class VL as an algebraic variant of deterministic log-space L. In the uniform setting, we show that our definition coincides with that of VPSPACE at polynomial width.

Further, todefinealgebraicvariants ofnon-deterministic space-bounded classes, we introduce the notion of "read-once" certificates for arithmetic circuits. We show that polynomial-size algebraic branching programs can be expressed as a read-once exponential sum over polynomials in VL, i.e. VBP ∈ ΣR ċ VL. We also show that ΣR ċ VBP = VBP, i.e. VBPs are stable under read-once exponential sums. Further, we show that read-once exponential sumsover a restricted class of constant-width arithmetic circuits are within VQP, and this is the largest known such subclass of poly-log-width circuits with this property.

/pdf/small-space-analogues-of-valiant-s-classes-5dsxi3fugo.pdf

Small-space analogues of Valiant's classes

We study structural properties of restricted width arithmetical circuits. It is shown that syntactically multilinear arithmetical circuits of constant width can be efficiently simulated by syntactically multilinear algebraic branching programs of constant width, i.e. that sm-VSC 0 *** sm-VBWBP. Also, we obtain a direct characterization of poly -size arithmetical formulas in terms of multiplicatively disjoint constant width circuits, i.e. MD-VSC0 = VNC1.

For log -width weakly-skew circuits a width efficient multilinearity preserving simulation by algebraic branching programs is given, i.e. weaklyskew-sm-VSC1 *** sm-VBP[width=log2 n ].

Finally, coefficient functions are considered, and closure properties are observed for sm-VSC i , and in general for a variety of other syntactic multilinear restrictions of algebraic complexity classes.

Simulation of Arithmetical Circuits by Branching Programs with Preservation of Constant Width and Syntactic Multilinearity

Euclidean optimization problems such as TSP and minimum-length matching admit fast partitioning algorithms that compute near-optimal solutions on typical instances.

/pdf/smoothed-analysis-of-partitioning-algorithms-for-euclidean-ir8exb9ytb.pdf

B. V. Raghavendra Rao

Papers

Faster algorithms for finding and counting subgraphs

Faster Algorithms for Finding and Counting Subgraphs

Small-space analogues of Valiant's classes

Simulation of Arithmetical Circuits by Branching Programs with Preservation of Constant Width and Syntactic Multilinearity

Smoothed Analysis of Partitioning Algorithms for Euclidean Functionals