Saket Saurabh

Lower bounds based on the Exponential Time Hypothesis

The Exponential Time Hypothesis (ETH) is a conjecture stating that, roughly speaking, n-variable 3-SAT cannot be solved in time 2o(n). In this chapter, we prove lower bounds based on ETH for the time needed to solve various problems. In many cases, these lower bounds match (up to small factors) the running time of the best known algorithms for the problem.

Planar F-Deletion: Approximation, Kernelization and Optimal FPT Algorithms

Let F be a finite set of graphs. In the F-DELETION problem, we are given an n-vertex graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. F-DELETION is a generic problem and by selecting different sets of forbidden minors F, one can obtain various fundamental problems such as VERTEX COVER, FEEDBACK VERTEX SET or TREEWIDTH η-DELETION. In this paper we obtain a number of generic algorithmic results about F-DELETION, when F contains at least one planar graph. The highlights of our work are · A constant factor approximation algorithm for the optimization version of F-DELETION; · A linear time and single exponential parameterized algorithm, that is, an algorithm running in time O(2O(k)n), for the parameterized version of F-DELETION where all graphs in F are connected; · A polynomial kernel for parameterized F-DELETION. These algorithms unify, generalize, and improve a multitude of results in the literature. Our main results have several direct applications, but also the methods we develop on the way have applicability beyond the scope of this paper. Our results - constant factor approximation, polynomial kernelization and FPT algorithms - are stringed together by a common theme of polynomial time preprocessing.

Parameterized algorithms for feedback set problems and their duals in tournaments

The parameterized feedback vertex (arc) set problem is to find whether there are k vertices (arcs) in a given graph whose removal makes the graph acyclic. The parameterized complexity of this problem in general directed graphs is a long standing open problem. We investigate the problems on tournaments, a well studied class of directed graphs. We consider both weighted and unweighted versions.We also address the parametric dual problems which are also natural optimization problems. We show that they are fixed parameter tractable not just in tournaments but in oriented directed graphs (where there is at most one directed arc between a pair of vertices). More specifically, the dual problem we show fixed parameter tractable are: Given an oriented directed graph, is there a subset of k vertices (arcs) that forms an acyclic directed subgraph of the graph?Our main results include: • an O((2.4143)knω)1 algorithm for weighted feedback vertex set problem, and an O((2.415)knω) algorithm for weighted feedback arc set problem in tournaments; • an O((e2k/k)kk2 + min{m Ig n, n2}) algorithm for the dual of feedback vertex set problem (maximum vertex induced acyclic graph) in oriented directed graphs, and an O(4kk + m) algorithm for the dual of feedback arc set problem (maximum arcinduced acyclic graph) in general directed graphs.We also show that the dual of feedback vertex set is W[1]--hard in general directed graphs and the feedback arc set problem is fixed parameter tractable in dense directed graphs. Our results are the first non-trivial results for these problems.

Kernel(s) for problems with no kernel: On out-trees with many leaves

The k-Leaf Out-Branching problem is to find an out-branching, that is a rooted oriented spanning tree, with at least k leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the k-Leaf-Out-Branching problem. We give the first polynomial kernel for Rooted k-Leaf-Out-Branching, a variant of k-Leaf-Out-Branching where the root of the tree searched for is also a part of the input. Our kernel with O(k3) vertices is obtained using extremal combinatorics. For the k-Leaf-Out-Branching problem, we show that no polynomial-sized kernel is possible unless coNP is in NP/poly. However, our positive results for Rooted k-Leaf-Out-Branching immediately imply that the seemingly intractable k-Leaf-Out-Branching problem admits a data reduction to n independent polynomial-sized kernels. These two results, tractability and intractability side by side, are the first ones separating Karp kernelization from Turing kernelization. This answers affirmatively an open problem regarding “cheat kernelization” raised by Mike Fellows and Jiong Guo independently.

Faster Parameterized Algorithms Using Linear Programming

We investigate the parameterized complexity of Vertex Cover parameterized by the difference between the size of the optimal solution and the value of the linear programming (LP) relaxation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that combining previously known preprocessing rules with the most straightforward branching algorithm yields an O*(2.618k) algorithm for the problem. Here, k is the excess of the vertex cover size over the LP optimum, and we write O*(f(k)) for a time complexity of the form O(f(k)nO(1)). We proceed to show that a more sophisticated branching algorithm achieves a running time of O*(2.3146k). Following this, using previously known as well as new reductions, we give O*(2.3146k) algorithms for the parameterized versions of Above Guarantee Vertex Cover, Odd Cycle Transversal, Split Vertex Deletion, and Almost 2-SAT, and O*(1.5214k) algorithms for König Vertex Deletion and Vertex Cover parameterized by the size of the smallest odd cycle transversal and König vertex deletion set. These algorithms significantly improve the best known bounds for these problems. The most notable improvement among these is the new bound for Odd Cycle Transversal—this is the first algorithm that improves on the dependence on k of the seminal O*(3k) algorithm of Reed, Smith, and Vetta. Finally, using our algorithm, we obtain a kernel for the standard parameterization of Vertex Cover with at most 2k − clog k vertices. Our kernel is simpler than previously known kernels achieving the same size bound.

Graph Layout Problems Parameterized by Vertex Cover

In the framework of parameterized complexity, one of the most commonly used structural parameters is the treewidth of the input graph. The reason for this is that most natural graph problems turn out to be fixed parameter tractable when parameterized by treewidth. However, Graph Layout problems are a notable exception. In particular, no fixed parameter tractable algorithms are known for the Cutwidth, Bandwidth, Imbalance and Distortion problems parameterized by treewidth. In fact, Bandwidth remains NP-complete even restricted to trees. A possible way to attack graph layout problems is to consider structural parameterizations that are stronger than treewidth. In this paper we study graph layout problems parameterized by the size of the minimum vertex cover of the input graph. We show that all the mentioned problems are fixed parameter tractable. Our basic ingredient is a classical algorithm for Integer Linear Programming when parameterized by dimension, designed by Lenstra and later improved by Kannan. We hope that our results will serve to re-emphasize the importance and utility of this algorithm.

Fast FAST

We present a randomized subexponential time, polynomial space parameterized algorithm for the k -Weighted Feedback Arc Set in Tournaments (k -FAST ) problem. We also show that our algorithm can be derandomized by slightly increasing the running time. To derandomize our algorithm we construct a new kind of universal hash functions, that we coin universal coloring families . For integers m ,k and r , a family

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

{\mathcal F}

The Complexity Ecology of Parameters: An Illustration Using Bounded Max Leaf Number

of functions from [m ] to [r ] is called a universal (m ,k ,r )-coloring family if for any graph G on the set of vertices [m ] with at most k edges, there exists an

On Problems as Hard as CNF-SAT

f \in {\mathcal F}