M. S. Ramanujan

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.

On the Kernelization Complexity of Colorful Motifs

The Colorful Motif problem asks if, given a vertex-colored graph G, there exists a subset S of vertices of G such that the graph induced by G on S is connected and contains every color in the graph exactly once. The problem is motivated by applications in computational biology and is also well-studied from the theoretical point of view. In particular, it is known to be NP-complete even on trees of maximum degree three [Fellows et al, ICALP 2007]. In their pioneering paper that introduced the color-coding technique, Alon et al. [STOC 1995] show, inter alia, that the problem is FPT on general graphs. More recently, Cygan et al. [WG 2010] showed that Colorful Motif is NP-complete on comb graphs, a special subclass of the set of trees of maximum degree three. They also showed that the problem is not likely to admit polynomial kernels on forests.

LP can be a cure for Parameterized Problems

We investigate the parameterized complexity of Vertex Cover parameterized above the optimum value of the linear programming (LP) relaxation of the integer linear programming formulation of the problem. By carefully analyzing the change in the LP value in the branching steps, we argue that even the most straightforward branching algorithm (after some preprocessing) results in an O (2.6181 r ) algorithm for the problem where r is the excess of the vertex cover size over the LP optimum. We write O (f(k)) for a time complexity of the form O(f(k)n O(1) ), where f(k) grows exponentially with k.

Parameterized tractability of multiway cut with parity constraints

In this paper, we study a parity based generalization of the classical Multiway Cut problem. Formally, we study the Parity Multiway Cut problem, where the input is a graph G, vertex subsets Te and To (T=Te∪To) called terminals, a positive integer k and the objective is to test whether there exists a k-sized vertex subset S such that S intersects all odd paths from v∈To to T∖{v} and all even paths from v∈Te to T∖{v}. When Te=To, this is precisely the classical Multiway Cut problem. If To=∅ then this is the Even Multiway Cut problem and if Te=∅ then this is the Odd Multiway Cut problem. We remark that even the problem of deciding whether there is a set of at most k vertices that intersects all odd paths between a pair of vertices s and t is NP-complete. Our primary motivation for studying this problem is the recently initiated parameterized study of parity versions of graphs minors (Kawarabayashi, Reed and Wollan, FOCS 2011) and separation problems similar to Multiway Cut. The area of design of parameterized algorithms for graph separation problems has seen a lot of recent activity, which includes algorithms for Multi-Cut on undirected graphs (Marx and Razgon, STOC 2011, Bousquet, Daligault and Thomasse, STOC 2011), k-way cut (Kawarabayashi and Thorup, FOCS 2011), and Multiway Cut on directed graphs (Chitnis, Hajiaghayi and Marx, SODA 2012). A second motivation is that this problem serves as a good example to illustrate the application of a generalization of important separators which we introduce, and can be applied even when most of the recently develped tools fail to apply. We believe that this could be a useful tool for several other separation problems as well. We obtain this generalization by dividing the graph into slices with small boundaries and applying a divide and conquer paradigm over these slices. We show that Parity Multiway Cut is fixed parameter tractable (FPT) by giving an algorithm that runs in time

Linear Time Parameterized Algorithms for Subset Feedback Vertex Set

f(k)n^{{\mathcal{O}}(1)}

Faster parameterized algorithms for deletion to split graphs

. More precisely, we show that instances of this problem with solutions of size

Linear-Time Parameterized Algorithms via Skew-Symmetric Multicuts

{\cal O}(\log \log n)

Parameterized Complexity of Directed Steiner Tree on Sparse Graphs

can be solved in polynomial time. Along with this new notion of generalized important separators, our algorithm also combines several ideas used in previous parameterized algorithms for graph separation problems including the notion of important separators and randomized selection of important sets to simplify the input instance.

Fixed-Parameter Tractability of Workflow Satisfiability in the Presence of Seniority Constraints

In the Subset Feedback Vertex Set (Subset FVS) problem, the input is a graph \(G\) on \(n\) vertices and \(m\) edges, a subset of vertices \(T\), referred to as terminals, and an integer \(k\). The objective is to determine whether there exists a set of at most \(k\) vertices intersecting every cycle that contains a terminal. The study of parameterized algorithms for this generalization of the Feedback Vertex Set problem has received significant attention over the last few years. In fact the parameterized complexity of this problem was open until 2011, when two groups independently showed that the problem is fixed parameter tractable (FPT). Using tools from graph minors Kawarabayashi and Kobayashi obtained an algorithm for Subset FVS running in time \({\mathcal {O}}(f(k)\cdot n^2 m)\) [SODA 2012, JCTB 2012]. Independently, Cygan et al. [ICALP 2011, SIDMA 2013] designed an algorithm for Subset FVS running in time \(2^{{\mathcal {O}}(k \log k)}\cdot n^{{\mathcal {O}}(1)}\). More recently, Wahlstrom obtained the first single exponential time algorithm for Subset FVS, running in time \(4^{k}\cdot n^{{\mathcal {O}}(1)}\) [SODA 2014]. While the \(2^{{\mathcal {O}}(k)}\) dependence on the parameter \(k\) is optimal under the Exponential Time Hypothesis (ETH), the dependence of this algorithm as well as those preceding it, on the input size is far from linear.

Lossy kernelization

An undirected graph is said to be split if its vertex set can be partitioned into two sets such that the subgraph induced on one of them is a complete graph and the subgraph induced on the other is an independent set. We study the problem of deleting the minimum number of vertices or edges from a given input graph so that the resulting graph is split.We initiate a systematic study and give efficient fixed-parameter algorithms and polynomial sized kernels for the problem. More precisely, 1 for Split Vertex Deletion, the problem of determining whether there are k vertices whose deletion results in a split graph, we give an