Related papers: Abusing the Tutte Matrix: An Algebraic Instance Co…
The Odd Cycle Transversal problem (OCT) asks whether a given graph can be made bipartite by deleting at most $k$ of its vertices. In a breakthrough result Reed, Smith, and Vetta (Operations Research Letters, 2004) gave a $\BigOh(4^kkmn)$…
The NP-complete $k$-Path problem asks whether a given undirected graph has a (simple) path of length at least $k$. We prove that $k$-Path has polynomial-size Turing kernels when restricted to planar graphs, graphs of bounded degree,…
The Path Contraction and Cycle Contraction problems take as input an undirected graph $G$ with $n$ vertices, $m$ edges and an integer $k$ and determine whether one can obtain a path or a cycle, respectively, by performing at most $k$ edge…
We give algorithms with running time $2^{O({\sqrt{k}\log{k}})} \cdot n^{O(1)}$ for the following problems. Given an $n$-vertex unit disk graph $G$ and an integer $k$, decide whether $G$ contains (1) a path on exactly/at least $k$ vertices,…
We show that computing the Tutte polynomial of a linear matroid of dimension $k$ on $k^{O(1)}$ points over a field of $k^{O(1)}$ elements requires $k^{\Omega(k)}$ time unless the \#ETH---a counting extension of the Exponential Time…
Research of cycles through specific vertices is a central topic in graph theory. In this context, we focus on a well-studied computational problem, \textsc{$T$-Cycle}: given an undirected $n$-vertex graph $G$ and a set of $k$ vertices…
Let $n$ be the size of a parameterized problem and $k$ the parameter. We present kernels for Feedback Vertex Set, Path Contraction and Cluster Editing/Deletion whose sizes are all polynomial in $k$ and that are computable in polynomial time…
Subexponential parameterized algorithms are known for a wide range of natural problems on planar graphs, but the techniques are usually highly problem specific. The goal of this paper is to introduce a framework for obtaining…
For a family of graphs $\cal F$, the $\mathcal{F}$-Contraction problem takes as an input a graph $G$ and an integer $k$, and the goal is to decide if there exists $S \subseteq E(G)$ of size at most $k$ such that $G/S$ belongs to $\cal F$.…
In this work, we initiate the complexity study of Biclique Contraction and Balanced Biclique Contraction. In these problems, given as input a graph G and an integer k, the objective is to determine whether one can contract at most k edges…
A large number of NP-hard graph problems can be solved in $f(w)n^{O(1)}$ time and space when the input graph is provided together with a tree decomposition of width $w$, in many cases with a modest exponential dependence $f(w)$ on $w$.…
Following recent advances in combining approximation algorithms with fixed-parameter tractability (FPT), we study FPT-time approximation algorithms for minimum-norm $k$-clustering problems, parameterized by the number $k$ of open…
The orbit problem is at the heart of symmetry reduction methods for model checking concurrent systems. It asks whether two given configurations in a concurrent system (represented as finite strings over some finite alphabet) are in the same…
We consider the problem of finding a subcomplex K' of a simplicial complex K such that K' is homeomorphic to the 2-dimensional sphere, S^2. We study two variants of this problem. The first asks if there exists such a K' with at most k…
We introduce the problem of finding a satisfying assignment to a CNF formula that must further belong to a prescribed input subspace. Equivalent formulations of the problem include finding a point outside a union of subspaces (the…
We introduce a new framework for the analysis of preprocessing routines for parameterized counting problems. Existing frameworks that encapsulate parameterized counting problems permit the usage of exponential (rather than polynomial) time…
Local search is a widely-employed strategy for finding good solutions to Traveling Salesman Problem. We analyze the problem of determining whether the weight of a given cycle can be decreased by a popular $k$-opt move. Earlier work has…
The NP-hard Odd Cycle Transversal problem asks for a minimum vertex set whose removal from an undirected input graph $G$ breaks all odd cycles, and thereby yields a bipartite graph. The problem is well-known to be fixed-parameter tractable…
The three-in-a-tree problem asks for an induced tree of the input graph containing three mandatory vertices. In 2006, Chudnovsky and Seymour [Combinatorica, 2010] presented the first polynomial time algorithm for this problem, which has…
Many hard graph problems, such as Hamiltonian Cycle, become FPT when parameterized by treewidth, a parameter that is bounded only on sparse graphs. When parameterized by the more general parameter clique-width, Hamiltonian Cycle becomes…