arXiv subjects

Dieter Kratsch

Publications and source records attributed to Dieter Kratsch.

11 recordsLinked to original sources

arXiv2021

Refined Notions of Parameterized Enumeration Kernels with Applications to Matching Cut Enumeration

An enumeration kernel as defined by Creignou et al. [Theory Comput. Syst. 2017] for a parameterized enumeration problem consists of an algorithm that transforms each instance into one whose size is bounded by the parameter plus a solution-lifting algorithm that efficiently enumerates all solutions from the set of the solutions of the kernel. We propose to consider two new versions of enumeration kernels by asking that the solutions of the original instance can be enumerated in polynomial time or with polynomial delay from the kernel solutions. Using the NP-hard Matching Cut problem parameterized by structural parameters such as the vertex cover number or the cyclomatic number of the input graph, we show that the new enumeration kernels present a useful notion of data reduction for enumeration problems which allows to compactly represent the set of feasible solutions.

cs.DS↗

arXiv2018

Algorithms for outerplanar graph roots and graph roots of pathwidth at most 2

Deciding whether a given graph has a square root is a classical problem that has been studied extensively both from graph theoretic and from algorithmic perspectives. The problem is NP-complete in general, and consequently substantial effort has been dedicated to deciding whether a given graph has a square root that belongs to a particular graph class. There are both polynomial-time solvable and NP-complete cases, depending on the graph class. We contribute with new results in this direction. Given an arbitrary input graph G, we give polynomial-time algorithms to decide whether G has an outerplanar square root, and whether G has a square root that is of pathwidth at most 2.

cs.DS↗

arXiv2016

Space-Efficient Biconnected Components and Recognition of Outerplanar Graphs

We present space-efficient algorithms for computing cut vertices in a given graph with $n$ vertices and $m$ edges in linear time using $O(n+\min\{m,n\log \log n\})$ bits. With the same time and using $O(n+m)$ bits, we can compute the biconnected components of a graph. We use this result to show an algorithm for the recognition of (maximal) outerplanar graphs in $O(n\log \log n)$ time using $O(n)$ bits.

cs.DS↗

arXiv2016

Squares of Low Maximum Degree

A graph H is a square root of a graph G if G can be obtained from H by adding an edge between any two vertices in H that are of distance 2. The Square Root problem is that of deciding whether a given graph admits a square root. This problem is only known to be NP-complete for chordal graphs and polynomial-time solvable for non-trivial minor-closed graph classes and a very limited number of other graph classes. We prove that Square Root is O(n)-time solvable for graphs of maximum degree 5 and O(n^4)-time solvable for graphs of maximum degree at most 6.

cs.DS↗

arXiv2016

A Linear Kernel for Finding Square Roots of Almost Planar Graphs

A graph H is a square root of a graph G if G can be obtained from H by the addition of edges between any two vertices in H that are of distance 2 from each other. The Square Root problem is that of deciding whether a given graph admits a square root. We consider this problem for planar graphs in the context of the "distance from triviality" framework. For an integer k, a planar+kv graph (or k-apex graph) is a graph that can be made planar by the removal of at most k vertices. We prove that a generalization of Square Root, in which some edges are prescribed to be either in or out of any solution, has a kernel of size O(k) for planar+kv graphs, when parameterized by k. Our result is based on a new edge reduction rule which, as we shall also show, has a wider applicability for the Square Root problem.

cs.DS↗

arXiv2016

Enumeration and Maximum Number of Minimal Connected Vertex Covers in Graphs

Connected Vertex Cover is one of the classical problems of computer science, already mentioned in the monograph of Garey and Johnson. Although the optimization and decision variants of finding connected vertex covers of minimum size or weight are well studied, surprisingly there is no work on the enumeration or maximum number of minimal connected vertex covers of a graph. In this paper we show that the maximum number of minimal connected vertex covers of a graph is at most 1.8668^n, and these can be enumerated in time O(1.8668^n). For graphs of chordality at most 5, we are able to give a better upper bound, and for chordal graphs and distance-hereditary graphs we are able to give tight bounds on the maximum number of minimal connected vertex covers.

cs.DS↗

arXiv2015

Faster algorithms to enumerate hypergraph transversals

A transversal of a hypergraph is a set of vertices intersecting each hyperedge. We design and analyze new exponential-time algorithms to enumerate all inclusion-minimal transversals of a hypergraph. For each fixed k>2, our algorithms for hypergraphs of rank k, where the rank is the maximum size of a hyperedge, outperform the previous best. This also implies improved upper bounds on the maximum number of minimal transversals in n-vertex hypergraphs of rank k>2. Our main algorithm is a branching algorithm whose running time is analyzed with Measure and Conquer. It enumerates all minimal transversals of hypergraphs of rank 3 on n vertices in time O(1.6755^n). Our algorithm for hypergraphs of rank 4 is based on iterative compression. Our enumeration algorithms improve upon the best known algorithms for counting minimum transversals in hypergraphs of rank k for k>2 and for computing a minimum transversal in hypergraphs of rank k for k>5.

cs.DS↗

arXiv2015

Output-Polynomial Enumeration on Graphs of Bounded (Local) Linear MIM-Width

The linear induced matching width (LMIM-width) of a graph is a width parameter defined by using the notion of branch-decompositions of a set function on ternary trees. In this paper we study output-polynomial enumeration algorithms on graphs of bounded LMIM-width and graphs of bounded local LMIM-width. In particular, we show that all 1-minimal and all 1-maximal (σ,ρ)-dominating sets, and hence all minimal dominating sets, of graphs of bounded LMIM-width can be enumerated with polynomial (linear) delay using polynomial space. Furthermore, we show that all minimal dominating sets of a unit square graph can be enumerated in incremental polynomial time.

cs.DS↗

arXiv2014

Finding Shortest Paths between Graph Colourings

The $k$-colouring reconfiguration problem asks whether, for a given graph $G$, two proper $k$-colourings $α$ and $β$ of $G$, and a positive integer $\ell$, there exists a sequence of at most $\ell+1$ proper $k$-colourings of $G$ which starts with $α$ and ends with $β$ and where successive colourings in the sequence differ on exactly one vertex of $G$. We give a complete picture of the parameterized complexity of the $k$-colouring reconfiguration problem for each fixed $k$ when parameterized by $\ell$. First we show that the $k$-colouring reconfiguration problem is polynomial-time solvable for $k=3$, settling an open problem of Cereceda, van den Heuvel and Johnson. Then, for all $k \geq 4$, we show that the $k$-colouring reconfiguration problem, when parameterized by $\ell$, is fixed-parameter tractable (addressing a question of Mouawad, Nishimura, Raman, Simjour and Suzuki) but that it has no polynomial kernel unless the polynomial hierarchy collapses.

cs.CC↗

arXiv2013

Parameterized Algorithms for Finding Square Roots

We show that the following two problems are fixed-parameter tractable with parameter k: testing whether a connected n-vertex graph with m edges has a square root with at most n-1+k edges and testing whether such a graph has a square root with at least m-k edges. Our first result implies that squares of graphs obtained from trees by adding at most k edges can be recognized in polynomial time for every fixed k>=0; previously this result was known only for k=0. Our second result is equivalent to stating that deciding whether a graph can be modified into a square root of itself by at most k edge deletions is fixed-parameter tractable with parameter k.

cs.DS↗

arXiv2012

Generating All Minimal Edge Dominating Sets with Incremental-Polynomial Delay

For an arbitrary undirected simple graph G with m edges, we give an algorithm with running time O(m^4 |L|^2) to generate the set L of all minimal edge dominating sets of G. For bipartite graphs we obtain a better result; we show that their minimal edge dominating sets can be enumerated in time O(m^4 |L|). In fact our results are stronger; both algorithms generate the next minimal edge dominating set with incremental-polynomial delay O(m^5 |L|) and O(m^4 |L|) respectively, when L is the set of already generated minimal edge dominating sets. Our algorithms are tailored for and solve the equivalent problems of enumerating minimal (vertex) dominating sets of line graphs and line graphs of bipartite graphs, with incremental-polynomial delay, and consequently in output-polynomial time. Enumeration of minimal dominating sets in graphs has very recently been shown to be equivalent to enumeration of minimal transversals in hypergraphs. The question whether the minimal transversals of a hypergraph can be enumerated in output-polynomial time is a fundamental and challenging question in Output-Sensitive Enumeration; it has been open for several decades and has triggered extensive research in the field.

cs.DS↗