Eduard Eiben

Equimatchable Graphs on Surfaces

A graph G is equimatchable if each matching in G is a subset of a maximum-size matching and it is factor critical if G-v has a perfect matching for each vertex v of G. It is known that any 2-connected equimatchable graph is either bipartite or factor critical. We prove that for 2-connected factor-critical equimatchable graph G the graph Gi¾?VMi¾?{v} is either K2n or Kn,n for some n for any vertex v of G and any minimal matching M such that {v} is a component of Gi¾?VM. We use this result to improve the upper bounds on the maximum number of vertices of 2-connected equimatchable factor-critical graphs embeddable in the orientable surface of genus g to 4g+17 if gi¾?2 and to 12g+5 if gi¾?3. Moreover, for any nonnegative integer g we construct a 2-connected equimatchable factor-critical graph with genus g and more than 42g vertices, which establishes that the maximum size of such graphs is i¾?g. Similar bounds are obtained also for nonorientable surfaces. In the bipartite case for any nonnegative integers g, h, and k we provide a construction of arbitrarily large 2-connected equimatchable bipartite graphs with orientable genus g, respectively nonorientable genus h, and a genus embedding with face-width k. Finally, we prove that any d-degenerate 2-connected equimatchable factor-critical graph has at most 4d+1 vertices, where a graph is d-degenerate if every its induced subgraph contains a vertex of degree at most d.

Solving Problems on Graphs of High Rank-Width

A modulator in a graph is a vertex set whose deletion places the considered graph into some specified graph class. The cardinality of a modulator to various graph classes has long been used as a structural parameter which can be exploited to obtain fixed-parameter algorithms for a range of hard problems. Here we investigate what happens when a graph contains a modulator which is large but “well-structured” (in the sense of having bounded rank-width). Can such modulators still be exploited to obtain efficient algorithms? And is it even possible to find such modulators efficiently? We first show that the parameters derived from such well-structured modulators are more powerful for fixed-parameter algorithms than the cardinality of modulators and rank-width itself. Then, we develop a fixed-parameter algorithm for finding such well-structured modulators to every graph class which can be characterized by a finite set of forbidden induced subgraphs. We proceed by showing how well-structured modulators can be used to obtain efficient parameterized algorithms for Minimum Vertex Cover and Maximum Clique. Finally, we use the concept of well-structured modulators to develop an algorithmic meta-theorem for deciding problems expressible in monadic second order logic, and prove that this result is tight in the sense that it cannot be generalized to LinEMSO problems.

On the Complexity of Rainbow Coloring Problems

An edge-colored graph G is said to be rainbow connected if between each pair of vertices there exists a path which uses each color at most once. The rainbow connection number, denoted by \({{\mathrm{rc}}}(G)\), is the minimum number of colors needed to make G rainbow connected. Along with its variants, which consider vertex colorings and/or so-called strong colorings, the rainbow connection number has been studied from both the algorithmic and graph-theoretic points of view.

Counting Linear Extensions: Parameterizations by Treewidth

We consider the #P-complete problem of counting the number of linear extensions of a poset (#LE); a fundamental problem in order theory with applications in a variety of distinct areas. In particular, we study the complexity of #LE parameterized by the well-known decompositional parameter treewidth for two natural graphical representations of the input poset, i.e., the cover and the incomparability graph. Our main result shows that #LE is fixed-parameter intractable parameterized by the treewidth of the cover graph. This resolves an open problem recently posed in the Dagstuhl seminar on Exact Algorithms. On the positive side we show that #LE becomes fixed-parameter tractable parameterized by the treewidth of the incomparability graph.

On the complexity of rainbow coloring problems

Abstract An edge-colored graph G is said to be rainbow connected if between each pair of vertices there exists a path which uses each color at most once. The rainbow connection number , denoted by rc ( G ) , is the minimum number of colors needed to make G rainbow connected. Along with its variants, which consider vertex colorings and/or so-called strong colorings, the rainbow connection number has been studied from both the algorithmic and graph-theoretic points of view. In this paper we present a range of new results on the computational complexity of computing the four major variants of the rainbow connection number. In particular, we prove that the Strong Rainbow Vertex Coloring problem is NP -complete even on graphs of diameter 3 , and also when the number of colors is restricted to 2 . On the other hand, we show that if the number of colors is fixed then all of the considered problems can be solved in linear time on graphs of bounded treewidth. Moreover, we provide a linear-time algorithm which decides whether it is possible to obtain a rainbow coloring by saving a fixed number of colors from a trivial upper bound. Finally, we give a linear-time algorithm for computing the exact rainbow connection numbers for three variants of the problem on graphs of bounded vertex cover number.

Meta-kernelization using Well-structured Modulators

Kernelization investigates exact preprocessing algorithms with performance guarantees. The most prevalent type of parameters used in kernelization is the solution size for optimization problems; however, also structural parameters have been successfully used to obtain polynomial kernels for a wide range of problems. Many of these parameters can be defined as the size of a smallest modulator of the given graph into a fixed graph class (i.e., a set of vertices whose deletion puts the graph into the graph class). Such parameters admit the construction of polynomial kernels even when the solution size is large or not applicable. This work follows up on the research on meta-kernelization frameworks in terms of structural parameters. We develop a class of parameters which are based on a more general view on modulators: instead of size, the parameters employ a combination of rank-width and split decompositions to measure structure inside the modulator. This allows us to lift kernelization results from modulator-size to more general parameters, hence providing smaller kernels. We show (i) how such large but well-structured modulators can be efficiently approximated, (ii) how they can be used to obtain polynomial kernels for any graph problem expressible in Monadic Second Order logic, and (iii) how they allow the extension of previous results in the area of structural meta-kernelization.

A Structural Approach to Activity Selection

The general task of finding an assignment of agents to activities under certain stability and rationality constraints has led to the introduction of two prominent problems in the area of computational social choice: Group Activity Selection (GASP) and Stable Invitations (SIP). Here we introduce and study the Comprehensive Activity Selection Problem, which naturally generalizes both of these problems. In particular, we apply the parameterized complexity paradigm, which has already been successfully employed for SIP and GASP. While previous work has focused strongly on parameters such as solution size or number of activities, here we focus on parameters which capture the complexity of agent-to-agent interactions. Our results include a comprehensive complexity map for CAS under various restrictions on the number of activities in combination with restrictions on the complexity of agent interactions.

Unary Integer Linear Programming with Structural Restrictions

Recently a number of algorithmic results have appeared which show the tractability of Integer Linear Programming (ILP) instances under strong restrictions on variable domains and/or coefficients (AAAI 2016, AAAI 2017, IJCAI 2017). In this paper, we target ILPs where neither the variable domains nor the coefficients are restricted by a fixed constant or parameter; instead, we only require that our instances can be encoded in unary. We provide new algorithms and lower bounds for such ILPs by exploiting the structure of their variable interactions, represented as a graph. Our first set of results focuses on solving ILP instances through the use of a graph parameter called clique-width, which can be seen as an extension of treewidth which also captures well-structured dense graphs. In particular, we obtain a polynomial-time algorithm for instances of bounded clique-width whose domain and coefficients are polynomially bounded by the input size, and we complement this positive result by a number of algorithmic lower bounds. Afterwards, we turn our attention to ILPs with acyclic variable interactions. In this setting, we obtain a complexity map for the problem with respect to the graph representation used and restrictions on the encoding.

Lossy Kernels for Hitting Subgraphs

In this paper, we study the Connected H-hitting Set and Dominating Set problems from the perspective of approximate kernelization, a framework recently introduced by Lokshtanov et al. [STOC 2017]. For the Connected H-hitting set problem, we obtain an \alpha-approximate kernel for every \alpha>1 and complement it with a lower bound for the natural weighted version. We then perform a refined analysis of the tradeoff between the approximation factor and kernel size for the Dominating Set problem on d-degenerate graphs and provide an interpolation of approximate kernels between the known d^2-approximate kernel of constant size and 1-approximate kernel of size k^{O(d^2)}.

A Single-Exponential Fixed-Parameter Algorithm for Distance-Hereditary Vertex Deletion

Vertex deletion problems ask whether it is possible to delete at most