Torsten Tholey

Solving the 2-Disjoint Paths Problem in Nearly Linear Time

AbstractGiven four distinct vertices s1,s2,t1, and t2 of a graph G, the 2-disjoint paths problem is to determine two disjoint paths, p1 from s1 to t1 and p2 from s2 to t2, if such paths exist. Disjoint can mean vertex- or edge-disjoint. Both, the edge- and the vertex-disjoint version of the problem, are NP-hard in the case of directed graphs. For undirected graphs, we show that the O(mn)-time algorithm of Shiloach can be modified to solve the 2-vertex-disjoint paths problem in only O(n + mα(m,n)) time, where m is the number of edges in G, n is the number of vertices in G, and where α denotes the inverse of the Ackermann function. Our result also improves the running time for the 2-edge-disjoint paths problem on undirected graphs as well as the running times for the 2-vertex- and the 2-edge-disjoint paths problem on dags.

Approximation Algorithms for Intersection Graphs

We study three complexity parameters that, for each vertex v, are an upper bound for the number of cliques that are sufficient to cover a subset S(v) of its neighbors. We call a graph k-perfectly groupable if S(v) consists of all neighbors, k-simplicial if S(v) consists of the neighbors with a higher number after assigning distinct numbers to all vertices, and k-perfectly orientable if S(v) consists of the endpoints of all outgoing edges from v for an orientation of all edges. These parameters measure in some sense how chordal-like a graph is—the last parameter was not previously considered in literature. The similarity to chordal graphs is used to construct simple polynomial-time approximation algorithms with constant approximation ratio for many NP-hard problems, when restricted to graphs for which at least one of the three complexity parameters is bounded by a constant. As applications we present approximation algorithms with constant approximation ratio for maximum weighted independent set, minimum (independent) dominating set, minimum vertex coloring, maximum weighted clique, and minimum clique partition for large classes of intersection graphs.

The complexity of minimum convex coloring

A coloring of the vertices of a graph is called convex if each subgraph induced by all vertices of the same color is connected. We consider three variants of recoloring a colored graph with minimal cost such that the resulting coloring is convex. Two variants of the problem are shown to be NP-hard on trees even if in the initial coloring each color is used to color only a bounded number of vertices. For graphs of bounded treewidth, we present a polynomial-time (2+@e)-approximation algorithm for these two variants and a polynomial-time algorithm for the third variant. Our results also show that, unless NP@?DTIME(n^O^(^l^o^g^l^o^g^n^)), there is no polynomial-time approximation algorithm with a ratio of size (1-o(1))lnlnN for the following problem: given pairs of vertices in an undirected N-vertex graph of bounded treewidth, determine the minimal possible number l for which all except l pairs can be connected by disjoint paths.

Finding disjoint paths on directed acyclic graphs

Given k + 1 pairs of vertices (s1,s2),(u1,v1),...,(uk,vk) of a directed acyclic graph, we show that a modified version of a data structure of Suurballe and Tarjan can output, for each pair (ul,vl) with 1 ≤ l≤ k, a tuple (s1,t1,s2,t2) with {t1,t2} = {ul,vl} in constant time such that there are two disjoint paths p1, from s1 to t1, and p2, from s2 to t2, if such a tuple exists. Disjoint can mean vertex- as well as edge-disjoint. As an application we show that the presented data structure can be used to improve the previous best known running time O(mn) for the so called 2-disjoint paths problem on directed acyclic graphs to O(m(log2+m/nn) + nlog3n). In this problem, given a tuple (s1,s2,t1,t2) of four vertices, we want to construct two disjoint paths p1, from s1 to t1, and p2, from s2 to t2, if such paths exist.

The k-Disjoint Paths Problem on Chordal Graphs

Algorithms based on a bottom-up traversal of a tree decomposition are used in literature to develop very efficient algorithms for graphs of bounded treewidth. However, such algorithms can also be used to efficiently solve problems on chordal graphs, which in general do not have a bounded treewidth. By combining this approach with a sparsification technique we obtain the first linear-time algorithm for chordal graphs that solves the k-disjoint paths problem. In this problem k pairs of vertices are to be connected by pairwise vertex-disjoint paths. We also present the first polynomial-time algorithm for chordal graphs capable of finding disjoint paths solving the k-disjoint paths problem with minimal total length. Finally, we prove that the version of the disjoint paths problem, where k is part of the input, is

The Complexity of Minimum Convex Coloring

\mathcal{NP}

Approximation algorithms for intersection graphs

-hard on chordal graphs.

A Dynamic Data Structure for Maintaining Disjoint Paths Information in Digraphs

A coloring of the vertices of a graph is called convex if each subgraph induced by all vertices of the same color is connected. We consider three variants of recoloring a colored graph with minimal cost such that the resulting coloring is convex. Two variants of the problem are shown to be

Efficient Minimal Perfect Hashing in Nearly Minimal Space

{\mathcal{NP}}

Approximate tree decompositions of planar graphs in linear time

-hard on trees even if in the initial coloring each color is used to color only a bounded number of vertices. For graphs of bounded treewidth, we present a polynomial-time (2 + e)-approximation algorithm for these two variants and a polynomial-time algorithm for the third variant. Our results also show that, unless