We give a complete characterization of parameter graphs H for which the problem of coloring H-free graphs is polynomial and for which it is NP-complete. We further initiate a study of this problem for two forbidden subgraphs.

Complexity of Coloring Graphs without Forbidden Induced Subgraphs

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The disjoint paths problem in quadratic time

We consider the following maximum disjoint paths problem (MDPP). We are given a large network, and pairs of nodes that wish to communicate over paths through the network-the goal is to simultaneously connect as many of these pairs as possible in such a way that no two communication paths share an edge in the network. This classical problem has been brought into focus recently in papers discussing applications to routing in high-speed networks, where the current lack of understanding of the MDPP is an obstacle to the design of practical heuristics. We consider the class of densely embedded, nearly-Eulerian graphs, which includes the two-dimensional mesh and other planar and locally planar interconnection networks. We obtain a constant-factor approximation algorithm for the maximum disjoint paths problem for this class of graphs; this improves on an O(log n)-approximation for the special case of the two-dimensional mesh due to Aumann-Rabani and the authors. For networks that are not explicitly required to be "high-capacity," this is the first constant-factor approximation for the MDPP in any class of graphs other than trees. We also consider the MDPP in the on-line setting, relevant to applications in which connection requests arrive over time and must be processed immediately. Here we obtain an asymptptically optimal O(log n)competitive on-line algorithm for the same class of graphs; this improves on an O(log n log log n) competitive algorithm for the special case of the mesh due to B. Awerbuch et al (1994).

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Disjoint paths in densely embedded graphs

Problem (1)

https://www-lipn.univ-paris13.fr/~letocart/SURVEY.pdf

Minimal multicut and maximal integer multiflow: A survey

In this paper we consider the disjoint paths problem. Given a graphG and a subsetS of the edge-set ofG the problem is to decide whether there exists a family ℱ of disjoint circuits inG each containing exactly one edge ofS such that every edge inS belongs to a circuit inC. By a well-known theorem of P. Seymour the edge-disjoint paths problem is polynomially solvable for Eulerian planar graphsG. We show that (assumingP≠NP) one can drop neither planarity nor the Eulerian condition onG without losing polynomial time solvability. We prove theNP-completeness of the planar edge-disjoint paths problem by showing theNP-completeness of the vertex disjoint paths problem for planar graphs with maximum vertex-degree three. This disproves (assumingP≠NP) a conjecture of A. Schrijver concerning the existence of a polynomial time algorithm for the planar vertex-disjoint paths problem. Furthermore we present a counterexample to a conjecture of A. Frank. This conjecture would have implied a polynomial algorithm for the planar edge-disjoint paths problem. Moreover we derive a complete characterization of all minorclosed classes of graphs for which the disjoint paths problem is polynomially solvable. Finally we show theNP-completeness of the half-integral relaxation of the edge-disjoint paths problem. This implies an answer to the long-standing question whether the edge-disjoint paths problem is polynomially solvable for Eulerian graphs.

On the complexity of the disjoint paths problem

The computational complexity of a number of problems concerning induced structures in graphs is studied, and compared with the complexity of corresponding problems concerning non-induced structures. The effect on these problems of restricting the input to planar graphs is also considered. The principal results include: (1) Induced Maximum Matching and Induced Directed Path are NP-complete for planar graphs, (2) for every fixed graphH, InducedH-Minor Testing can be accomplished for planar graphs in time0(n), and (3) there are graphsH for which InducedH-Minor Testing is NP-complete for unrestricted input. Some useful structural theorems concerning induced minors are presented, including a bound on the treewidth of planar graphs that exclude a planar induced minor.

The complexity of induced minors and related problems

Weakly transitive orientations, Hasse diagrams and string graphs

The three-dimensional strongly screened vortex glass model is studied numerically using methods from combinatorial optimization. We focus on the effect of disorder strength on the ground state and find the existence of a disorder-driven normal-to-superconducting phase transition. The transition turns out to be a geometrical phase transition with percolating vortex loops in the ground state configuration. We determine the critical exponents and provide evidence for a new universality class of correlated percolation.

Superconductor-to-normal phase transition in a vortex glass model: numerical evidence for a new percolation universality class

The vortex-glass model for a disordered high-${T}_{c}$ superconductor in an external magnetic field is studied in the strong screening limit. With exact ground state (i.e., $T=0)$ calculations we show that (1) the ground state of the vortex configuration varies drastically with infinitesimal variations of the strength of the external field; (2) the minimum energy of global excitation loops of length scale L do not depend on the strength of the external field; however, (3) the excitation loops themself depend sensibly on the field. From (2) we infer the absence of a true superconducting state at any finite-temperature independent of the external field.

Frank Pfeiffer

Papers

On the complexity of the disjoint paths problem

The complexity of induced minors and related problems

Weakly transitive orientations, Hasse diagrams and string graphs

Superconductor-to-normal phase transition in a vortex glass model: numerical evidence for a new percolation universality class

Numerical study of the strongly screened vortex glass model in an external field