115,650 research outputs found
Berge's conjecture on directed path partitions—a survey
AbstractBerge's conjecture from 1982 on path partitions in directed graphs generalizes and extends Dilworth's theorem and the Greene–Kleitman theorem which are well known for partially ordered sets. The conjecture relates path partitions to a collection of k independent sets, for each k⩾1. The conjecture is still open and intriguing for all k>1.11Only recently it was proved Berger and Ben-Arroyo Hartman [56] for k=2 (added in proof). In this paper, we will survey partial results on the conjecture, look into different proof techniques for these results, and relate the conjecture to other theorems, conjectures and open problems of Berge and other mathematicians
A survey on algorithmic aspects of modular decomposition
The modular decomposition is a technique that applies but is not restricted
to graphs. The notion of module naturally appears in the proofs of many graph
theoretical theorems. Computing the modular decomposition tree is an important
preprocessing step to solve a large number of combinatorial optimization
problems. Since the first polynomial time algorithm in the early 70's, the
algorithmic of the modular decomposition has known an important development.
This paper survey the ideas and techniques that arose from this line of
research
How the structure of precedence constraints may change the complexity class of scheduling problems
This survey aims at demonstrating that the structure of precedence
constraints plays a tremendous role on the complexity of scheduling problems.
Indeed many problems can be NP-hard when considering general precedence
constraints, while they become polynomially solvable for particular precedence
constraints. We also show that there still are many very exciting challenges in
this research area
- …