115 research outputs found
Binarisation for Valued Constraint Satisfaction Problems
We study methods for transforming valued constraint satisfaction problems (VCSPs) to binary VCSPs. First, we show that the standard dual encoding preserves many aspects of the algebraic properties that capture the computational complexity of VCSPs. Second, we extend the reduction of CSPs to binary CSPs described by Bul´ın et al. [Log. Methods Comput. Sci., 11 (2015)] to VCSPs. This reduction establishes that VCSPs over a fixed valued constraint language are polynomial-time equivalent to minimum-cost homomorphism problems over a fixed digraph
Algorithmic Graph Theory
The main focus of this workshop was on mathematical techniques needed for the development of efficient solutions and algorithms for computationally difficult graph problems. The techniques studied at the workshhop included: the probabilistic method and randomized algorithms, approximation and optimization, structured families of graphs and approximation algorithms for large problems. The workshop Algorithmic Graph Theory was attended by 46 participants, many of them being young researchers. In 15 survey talks an overview of recent developments in Algorithmic Graph Theory was given. These talks were supplemented by 10 shorter talks and by two special sessions
Component twin-width as a parameter for BINARY-CSP and its semiring generalizations
We investigate the fine-grained and the parameterized complexity of several generalizations of binary constraint satisfaction problems (BINARY-CSPs), that subsume variants of graph colouring problems. Our starting point is the observation that several algorithmic approaches that resulted in complexity upper bounds for these problems, share a common structure. We thus explore an algebraic approach relying on semirings that unifies different generalizations of BINARY-CSPs (such as the counting, the list, and the weighted versions), and that facilitates a general algorithmic approach to efficiently solving them. The latter is inspired by the (component) twin-width parameter introduced by Bonnet et al., which we generalize via edge-labelled graphs in order to formulate it to arbitrary binary constraints. We consider input instances with bounded component twin-width, as well as constraint templates of bounded component twin-width, and obtain an FPT algorithm as well as an improved, exponential-time algorithm, for broad classes of binary constraints. We illustrate the advantages of this framework by instantiating our general algorithmic approach on several classes of problems (e.g., the H-coloring problem and its variants), and showing that it improves the best complexity upper bounds in the literature for several well-known problems
Invariant percolation and measured theory of nonamenable groups
Using percolation techniques, Gaboriau and Lyons recently proved that every
countable, discrete, nonamenable group contains measurably the free
group on two generators: there exists a probability
measure-preserving, essentially free, ergodic action of on such that almost every -orbit of the
Bernoulli shift splits into -orbits. A combination of this result
and works of Ioana and Epstein shows that every countable, discrete,
nonamenable group admits uncountably many non-orbit equivalent actions.Comment: Bourbaki seminar, 33 page
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