42 research outputs found
Majority choosability of digraphs
A \emph{majority coloring} of a digraph is a coloring of its vertices such
that for each vertex , at most half of the out-neighbors of has the same
color as . A digraph is \emph{majority -choosable} if for any
assignment of lists of colors of size to the vertices there is a majority
coloring of from these lists. We prove that every digraph is majority
-choosable. This gives a positive answer to a question posed recently by
Kreutzer, Oum, Seymour, van der Zypen, and Wood in \cite{Kreutzer}. We obtain
this result as a consequence of a more general theorem, in which majority
condition is profitably extended. For instance, the theorem implies also that
every digraph has a coloring from arbitrary lists of size three, in which at
most of the out-neighbors of any vertex share its color. This solves
another problem posed in \cite{Kreutzer}, and supports an intriguing conjecture
stating that every digraph is majority -colorable
Algorithm for the Stochastic Generalized Transportation Problem
The equalization method for the stochastic generalized transportation problem has been presented. The algorithm allows us to find the optimal solution to the problem of minimizing the expected total cost in the generalized transportation problem with random demand. After a short introduction and literature review, the algorithm is presented. It is a version of the method proposed by the author for the nonlinear generalized transportation problem. It is shown that this version of the method generates a sequence of solutions convergent to the KKT point. This guarantees the global optimality of the obtained solution, as the expected cost functions are convex and twice differentiable. The computational experiments performed for test problems of reasonable size show that the method is fast. (original abstract