4 research outputs found
A laminarity property of the polyhedron described by a weakly posi-modular set function
AbstractRecently, Nagamochi and Ibaraki have introduced a concept of posi-modular set function and considered the structure of the polyhedron described by an intersecting submodular and posi-modular function. They showed that the facets of the polyhedron form a laminar family. We show that such a laminarity property also holds for a much more general class of set functions, called weakly posi-modular set functions, without submodularity
Posimodular Function Optimization
Given a posimodular function on a finite set , we
consider the problem of finding a nonempty subset of that minimizes
. Posimodular functions often arise in combinatorial optimization such as
undirected cut functions. In this paper, we show that any algorithm for the
problem requires oracle calls to , where
. It contrasts to the fact that the submodular function minimization,
which is another generalization of cut functions, is polynomially solvable.
When the range of a given posimodular function is restricted to be
for some nonnegative integer , we show that
oracle calls are necessary, while we propose an
-time algorithm for the problem. Here, denotes the
time needed to evaluate the function value for a given .
We also consider the problem of maximizing a given posimodular function. We
show that oracle calls are necessary for solving the problem,
and that the problem has time complexity when
is the range of for some constant .Comment: 18 page