1,208 research outputs found
Constrained Monotone Function Maximization and the Supermodular Degree
The problem of maximizing a constrained monotone set function has many
practical applications and generalizes many combinatorial problems.
Unfortunately, it is generally not possible to maximize a monotone set function
up to an acceptable approximation ratio, even subject to simple constraints.
One highly studied approach to cope with this hardness is to restrict the set
function. An outstanding disadvantage of imposing such a restriction on the set
function is that no result is implied for set functions deviating from the
restriction, even slightly. A more flexible approach, studied by Feige and
Izsak, is to design an approximation algorithm whose approximation ratio
depends on the complexity of the instance, as measured by some complexity
measure. Specifically, they introduced a complexity measure called supermodular
degree, measuring deviation from submodularity, and designed an algorithm for
the welfare maximization problem with an approximation ratio that depends on
this measure.
In this work, we give the first (to the best of our knowledge) algorithm for
maximizing an arbitrary monotone set function, subject to a k-extendible
system. This class of constraints captures, for example, the intersection of
k-matroids (note that a single matroid constraint is sufficient to capture the
welfare maximization problem). Our approximation ratio deteriorates gracefully
with the complexity of the set function and k. Our work can be seen as
generalizing both the classic result of Fisher, Nemhauser and Wolsey, for
maximizing a submodular set function subject to a k-extendible system, and the
result of Feige and Izsak for the welfare maximization problem. Moreover, when
our algorithm is applied to each one of these simpler cases, it obtains the
same approximation ratio as of the respective original work.Comment: 23 page
Combinatorial Assortment Optimization
Assortment optimization refers to the problem of designing a slate of
products to offer potential customers, such as stocking the shelves in a
convenience store. The price of each product is fixed in advance, and a
probabilistic choice function describes which product a customer will choose
from any given subset. We introduce the combinatorial assortment problem, where
each customer may select a bundle of products. We consider a model of consumer
choice where the relative value of different bundles is described by a
valuation function, while individual customers may differ in their absolute
willingness to pay, and study the complexity of the resulting optimization
problem. We show that any sub-polynomial approximation to the problem requires
exponentially many demand queries when the valuation function is XOS, and that
no FPTAS exists even for succinctly-representable submodular valuations. On the
positive side, we show how to obtain constant approximations under a
"well-priced" condition, where each product's price is sufficiently high. We
also provide an exact algorithm for -additive valuations, and show how to
extend our results to a learning setting where the seller must infer the
customers' preferences from their purchasing behavior
- …