  We consider an online multi-weighted generalization of several classic online
optimization problems, called the online combinatorial assignment problem. We
are given an independence system over a ground set of elements and agents that
arrive online one by one. Upon arrival, each agent reveals a weight function
over the elements of the ground set. If the independence system is given by the
matchings of a hypergraph we recover the combinatorial auction problem, where
every node represents an item to be sold, and every edge represents a bundle of
items. For combinatorial auctions, Kesselheim et al. showed upper bounds of
O(loglog(k)/log(k)) and $O(\log \log(n)/\log(n))$ on the competitiveness of any
online algorithm, even in the random order model, where $k$ is the maximum
bundle size and $n$ is the number of items. We provide an exponential
improvement on these upper bounds to show that the competitiveness of any
online algorithm in the prophet IID setting is upper bounded by $O(\log(k)/k)$,
and $O(\log(n)/\sqrt{n})$. Furthermore, using linear programming, we provide
new and improved guarantees for the $k$-bounded online combinatorial auction
problem (i.e., bundles of size at most $k$). We show a
$(1-e^{-k})/k$-competitive algorithm in the prophet IID model, a
$1/(k+1)$-competitive algorithm in the prophet-secretary model using a single
sample per agent, and a $k^{-k/(k-1)}$-competitive algorithm in the secretary
model. Our algorithms run in polynomial time and work in more general
independence systems where the offline combinatorial assignment problem admits
the existence of a polynomial-time randomized algorithm that we call
certificate sampler. We show that certificate samplers have a nice interplay
with random order models, and we also provide new polynomial-time competitive
algorithms for some classes of matroids, matroid intersections, and matchoids.
