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 \(O(\log \log (k)/\log (k))\) 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 by giving upper bounds of \(O(\log (k)/k)\) , and \(O(\log (n)/\sqrt{n})\) for the prophet IID setting. 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. These systems include some classes of matroids, matroid intersections, and matchoids.
