Zusammenfassung
We study various generalizations of the secretary problem with submodular
objective functions. Generally, a set of requests is revealed step-by-step to
an algorithm in random order. For each request, one option has to be selected
so as to maximize a monotone submodular function while ensuring feasibility.
For our results, we assume that we are given an offline algorithm computing an
$\alpha$-approximation for the respective problem. This way, we separate
computational limitations from the ones due to the online nature. When only
focusing on the online aspect, we can assume $\alpha = 1$.
In the submodular secretary problem, feasibility constraints are cardinality
constraints. That is, out of a randomly ordered stream of entities, one has to
select a subset size $k$. For this problem, we present a
$0.31\alpha$-competitive algorithm for all $k$, which asymptotically reaches
competitive ratio $\frac{\alpha}{e}$ for large $k$. In submodular secretary
matching, one side of a bipartite graph is revealed online. Upon arrival, each
node has to be matched permanently to an offline node or discarded irrevocably.
We give an $\frac{\alpha}{4}$-competitive algorithm. In both cases, we improve
over previously best known competitive ratios, using a generalization of the
algorithm for the classic secretary problem.
Furthermore, we give an $O(\alpha d^{-\frac{2}{B-1}})$-competitive algorithm
for submodular function maximization subject to linear packing constraints.
Here, $d$ is the column sparsity, that is the maximal number of none-zero
entries in a column of the constraint matrix, and $B$ is the minimal capacity
of the constraints. Notably, this bound is independent of the total number of
constraints. We improve the algorithm to be $O(\alpha
d^{-\frac{1}{B-1}})$-competitive if both $d$ and $B$ are known to the algorithm
beforehand.
