Submodular Secretary Problems: Cardinality, Matching, and Linear Constraints
Abstract
We study various generalizations of the secretary problem with submodular objective functions. Generally, a set of requests is revealed stepbystep 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 $O(\alpha d^{\frac{2}{B1}})$competitive algorithm for submodular function maximization subject to linear packing constraints. Here, $d$ is the column sparsity, that is the maximal number of nonezero 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 $O(\alpha d^{\frac{1}{B1}})$competitive if both $d$ and $B$ are known to the algorithm beforehand.
 Publication:

arXiv eprints
 Pub Date:
 July 2016
 arXiv:
 arXiv:1607.08805
 Bibcode:
 2016arXiv160708805K
 Keywords:

 Computer Science  Data Structures and Algorithms