Algorithms for Secretary Problems on Graphs and Hypergraphs

Abstract

We examine online matching problems with applications to Internet advertising reservation systems. Consider an edge-weighted bipartite graph G(L ∪ R, E). We develop an 8-competitive algorithm for the following secretary problem: Initially given R, and the size of L, the algorithm receives the vertices of L sequentially, in a random order. When a vertex l ∈ L is seen, all edges incident to l are revealed, together with their weights. The algorithm must immediately either match l to an available vertex of R, or decide that l will remain unmatched.

In [5], the authors show a 16-competitive algorithm for the transversal matroid secretary problem, which is the special case with weights on vertices, not edges. (Equivalently, one may assume that for each l ∈ L, the weights on all edges incident to l are identical.) We use a very similar algorithm, but simplify and improve the analysis to obtain a better competitive ratio for the more general problem. Our analysis is easily extended to obtain competitive algorithms for a class of similar problems, such as to find disjoint sets of edges in hypergraphs where edges arrive online. We also introduce secretary problems with adversarially chosen groups.

Finally, we give a 2e-competitive algorithm for the secretary problem on graphic matroids, where, with edges appearing online, the goal is to find a maximum-weight acyclic subgraph of a given graph.