# Algorithms for Secretary Problems on Graphs and Hypergraphs

## Abstract

We examine several online matching problems, with applications to Internet
advertising reservation systems. Consider an edge-weighted bipartite graph G, with
partite sets L, R. 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 \in 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. Dimitrov and Plaxton 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 \in L, the
weights on all edges incident to l are identical.) We use a similar algorithm, but
simplify and improve the analysis to obtain a better competitive ratio for the more
general problem. Perhaps of more interest is the fact that our analysis is easily
extended to obtain competitive algorithms for 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.