Almost Optimal Stochastic Weighted Matching With Few Queries
release_stpop6pbhrb77gc74pyrdezjyy
by
Soheil Behnezhad, Nima Reyhani
2017
Abstract
We consider the stochastic matching problem. An edge-weighted general
(i.e., not necessarily bipartite) graph G(V, E) is given in the input, where
each edge in E is realized independently with probability p; the
realization is initially unknown, however, we are able to query the edges
to determine whether they are realized. The goal is to query only a small
number of edges to find a realized matching that is sufficiently close to
the maximum matching among all realized edges. This problem has received a
considerable attention during the past decade due to its numerous real-world
applications in kidney-exchange, matchmaking services, online labor markets,
and advertisements.
Our main result is an adaptive algorithm that for any arbitrarily small
ϵ > 0, finds a (1-ϵ)-approximation in expectation, by
querying only O(1) edges per vertex. We further show that our approach leads
to a (1/2-ϵ)-approximate non-adaptive algorithm that also
queries only O(1) edges per vertex. Prior to our work, no nontrivial
approximation was known for weighted graphs using a constant per-vertex budget.
The state-of-the-art adaptive (resp. non-adaptive) algorithm of Maehara and
Yamaguchi [SODA 2018] achieves a (1-ϵ)-approximation (resp.
(1/2-ϵ)-approximation) by querying up to O(wn) edges per
vertex where w denotes the maximum integer edge-weight. Our result is a
substantial improvement over this bound and has an appealing message: No matter
what the structure of the input graph is, one can get arbitrarily close to the
optimum solution by querying only a constant number of edges per vertex.
To obtain our results, we introduce novel properties of a generalization of
augmenting paths to weighted matchings that may be of independent
interest.
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