Bicriteria Online Matching: Maximizing Weight and Cardinality
Abstract
Inspired by online ad allocation problems,
many results have been developed for
online matching problems.
Most of the previous work deals with a single objective,
but, in practice, there is a need to optimize multiple objectives.
Here, as an illustrative example motivated by display ads allocation,
we study a bi-objective online matching problem.
In particular, we consider a set of fixed nodes (ads) with capacity
constraints, and a set of online items (pageviews) arriving one by
one. Upon arrival of an online item $i$, a set of eligible fixed
neighbors (ads) for the item is revealed, together with a weight
$w_{ia}$ for eligible neighbor $a$. The problem is to assign each item to
an eligible neighbor online, while respecting the capacity
constraints; the goal is to maximize both the total weight of the
matching and the cardinality.
In this paper, we present both approximation algorithms and
hardness results for this problem.
An $(\alpha, \beta)$-approximation for this problem is a matching with
weight at least $\alpha$ fraction of the maximum
weighted matching, and cardinality
at least $\beta$ fraction of maximum cardinality matching.
We present a parametrized approximation
algorithm that allows a smooth tradeoff curve between the two
objectives: when the capacities of fixed nodes are
large, we give a $p(1- 1/e^{1/p}), (1-p)(1-1/e^{1/1-p})$-approximation
for any $0 \le p \le 1$, and prove a `hardness curve' combining several
inapproximability
results. These upper and lower bounds are always close (with a maximum
gap of $9\%$), and exactly coincide at the point $(0.43, 0.43)$.
For small capacities, we present a
smooth parametrized approximation curve for the problem
between $(0,1-1/e)$ and $(1/2,0)$ passing
through a $(1/3,0.3698)$-approximation.