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 (α, β)-approximation for this problem is a matching with weight at least α fraction of the maximum weighted matching, and cardinality at least β 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 ≤ p ≤ 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.