5 research outputs found
An Economic-Based Analysis of RANKING for Online Bipartite Matching
We give a simple proof showing that the RANKING algorithm introduced by Karp,
Vazirani and Vazirani \cite{DBLP:conf/stoc/KarpVV90} is
competitive for the online bipartite matching problem. Our proof resembles the
proof given by Devanur, Jain and Kleinberg [2013], but does not make an
explicit use of linear programming duality; instead, it is based on an economic
interpretation of the matching problem. In our interpretation, one set of
vertices represent items that are assigned prices, and the other set of
vertices represent unit-demand buyers that arrive sequentially and choose their
most-demanded items
The Invisible Hand of Dynamic Market Pricing
Walrasian prices, if they exist, have the property that one can assign every
buyer some bundle in her demand set, such that the resulting assignment will
maximize social welfare. Unfortunately, this assumes carefully breaking ties
amongst different bundles in the buyer demand set. Presumably, the shopkeeper
cleverly convinces the buyer to break ties in a manner consistent with
maximizing social welfare. Lacking such a shopkeeper, if buyers arrive
sequentially and simply choose some arbitrary bundle in their demand set, the
social welfare may be arbitrarily bad. In the context of matching markets, we
show how to compute dynamic prices, based upon the current inventory, that
guarantee that social welfare is maximized. Such prices are set without knowing
the identity of the next buyer to arrive. We also show that this is impossible
in general (e.g., for coverage valuations), but consider other scenarios where
this can be done. We further extend our results to Bayesian and bounded
rationality models
Online Matching with General Arrivals
The online matching problem was introduced by Karp, Vazirani and Vazirani
nearly three decades ago. In that seminal work, they studied this problem in
bipartite graphs with vertices arriving only on one side, and presented optimal
deterministic and randomized algorithms for this setting. In comparison, more
general arrival models, such as edge arrivals and general vertex arrivals, have
proven more challenging and positive results are known only for various
relaxations of the problem. In particular, even the basic question of whether
randomization allows one to beat the trivially-optimal deterministic
competitive ratio of for either of these models was open. In this
paper, we resolve this question for both these natural arrival models, and show
the following.
1. For edge arrivals, randomization does not help --- no randomized algorithm
is better than competitive.
2. For general vertex arrivals, randomization helps --- there exists a
randomized -competitive online matching algorithm
Online Stochastic Max-Weight Matching: prophet inequality for vertex and edge arrival models
We provide prophet inequality algorithms for online weighted matching in
general (non-bipartite) graphs, under two well-studied arrival models, namely
edge arrival and vertex arrival. The weight of each edge is drawn independently
from an a-priori known probability distribution. Under edge arrival, the weight
of each edge is revealed upon arrival, and the algorithm decides whether to
include it in the matching or not. Under vertex arrival, the weights of all
edges from the newly arriving vertex to all previously arrived vertices are
revealed, and the algorithm decides which of these edges, if any, to include in
the matching. To study these settings, we introduce a novel unified framework
of batched prophet inequalities that captures online settings where elements
arrive in batches; in particular it captures matching under the two
aforementioned arrival models. Our algorithms rely on the construction of
suitable online contention resolution scheme (OCRS). We first extend the
framework of OCRS to batched-OCRS, we then establish a reduction from batched
prophet inequality to batched OCRS, and finally we construct batched OCRSs with
selectable ratios of 0.337 and 0.5 for edge and vertex arrival models,
respectively. Both results improve the state of the art for the corresponding
settings. For the vertex arrival, our result is tight. Interestingly, a
pricing-based prophet inequality with comparable competitive ratios is unknown.Comment: 29 pages, 2 figure
Prophet Inequality Matching Meets Probing with Commitment
Within the context of stochastic probing with commitment, we consider the
online stochastic matching problem for bipartite graphs where edges adjacent to
an online node must be probed to determine if they exist, based on known edge
probabilities. If a probed edge exists, it must be used in the matching (if
possible). In addition to improving upon existing stochastic bipartite matching
results, our results can also be seen as extensions to multi-item prophet
inequalities. We study this matching problem for given constraints on the
allowable sequences of probes adjacent to an online node. Our setting
generalizes the patience (or time-out) constraint which limits the number of
probes that can be made to edges. The generality of our setting leads to some
modelling and computational efficiency issues that are not encountered in
previous works. We establish new competitive bounds all of which generalize the
standard non-stochastic setting when edges do not need to be probed (i.e.,
exist with certainty). Specifically, we establish the following competitive
ratio results for a general formulation of edge constraints, arbitrary edge
weights, and arbitrary edge probabilities:
(1) A tight ratio when the stochastic graph is generated from a
known stochastic type graph where the online node is drawn
independently from a known distribution and is chosen
adversarially. We refer to this setting as the known i.d. stochastic matching
problem with adversarial arrivals.
(2) A ratio when the stochastic graph is generated from a known
stochastic type graph where the online node is drawn
independently from a known distribution and is a
random permutation. This is referred to as the known i.d. stochastic matching
problem with random order arrivals.Comment: Corrected typos and added details to various proof