9 research outputs found
A Match in Time Saves Nine: Deterministic Online Matching With Delays
We consider the problem of online Min-cost Perfect Matching with Delays
(MPMD) introduced by Emek et al. (STOC 2016). In this problem, an even number
of requests appear in a metric space at different times and the goal of an
online algorithm is to match them in pairs. In contrast to traditional online
matching problems, in MPMD all requests appear online and an algorithm can
match any pair of requests, but such decision may be delayed (e.g., to find a
better match). The cost is the sum of matching distances and the introduced
delays.
We present the first deterministic online algorithm for this problem. Its
competitive ratio is , where is the
number of requests. This is polynomial in the number of metric space points if
all requests are given at different points. In particular, the bound does not
depend on other parameters of the metric, such as its aspect ratio. Unlike
previous (randomized) solutions for the MPMD problem, our algorithm does not
need to know the metric space in advance
Stable Secretaries
We define and study a new variant of the secretary problem. Whereas in the
classic setting multiple secretaries compete for a single position, we study
the case where the secretaries arrive one at a time and are assigned, in an
on-line fashion, to one of multiple positions. Secretaries are ranked according
to talent, as in the original formulation, and in addition positions are ranked
according to attractiveness. To evaluate an online matching mechanism, we use
the notion of blocking pairs from stable matching theory: our goal is to
maximize the number of positions (or secretaries) that do not take part in a
blocking pair. This is compared with a stable matching in which no blocking
pair exists. We consider the case where secretaries arrive randomly, as well as
that of an adversarial arrival order, and provide corresponding upper and lower
bounds.Comment: Accepted for presentation at the 18th ACM conference on Economics and
Computation (EC 2017
Tight Bounds for Online Matching in Bounded-Degree Graphs with Vertex Capacities
We study the b-matching problem in bipartite graphs G = (S,R,E). Each vertex s ? S is a server with individual capacity b_s. The vertices r ? R are requests that arrive online and must be assigned instantly to an eligible server. The goal is to maximize the size of the constructed matching. We assume that G is a (k,d)-graph [J. Naor and D. Wajc, 2018], where k specifies a lower bound on the degree of each server and d is an upper bound on the degree of each request. This setting models matching problems in timely applications.
We present tight upper and lower bounds on the performance of deterministic online algorithms. In particular, we develop a new online algorithm via a primal-dual analysis. The optimal competitive ratio tends to 1, for arbitrary k ? d, as the server capacities increase. Hence, nearly optimal solutions can be computed online. Our results also hold for the vertex-weighted problem extension, and thus for AdWords and auction problems in which each bidder issues individual, equally valued bids.
Our bounds improve the previous best competitive ratios. The asymptotic competitiveness of 1 is a significant improvement over the previous factor of 1-1/e^{k/d}, for the interesting range where k/d ? 1 is small. Recall that 1-1/e ? 0.63. Matching problems that admit a competitive ratio arbitrarily close to 1 are rare. Prior results rely on randomization or probabilistic input models
Improved Competitive Ratio for Edge-Weighted Online Stochastic Matching
We consider the edge-weighted online stochastic matching problem, in which an
edge-weighted bipartite graph G=(I\cup J, E) with offline vertices J and online
vertex types I is given. The online vertices have types sampled from I with
probability proportional to the arrival rates of online vertex types. The
online algorithm must make immediate and irrevocable matching decisions with
the objective of maximizing the total weight of the matching. For the problem
with general arrival rates, Feldman et al. (FOCS 2009) proposed the Suggested
Matching algorithm and showed that it achieves a competitive ratio of 1-1/e
\approx 0.632. The ratio has recently been improved to 0.645 by Yan (2022), who
proposed the Multistage Suggested Matching (MSM) algorithm. In this paper, we
propose the Evolving Suggested Matching (ESM) algorithm, and show that it
achieves a competitive ratio of 0.650.Comment: To appear in WINE202
A Randomness Threshold for Online Bipartite Matching, via Lossless Online Rounding
Over three decades ago, Karp, Vazirani and Vazirani (STOC'90) introduced the
online bipartite matching problem. They observed that deterministic algorithms'
competitive ratio for this problem is no greater than , and proved that
randomized algorithms can do better. A natural question thus arises: \emph{how
random is random}? i.e., how much randomness is needed to outperform
deterministic algorithms? The \textsc{ranking} algorithm of Karp et
al.~requires random bits, which, ignoring polylog terms,
remained unimproved. On the other hand, Pena and Borodin (TCS'19) established a
lower bound of random bits for any
competitive ratio.
We close this doubly-exponential gap, proving that, surprisingly, the lower
bound is tight. In fact, we prove a \emph{sharp threshold} of random bits for the randomness necessary and sufficient to
outperform deterministic algorithms for this problem, as well as its
vertex-weighted generalization. This implies the same threshold for the advice
complexity (nondeterminism) of these problems.
Similar to recent breakthroughs in the online matching literature, for
edge-weighted matching (Fahrbach et al.~FOCS'20) and adwords (Huang et
al.~FOCS'20), our algorithms break the barrier of by randomizing matching
choices over two neighbors. Unlike these works, our approach does not rely on
the recently-introduced OCS machinery, nor the more established randomized
primal-dual method. Instead, our work revisits a highly-successful online
design technique, which was nonetheless under-utilized in the area of online
matching, namely (lossless) online rounding of fractional algorithms. While
this technique is known to be hopeless for online matching in general, we show
that it is nonetheless applicable to carefully designed fractional algorithms
with additional (non-convex) constraints
Online Algorithms for Matching Platforms with Multi-Channel Traffic
Two-sided platforms rely on their recommendation algorithms to help visitors
successfully find a match. However, on platforms such as VolunteerMatch (VM) --
which has facilitated millions of connections between volunteers and nonprofits
-- a sizable fraction of website traffic arrives directly to a nonprofit's
volunteering page via an external link, thus bypassing the platform's
recommendation algorithm. We study how such platforms should account for this
external traffic in the design of their recommendation algorithms, given the
goal of maximizing successful matches. We model the platform's problem as a
special case of online matching, where (using VM terminology) volunteers arrive
sequentially and probabilistically match with one opportunity, each of which
has finite need for volunteers. In our framework, external traffic is
interested only in their targeted opportunity; by contrast, internal traffic
may be interested in many opportunities, and the platform's online algorithm
selects which opportunity to recommend. In evaluating different algorithms, we
parameterize the competitive ratio based on the amount of external traffic.
After demonstrating the shortcomings of a commonly-used algorithm that is
optimal in the absence of external traffic, we propose a new algorithm --
Adaptive Capacity (AC) -- which accounts for matches differently based on
whether they originate from internal or external traffic. We provide a lower
bound on AC's competitive ratio that is increasing in the amount of external
traffic and that is close to the parameterized upper bound we establish on the
competitive ratio of any online algorithm. Our analysis utilizes a path-based,
pseudo-rewards approach, which we further generalize to settings where the
platform can recommend a ranked set of opportunities. Beyond our theoretical
results, we show the strong performance of AC in a case study motivated by VM
data
LIPIcs, Volume 244, ESA 2022, Complete Volume
LIPIcs, Volume 244, ESA 2022, Complete Volum