2,619 research outputs found
The Dual Polynomial of Bipartite Perfect Matching
We obtain a description of the Boolean dual function of the Bipartite Perfect
Matching decision problem, as a multilinear polynomial over the Reals. We show
that in this polynomial, both the number of monomials and the magnitude of
their coefficients are at most exponential in . As an
application, we obtain a new upper bound of on the approximate degree of the bipartite perfect matching function,
improving the previous best known bound of . We deduce
that, beyond a factor, the polynomial method
cannot be used to improve the lower bound on the bounded-error quantum query
complexity of bipartite perfect matching
Web Service Retrieval by Structured Models
Much of the information available on theWorldWideWeb cannot effectively be found by the help of search engines because the information is dynamically generated on a userâs request.This applies to online decision support services as well as Deep Web information. We present in this paper a retrieval system that uses a variant of structured modeling to describe such information services, and similarity of models for retrieval. The computational complexity of the similarity problem is discussed, and graph algorithms for retrieval on repositories of service descriptions are introduced. We show how bounds for combinatorial optimization problems can provide filter algorithms in a retrieval context. We report about an evaluation of the retrieval system in a classroom experiment and give computational results on a benchmark library.Economics ;
Diverse Weighted Bipartite b-Matching
Bipartite matching, where agents on one side of a market are matched to
agents or items on the other, is a classical problem in computer science and
economics, with widespread application in healthcare, education, advertising,
and general resource allocation. A practitioner's goal is typically to maximize
a matching market's economic efficiency, possibly subject to some fairness
requirements that promote equal access to resources. A natural balancing act
exists between fairness and efficiency in matching markets, and has been the
subject of much research.
In this paper, we study a complementary goal---balancing diversity and
efficiency---in a generalization of bipartite matching where agents on one side
of the market can be matched to sets of agents on the other. Adapting a
classical definition of the diversity of a set, we propose a quadratic
programming-based approach to solving a supermodular minimization problem that
balances diversity and total weight of the solution. We also provide a scalable
greedy algorithm with theoretical performance bounds. We then define the price
of diversity, a measure of the efficiency loss due to enforcing diversity, and
give a worst-case theoretical bound. Finally, we demonstrate the efficacy of
our methods on three real-world datasets, and show that the price of diversity
is not bad in practice
Almost Optimal Stochastic Weighted Matching With Few Queries
We consider the {\em stochastic matching} problem. An edge-weighted general
(i.e., not necessarily bipartite) graph is given in the input, where
each edge in is {\em realized} independently with probability ; the
realization is initially unknown, however, we are able to {\em query} the edges
to determine whether they are realized. The goal is to query only a small
number of edges to find a {\em 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 {\em adaptive} algorithm that for any arbitrarily small
, finds a -approximation in expectation, by
querying only edges per vertex. We further show that our approach leads
to a -approximate {\em non-adaptive} algorithm that also
queries only 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 -approximation (resp.
-approximation) by querying up to edges per
vertex where 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
{\em augmenting paths} to weighted matchings that may be of independent
interest
Nearly Optimal Communication and Query Complexity of Bipartite Matching
We settle the complexities of the maximum-cardinality bipartite matching
problem (BMM) up to poly-logarithmic factors in five models of computation: the
two-party communication, AND query, OR query, XOR query, and quantum edge query
models. Our results answer open problems that have been raised repeatedly since
at least three decades ago [Hajnal, Maass, and Turan STOC'88; Ivanyos, Klauck,
Lee, Santha, and de Wolf FSTTCS'12; Dobzinski, Nisan, and Oren STOC'14; Nisan
SODA'21] and tighten the lower bounds shown by Beniamini and Nisan [STOC'21]
and Zhang [ICALP'04]. We also settle the communication complexity of the
generalizations of BMM, such as maximum-cost bipartite -matching and
transshipment; and the query complexity of unique bipartite perfect matching
(answering an open question by Beniamini [2022]). Our algorithms and lower
bounds follow from simple applications of known techniques such as cutting
planes methods and set disjointness.Comment: Accepted in FOCS 202
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