105 research outputs found
Revisiting a Cutting-Plane Method for Perfect Matchings
In 2016, Chandrasekaran, V\'egh, and Vempala published a method to solve the
minimum-cost perfect matching problem on an arbitrary graph by solving a
strictly polynomial number of linear programs. However, their method requires a
strong uniqueness condition, which they imposed by using perturbations of the
form . On large graphs (roughly ), these
perturbations lead to cost values that exceed the precision of floating-point
formats used by typical linear programming solvers for numerical calculations.
We demonstrate, by a sequence of counterexamples, that perturbations are
required for the algorithm to work, motivating our formulation of a general
method that arrives at the same solution to the problem as Chandrasekaran et
al. but overcomes the limitations described above by solving multiple linear
programs without using perturbations. We then give an explicit algorithm that
exploits are method, and show that this new algorithm still runs in strongly
polynomial time.Comment: 19 page
Nash-Bargaining-Based Models for Matching Markets: One-Sided and Two-Sided; Fisher and Arrow-Debreu
This paper addresses two deficiencies of models in the area of matching-based market design. The first arises from the recent realization that the most prominent solution that uses cardinal utilities, namely the Hylland-Zeckhauser (HZ) mechanism [Hylland and Zeckhauser, 1979], is intractable; computation of even an approximate equilibrium is PPAD-complete [Vazirani and Yannakakis, 2021; Chen et al., 2021]. The second is the extreme paucity of models that use cardinal utilities, in sharp contrast with general equilibrium theory.
Our paper addresses both these issues by proposing Nash-bargaining-based matching market models. Since the Nash bargaining solution is captured by a convex program, efficiency follow; in addition, it possesses a number of desirable game-theoretic properties. Our approach yields a rich collection of models: for one-sided as well as two-sided markets, for Fisher as well as Arrow-Debreu settings, and for a wide range of utility functions, all the way from linear to Leontief.
We also give very fast implementations for these models which solve large instances, with n = 2000, in one hour on a PC, even for a two-sided matching market. A number of new ideas were needed, beyond the standard methods, to obtain these implementations
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
Recursive Frank-Wolfe algorithms
In the last decade there has been a resurgence of interest in Frank-Wolfe
(FW) style methods for optimizing a smooth convex function over a polytope.
Examples of recently developed techniques include {\em Decomposition-invariant
Conditional Gradient} (DiCG), {\em Blended Condition Gradient} (BCG), and {\em
Frank-Wolfe with in-face directions} (IF-FW) methods. We introduce two
extensions of these techniques. First, we augment DiCG with the {\em working
set} strategy, and show how to optimize over the working set using {\em shadow
simplex steps}. Second, we generalize in-face Frank-Wolfe directions to
polytopes in which faces cannot be efficiently computed, and also describe a
generic recursive procedure that can be used in conjunction with several
FW-style techniques. Experimental results indicate that these extensions are
capable of speeding up original algorithms by orders of magnitude for certain
applications
Revisiting the Evolution and Application of Assignment Problem: A Brief Overview
The assignment problem (AP) is incredibly challenging that can model many real-life problems. This paper provides a limited review of the recent developments that have appeared in the literature, meaning of assignment problem as well as solving techniques and will provide a review on  a lot of research studies on different types of assignment problem taking place in present day real life situation in order to capture the variations in different types of assignment techniques. Keywords: Assignment problem, Quadratic Assignment, Vehicle Routing, Exact Algorithm, Bound, Heuristic etc
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