1,576 research outputs found
Bi-Criteria and Approximation Algorithms for Restricted Matchings
In this work we study approximation algorithms for the \textit{Bounded Color
Matching} problem (a.k.a. Restricted Matching problem) which is defined as
follows: given a graph in which each edge has a color and a profit
, we want to compute a maximum (cardinality or profit)
matching in which no more than edges of color are
present. This kind of problems, beside the theoretical interest on its own
right, emerges in multi-fiber optical networking systems, where we interpret
each unique wavelength that can travel through the fiber as a color class and
we would like to establish communication between pairs of systems. We study
approximation and bi-criteria algorithms for this problem which are based on
linear programming techniques and, in particular, on polyhedral
characterizations of the natural linear formulation of the problem. In our
setting, we allow violations of the bounds and we model our problem as a
bi-criteria problem: we have two objectives to optimize namely (a) to maximize
the profit (maximum matching) while (b) minimizing the violation of the color
bounds. We prove how we can "beat" the integrality gap of the natural linear
programming formulation of the problem by allowing only a slight violation of
the color bounds. In particular, our main result is \textit{constant}
approximation bounds for both criteria of the corresponding bi-criteria
optimization problem
Distributed local approximation algorithms for maximum matching in graphs and hypergraphs
We describe approximation algorithms in Linial's classic LOCAL model of
distributed computing to find maximum-weight matchings in a hypergraph of rank
. Our main result is a deterministic algorithm to generate a matching which
is an -approximation to the maximum weight matching, running in rounds. (Here, the
notations hides and factors).
This is based on a number of new derandomization techniques extending methods
of Ghaffari, Harris & Kuhn (2017).
As a main application, we obtain nearly-optimal algorithms for the
long-studied problem of maximum-weight graph matching. Specifically, we get a
approximation algorithm using randomized time and deterministic time.
The second application is a faster algorithm for hypergraph maximal matching,
a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of
local graph algorithms. This gives an algorithm for -edge-list
coloring in rounds deterministically or
rounds randomly. Another consequence (with
additional optimizations) is an algorithm which generates an edge-orientation
with out-degree at most for a graph of
arboricity ; for fixed this runs in
rounds deterministically or rounds randomly
Classified Stable Matching
We introduce the {\sc classified stable matching} problem, a problem
motivated by academic hiring. Suppose that a number of institutes are hiring
faculty members from a pool of applicants. Both institutes and applicants have
preferences over the other side. An institute classifies the applicants based
on their research areas (or any other criterion), and, for each class, it sets
a lower bound and an upper bound on the number of applicants it would hire in
that class. The objective is to find a stable matching from which no group of
participants has reason to deviate. Moreover, the matching should respect the
upper/lower bounds of the classes.
In the first part of the paper, we study classified stable matching problems
whose classifications belong to a fixed set of ``order types.'' We show that if
the set consists entirely of downward forests, there is a polynomial-time
algorithm; otherwise, it is NP-complete to decide the existence of a stable
matching.
In the second part, we investigate the problem using a polyhedral approach.
Suppose that all classifications are laminar families and there is no lower
bound. We propose a set of linear inequalities to describe stable matching
polytope and prove that it is integral. This integrality allows us to find
various optimal stable matchings using Ellipsoid algorithm. A further
ramification of our result is the description of the stable matching polytope
for the many-to-many (unclassified) stable matching problem. This answers an
open question posed by Sethuraman, Teo and Qian
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