27,605 research outputs found
Rank Maximal Matchings -- Structure and Algorithms
Let G = (A U P, E) be a bipartite graph where A denotes a set of agents, P
denotes a set of posts and ranks on the edges denote preferences of the agents
over posts. A matching M in G is rank-maximal if it matches the maximum number
of applicants to their top-rank post, subject to this, the maximum number of
applicants to their second rank post and so on.
In this paper, we develop a switching graph characterization of rank-maximal
matchings, which is a useful tool that encodes all rank-maximal matchings in an
instance. The characterization leads to simple and efficient algorithms for
several interesting problems. In particular, we give an efficient algorithm to
compute the set of rank-maximal pairs in an instance. We show that the problem
of counting the number of rank-maximal matchings is #P-Complete and also give
an FPRAS for the problem. Finally, we consider the problem of deciding whether
a rank-maximal matching is popular among all the rank-maximal matchings in a
given instance, and give an efficient algorithm for the problem
Distributed Approximation of Maximum Independent Set and Maximum Matching
We present a simple distributed -approximation algorithm for maximum
weight independent set (MaxIS) in the model which completes
in rounds, where is the maximum
degree, is the number of rounds needed to compute a maximal
independent set (MIS) on , and is the maximum weight of a node. %Whether
our algorithm is randomized or deterministic depends on the \texttt{MIS}
algorithm used as a black-box.
Plugging in the best known algorithm for MIS gives a randomized solution in
rounds, where is the number of nodes.
We also present a deterministic -round algorithm based
on coloring.
We then show how to use our MaxIS approximation algorithms to compute a
-approximation for maximum weight matching without incurring any additional
round penalty in the model. We use a known reduction for
simulating algorithms on the line graph while incurring congestion, but we show
our algorithm is part of a broad family of \emph{local aggregation algorithms}
for which we describe a mechanism that allows the simulation to run in the
model without an additional overhead.
Next, we show that for maximum weight matching, relaxing the approximation
factor to () allows us to devise a distributed algorithm
requiring rounds for any constant
. For the unweighted case, we can even obtain a
-approximation in this number of rounds. These algorithms are
the first to achieve the provably optimal round complexity with respect to
dependency on
Parallel and Distributed Algorithms for the Housing Allocation Problem
We give parallel and distributed algorithms for the housing allocation
problem. In this problem, there is a set of agents and a set of houses. Each
agent has a strict preference list for a subset of houses. We need to find a
matching such that some criterion is optimized. One such criterion is Pareto
Optimality. A matching is Pareto optimal if no coalition of agents can be
strictly better off by exchanging houses among themselves. We also study the
housing market problem, a variant of the housing allocation problem, where each
agent initially owns a house. In addition to Pareto optimality, we are also
interested in finding the core of a housing market. A matching is in the core
if there is no coalition of agents that can be better off by breaking away from
other agents and switching houses only among themselves.
In the first part of this work, we show that computing a Pareto optimal
matching of a house allocation is in {\bf CC} and computing the core of a
housing market is {\bf CC}-hard. Given a matching, we also show that verifying
whether it is in the core can be done in {\bf NC}. We then give an algorithm to
show that computing a maximum Pareto optimal matching for the housing
allocation problem is in {\bf RNC}^2 and quasi-{\bf NC}^2. In the second part
of this work, we present a distributed version of the top trading cycle
algorithm for finding the core of a housing market. To that end, we first
present two algorithms for finding all the disjoint cycles in a functional
graph: a Las Vegas algorithm which terminates in rounds with high
probability, where is the length of the longest cycle, and a deterministic
algorithm which terminates in rounds, where is the
number of nodes in the graph. Both algorithms work in the synchronous
distributed model and use messages of size
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
(Near) Optimal Adaptivity Gaps for Stochastic Multi-Value Probing
Consider a kidney-exchange application where we want to find a max-matching in a random graph. To find whether an edge e exists, we need to perform an expensive test, in which case the edge e appears independently with a known probability p_e. Given a budget on the total cost of the tests, our goal is to find a testing strategy that maximizes the expected maximum matching size.
The above application is an example of the stochastic probing problem. In general the optimal stochastic probing strategy is difficult to find because it is adaptive - decides on the next edge to probe based on the outcomes of the probed edges. An alternate approach is to show the adaptivity gap is small, i.e., the best non-adaptive strategy always has a value close to the best adaptive strategy. This allows us to focus on designing non-adaptive strategies that are much simpler. Previous works, however, have focused on Bernoulli random variables that can only capture whether an edge appears or not. In this work we introduce a multi-value stochastic probing problem, which can also model situations where the weight of an edge has a probability distribution over multiple values.
Our main technical contribution is to obtain (near) optimal bounds for the (worst-case) adaptivity gaps for multi-value stochastic probing over prefix-closed constraints. For a monotone submodular function, we show the adaptivity gap is at most 2 and provide a matching lower bound. For a weighted rank function of a k-extendible system (a generalization of intersection of k matroids), we show the adaptivity gap is between O(k log k) and k. None of these results were known even in the Bernoulli case where both our upper and lower bounds also apply, thereby resolving an open question of Gupta et al. [Gupta et al., 2017]
On Randomized Algorithms for Matching in the Online Preemptive Model
We investigate the power of randomized algorithms for the maximum cardinality
matching (MCM) and the maximum weight matching (MWM) problems in the online
preemptive model. In this model, the edges of a graph are revealed one by one
and the algorithm is required to always maintain a valid matching. On seeing an
edge, the algorithm has to either accept or reject the edge. If accepted, then
the adjacent edges are discarded. The complexity of the problem is settled for
deterministic algorithms.
Almost nothing is known for randomized algorithms. A lower bound of
is known for MCM with a trivial upper bound of . An upper bound of
is known for MWM. We initiate a systematic study of the same in this paper with
an aim to isolate and understand the difficulty. We begin with a primal-dual
analysis of the deterministic algorithm due to McGregor. All deterministic
lower bounds are on instances which are trees at every step. For this class of
(unweighted) graphs we present a randomized algorithm which is
-competitive. The analysis is a considerable extension of the
(simple) primal-dual analysis for the deterministic case. The key new technique
is that the distribution of primal charge to dual variables depends on the
"neighborhood" and needs to be done after having seen the entire input. The
assignment is asymmetric: in that edges may assign different charges to the two
end-points. Also the proof depends on a non-trivial structural statement on the
performance of the algorithm on the input tree.
The other main result of this paper is an extension of the deterministic
lower bound of Varadaraja to a natural class of randomized algorithms which
decide whether to accept a new edge or not using independent random choices
The Price of Information in Combinatorial Optimization
Consider a network design application where we wish to lay down a
minimum-cost spanning tree in a given graph; however, we only have stochastic
information about the edge costs. To learn the precise cost of any edge, we
have to conduct a study that incurs a price. Our goal is to find a spanning
tree while minimizing the disutility, which is the sum of the tree cost and the
total price that we spend on the studies. In a different application, each edge
gives a stochastic reward value. Our goal is to find a spanning tree while
maximizing the utility, which is the tree reward minus the prices that we pay.
Situations such as the above two often arise in practice where we wish to
find a good solution to an optimization problem, but we start with only some
partial knowledge about the parameters of the problem. The missing information
can be found only after paying a probing price, which we call the price of
information. What strategy should we adopt to optimize our expected
utility/disutility?
A classical example of the above setting is Weitzman's "Pandora's box"
problem where we are given probability distributions on values of
independent random variables. The goal is to choose a single variable with a
large value, but we can find the actual outcomes only after paying a price. Our
work is a generalization of this model to other combinatorial optimization
problems such as matching, set cover, facility location, and prize-collecting
Steiner tree. We give a technique that reduces such problems to their non-price
counterparts, and use it to design exact/approximation algorithms to optimize
our utility/disutility. Our techniques extend to situations where there are
additional constraints on what parameters can be probed or when we can
simultaneously probe a subset of the parameters.Comment: SODA 201
Size versus truthfulness in the House Allocation problem
We study the House Allocation problem (also known as the Assignment problem),
i.e., the problem of allocating a set of objects among a set of agents, where
each agent has ordinal preferences (possibly involving ties) over a subset of
the objects. We focus on truthful mechanisms without monetary transfers for
finding large Pareto optimal matchings. It is straightforward to show that no
deterministic truthful mechanism can approximate a maximum cardinality Pareto
optimal matching with ratio better than 2. We thus consider randomised
mechanisms. We give a natural and explicit extension of the classical Random
Serial Dictatorship Mechanism (RSDM) specifically for the House Allocation
problem where preference lists can include ties. We thus obtain a universally
truthful randomised mechanism for finding a Pareto optimal matching and show
that it achieves an approximation ratio of . The same bound
holds even when agents have priorities (weights) and our goal is to find a
maximum weight (as opposed to maximum cardinality) Pareto optimal matching. On
the other hand we give a lower bound of on the approximation
ratio of any universally truthful Pareto optimal mechanism in settings with
strict preferences. In the case that the mechanism must additionally be
non-bossy with an additional technical assumption, we show by utilising a
result of Bade that an improved lower bound of holds. This
lower bound is tight since RSDM for strict preference lists is non-bossy. We
moreover interpret our problem in terms of the classical secretary problem and
prove that our mechanism provides the best randomised strategy of the
administrator who interviews the applicants.Comment: To appear in Algorithmica (preliminary version appeared in the
Proceedings of EC 2014
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