1,574 research outputs found
"Almost-stable" matchings in the Hospitals / Residents problem with Couples
The Hospitals / Residents problem with Couples (hrc) models the allocation of intending junior doctors to hospitals where couples are allowed to submit joint preference lists over pairs of (typically geographically close) hospitals. It is known that a stable matching need not exist, so we consider min bp hrc, the problem of finding a matching that admits the minimum number of blocking pairs (i.e., is “as stable as possible”). We show that this problem is NP-hard and difficult to approximate even in the highly restricted case that each couple finds only one hospital pair acceptable. However if we further assume that the preference list of each single resident and hospital is of length at most 2, we give a polynomial-time algorithm for this case. We then present the first Integer Programming (IP) and Constraint Programming (CP) models for min bp hrc. Finally, we discuss an empirical evaluation of these models applied to randomly-generated instances of min bp hrc. We find that on average, the CP model is about 1.15 times faster than the IP model, and when presolving is applied to the CP model, it is on average 8.14 times faster. We further observe that the number of blocking pairs admitted by a solution is very small, i.e., usually at most 1, and never more than 2, for the (28,000) instances considered
Higher-order Projected Power Iterations for Scalable Multi-Matching
The matching of multiple objects (e.g. shapes or images) is a fundamental
problem in vision and graphics. In order to robustly handle ambiguities, noise
and repetitive patterns in challenging real-world settings, it is essential to
take geometric consistency between points into account. Computationally, the
multi-matching problem is difficult. It can be phrased as simultaneously
solving multiple (NP-hard) quadratic assignment problems (QAPs) that are
coupled via cycle-consistency constraints. The main limitations of existing
multi-matching methods are that they either ignore geometric consistency and
thus have limited robustness, or they are restricted to small-scale problems
due to their (relatively) high computational cost. We address these
shortcomings by introducing a Higher-order Projected Power Iteration method,
which is (i) efficient and scales to tens of thousands of points, (ii)
straightforward to implement, (iii) able to incorporate geometric consistency,
(iv) guarantees cycle-consistent multi-matchings, and (iv) comes with
theoretical convergence guarantees. Experimentally we show that our approach is
superior to existing methods
Fully Dynamic Matching in Bipartite Graphs
Maximum cardinality matching in bipartite graphs is an important and
well-studied problem. The fully dynamic version, in which edges are inserted
and deleted over time has also been the subject of much attention. Existing
algorithms for dynamic matching (in general graphs) seem to fall into two
groups: there are fast (mostly randomized) algorithms that do not achieve a
better than 2-approximation, and there slow algorithms with \O(\sqrt{m})
update time that achieve a better-than-2 approximation. Thus the obvious
question is whether we can design an algorithm -- deterministic or randomized
-- that achieves a tradeoff between these two: a approximation
and a better-than-2 approximation simultaneously. We answer this question in
the affirmative for bipartite graphs.
Our main result is a fully dynamic algorithm that maintains a 3/2 + \eps
approximation in worst-case update time O(m^{1/4}\eps^{/2.5}). We also give
stronger results for graphs whose arboricity is at most \al, achieving a (1+
\eps) approximation in worst-case time O(\al (\al + \log n)) for constant
\eps. When the arboricity is constant, this bound is and when the
arboricity is polylogarithmic the update time is also polylogarithmic.
The most important technical developement is the use of an intermediate graph
we call an edge degree constrained subgraph (EDCS). This graph places
constraints on the sum of the degrees of the endpoints of each edge: upper
bounds for matched edges and lower bounds for unmatched edges. The main
technical content of our paper involves showing both how to maintain an EDCS
dynamically and that and EDCS always contains a sufficiently large matching. We
also make use of graph orientations to help bound the amount of work done
during each update.Comment: Longer version of paper that appears in ICALP 201
Welfare Maximization and Truthfulness in Mechanism Design with Ordinal Preferences
We study mechanism design problems in the {\em ordinal setting} wherein the
preferences of agents are described by orderings over outcomes, as opposed to
specific numerical values associated with them. This setting is relevant when
agents can compare outcomes, but aren't able to evaluate precise utilities for
them. Such a situation arises in diverse contexts including voting and matching
markets.
Our paper addresses two issues that arise in ordinal mechanism design. To
design social welfare maximizing mechanisms, one needs to be able to
quantitatively measure the welfare of an outcome which is not clear in the
ordinal setting. Second, since the impossibility results of Gibbard and
Satterthwaite~\cite{Gibbard73,Satterthwaite75} force one to move to randomized
mechanisms, one needs a more nuanced notion of truthfulness.
We propose {\em rank approximation} as a metric for measuring the quality of
an outcome, which allows us to evaluate mechanisms based on worst-case
performance, and {\em lex-truthfulness} as a notion of truthfulness for
randomized ordinal mechanisms. Lex-truthfulness is stronger than notions
studied in the literature, and yet flexible enough to admit a rich class of
mechanisms {\em circumventing classical impossibility results}. We demonstrate
the usefulness of the above notions by devising lex-truthful mechanisms
achieving good rank-approximation factors, both in the general ordinal setting,
as well as structured settings such as {\em (one-sided) matching markets}, and
its generalizations, {\em matroid} and {\em scheduling} markets.Comment: Some typos correcte
Partial-Matching and Hausdorff RMS Distance Under Translation: Combinatorics and Algorithms
We consider the RMS distance (sum of squared distances between pairs of
points) under translation between two point sets in the plane, in two different
setups. In the partial-matching setup, each point in the smaller set is matched
to a distinct point in the bigger set. Although the problem is not known to be
polynomial, we establish several structural properties of the underlying
subdivision of the plane and derive improved bounds on its complexity. These
results lead to the best known algorithm for finding a translation for which
the partial-matching RMS distance between the point sets is minimized. In
addition, we show how to compute a local minimum of the partial-matching RMS
distance under translation, in polynomial time. In the Hausdorff setup, each
point is paired to its nearest neighbor in the other set. We develop algorithms
for finding a local minimum of the Hausdorff RMS distance in nearly linear time
on the line, and in nearly quadratic time in the plane. These improve
substantially the worst-case behavior of the popular ICP heuristics for solving
this problem.Comment: 31 pages, 6 figure
Computational complexity of -stable matchings
We study deviations by a group of agents in the three main types of matching
markets: the house allocation, the marriage, and the roommates models. For a
given instance, we call a matching -stable if no other matching exists that
is more beneficial to at least out of the agents. The concept
generalizes the recently studied majority stability. We prove that whereas the
verification of -stability for a given matching is polynomial-time solvable
in all three models, the complexity of deciding whether a -stable matching
exists depends on and is characteristic to each model.Comment: SAGT 202
Metatheorems for Dynamic Weighted Matching
We consider the maximum weight matching (MWM) problem in dynamic graphs. We provide two reductions. The first reduces the dynamic MWM problem on m-edge, n-node graphs with weights bounded by N to the problem with weights bounded by (n/eps)^2, so that if the MWM problem can be alpha-approximated with update time t(m,n,N), then it can also be (1+eps)alpha-approximated with update time O(t(m,n,(n/eps)^2)log^2 n+log n loglog N)). The second reduction reduces the dynamic MWM problem to the dynamic maximum cardinality matching (MCM) problem in which the graph is unweighted. This reduction shows that if there is an alpha-approximation algorithm for MCM with update time t(m,n) in m-edge n-node graphs, then there is also a (2+eps)alpha-approximation algorithm for MWM with update time O(t(m,n)eps^{-2}log^2 N). We also obtain better bounds in our reductions if the ratio between the largest and the smallest edge weight is small. Combined with recent work on MCM, these two reductions substantially improve upon the state-of-the-art of dynamic MWM algorithms
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