10 research outputs found
Weakly-Popular and Super-Popular Matchings with Ties and Their Connection to Stable Matchings
In this paper, we study a slightly different definition of popularity in
bipartite graphs with two-sided preferences, when ties are present
in the preference lists. This is motivated by the observation that if an agent
is indifferent between his original partner in matching and his new
partner in matching , then he may probably still prefer to stay
with his original partner, as change requires effort, so he votes for in
this case, instead of being indifferent.
We show that this alternative definition of popularity, which we call
weak-popularity allows us to guarantee the existence of such a matching and
also to find a weakly-popular matching in polynomial-time that has size at
least the size of the maximum weakly popular matching. We also
show that this matching is at least times the size of the maximum
(weakly) stable matching, so may provide a more desirable solution than the
current best (and tight under certain assumptions) -approximation
for such a stable matching. We also show that unfortunately, finding a maximum
size weakly popular matching is NP-hard, even with one-sided ties and that
assuming some complexity theoretic assumptions, the -approximation
bound is tight.
Then, we study a more general model than weak-popularity, where for each
edge, we can specify independently for both endpoints the size of improvement
the endpoint needs to vote in favor of a new matching . We show that even in
this more general model, a so-called -popular matching always exists
and that the same positive results still hold.
Finally, we define an other, stronger variant of popularity, called
super-popularity, where even a weak improvement is enough to vote in favor of a
new matching. We show that for this case, even the existence problem is
NP-hard
Popular and Dominant Matchings with Uncertain, Multilayer and Aggregated Preferences
We study the Popular Matching problem in multiple models, where the
preferences of the agents in the instance may change or may be
unknown/uncertain. In particular, we study an Uncertainty model, where each
agent has a possible set of preferences, a Multilayer model, where there are
layers of preference profiles, a Robust model, where any agent may move some
other agents up or down some places in his preference list and an Aggregated
Preference model, where votes are summed over multiple instances with different
preferences.
We study both one-sided and two-sided preferences in bipartite graphs. In the
one-sided model, we show that all our problems can be solved in polynomial time
by utilizing the structure of popular matchings. We also obtain nice structural
results. With two-sided preferences, we show that all four above models lead to
NP-hard questions for popular matchings. By utilizing the connection between
dominant matchings and stable matchings, we show that in the robust and
uncertainty model, a certainly dominant matching in all possible prefernce
profiles can be found in polynomial-time, whereas in the multilayer and
aggregated models, the problem remains NP-hard for dominant matchings too.
We also answer an open question about -robust stable matchings
A Simple 1.5-Approximation Algorithm for a Wide Range of Max-SMTI Generalizations
We give a simple approximation algorithm for a common generalization of many
previously studied extensions of the stable matching problem with ties. These
generalizations include the existence of critical vertices in the graph,
amongst whom we must match as much as possible, free edges, that cannot be
blocking edges and -stabilities, which mean that for an edge to block,
the improvement should be large enough on one or both sides. We also introduce
other notions to generalize these even further, which allows our framework to
capture many existing and future applications. We show that our edge
duplicating technique allows us to treat these different types of
generalizations simultaneously, while also making the algorithm, the proofs and
the analysis much simpler and shorter then in previous approaches. In
particular, we answer an open question by \cite{socialstable} about the
existence of a -approximation algorithm for the \smti\ problem
with free edges. This demonstrates well that this technique can grasp the
underlying essence of these problems quite well and have the potential to be
able to solve countless future applications as well
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
Solving the Maximum Popular Matching Problem with Matroid Constraints
We consider the problem of finding a maximum popular matching in a
many-to-many matching setting with two-sided preferences and matroid
constraints. This problem was proposed by Kamiyama (2020) and solved in the
special case where matroids are base orderable. Utilizing a newly shown matroid
exchange property, we show that the problem is tractable for arbitrary
matroids. We further investigate a different notion of popularity, where the
agents vote with respect to lexicographic preferences, and show that both
existence and verification problems become NP-hard, even in the -matching
case.Comment: 16 pages, 2 figure
Short Proof of a Theorem of Brylawski on the Coefficients of the Tutte Polynomial
In this short note we show that a system with a ground set of
size and (rank) function satisfying
for every set , the Tutte polynomial
written as
, satisfies that for any integer ,
we have
where , and we use the convention that when , the binomial
coefficient is interpreted as . This generalizes a
theorem of Brylawski on matroid rank functions and , and a theorem of
Gordon for with the same assumptions on the rank function. The proof
presented here is significantly shorter than the previous ones. We only use the
fact that the Tutte polynomial simplifies to
along the hyperbola .Comment: 4 page
Geometria a modern fizikában
Rövid ismertető a geometria és a modern fizika kapcsolatára, a kvantummechanika és a speciális relativitáselméleten keresztü
Complexity of Stable Matching Problems
INST: L_200The thesis explores the stable matching problem and its generalizations, with many of the theorems being new results of the author. The first section studies the original variant of the stable marriage problem and its structure. The second section explores the stable roommates problem and describes the novel algorithm of Irving to solve it. The rest of the thesis deals with more abstract generalizations, where most cases become NP or PPAD-hard, but the thesis also shows some tractable cases