940 research outputs found
Popular Matchings in Complete Graphs
Our input is a complete graph on vertices where each vertex
has a strict ranking of all other vertices in . Our goal is to construct a
matching in that is popular. A matching is popular if does not lose
a head-to-head election against any matching , where each vertex casts a
vote for the matching in where it gets assigned a better partner.
The popular matching problem is to decide whether a popular matching exists or
not. The popular matching problem in is easy to solve for odd .
Surprisingly, the problem becomes NP-hard for even , as we show here.Comment: Appeared at FSTTCS 201
Minimal Envy and Popular Matchings
We study ex-post fairness in the object allocation problem where objects are
valuable and commonly owned. A matching is fair from individual perspective if
it has only inevitable envy towards agents who received most preferred objects
-- minimal envy matching. A matching is fair from social perspective if it is
supported by majority against any other matching -- popular matching.
Surprisingly, the two perspectives give the same outcome: when a popular
matching exists it is equivalent to a minimal envy matching.
We show the equivalence between global and local popularity: a matching is
popular if and only if there does not exist a group of size up to 3 agents that
decides to exchange their objects by majority, keeping the remaining matching
fixed. We algorithmically show that an arbitrary matching is path-connected to
a popular matching where along the path groups of up to 3 agents exchange their
objects by majority. A market where random groups exchange objects by majority
converges to a popular matching given such matching exists.
When popular matching might not exist we define most popular matching as a
matching that is popular among the largest subset of agents. We show that each
minimal envy matching is a most popular matching and propose a polynomial-time
algorithm to find them
Popular Edges with Critical Nodes
In the popular edge problem, the input is a bipartite graph G = (A ? B,E) where A and B denote a set of men and a set of women respectively, and each vertex in A? B has a strict preference ordering over its neighbours. A matching M in G is said to be popular if there is no other matching M\u27 such that the number of vertices that prefer M\u27 to M is more than the number of vertices that prefer M to M\u27. The goal is to determine, whether a given edge e belongs to some popular matching in G. A polynomial-time algorithm for this problem appears in [Cseh and Kavitha, 2018].
We consider the popular edge problem when some men or women are prioritized or critical. A matching that matches all the critical nodes is termed as a feasible matching. It follows from [Telikepalli Kavitha, 2014; Kavitha, 2021; Nasre et al., 2021; Meghana Nasre and Prajakta Nimbhorkar, 2017] that, when G admits a feasible matching, there always exists a matching that is popular among all feasible matchings.
We give a polynomial-time algorithm for the popular edge problem in the presence of critical men or women. We also show that an analogous result does not hold in the many-to-one setting, which is known as the Hospital-Residents Problem in literature, even when there are no critical nodes
Popular Edges with Critical Nodes
In the popular edge problem, the input is a bipartite graph where and denote a set of men and a set of women respectively,
and each vertex in has a strict preference ordering over its
neighbours. A matching in is said to be {\em popular} if there is no
other matching such that the number of vertices that prefer to is
more than the number of vertices that prefer to . The goal is to
determine, whether a given edge belongs to some popular matching in . A
polynomial-time algorithm for this problem appears in \cite{CK18}. We consider
the popular edge problem when some men or women are prioritized or critical. A
matching that matches all the critical nodes is termed as a feasible matching.
It follows from \cite{Kavitha14,Kavitha21,NNRS21,NN17} that, when admits a
feasible matching, there always exists a matching that is popular among all
feasible matchings. We give a polynomial-time algorithm for the popular edge
problem in the presence of critical men or women. We also show that an
analogous result does not hold in the many-to-one setting, which is known as
the Hospital-Residents Problem in literature, even when there are no critical
nodes.Comment: Selected in ISAAC 2022 Conferenc
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