Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Preference Swaps for the Stable Matching Problem   An instance $I$ of the Stable Matching Problem (SMP) is given by a bipartite
graph with a preference list of neighbors for every vertex. A swap in $I$ is
the exchange of two consecutive vertices in a preference list. A swap can be
viewed as a smallest perturbation of $I$. Boehmer et al. (2021) designed a
polynomial-time algorithm to find the minimum number of swaps required to turn
a given maximal matching into a stable matching. To generalize this result to
the many-to-many version of SMP, we introduce a new representation of SMP as an
extended bipartite graph and reduce the problem to submodular minimization. It
is a natural problem to establish computational complexity of deciding whether
at most $k$ swaps are enough to turn $I$ into an instance where one of the
maximum matchings is stable. Using a hardness result of Gupta et al. (2020), we
prove that it is NP-hard to decide whether at most $k$ swaps are enough to turn
$I$ into an instance with a stable perfect matching. Moreover, this problem
parameterized by $k$ is W[1]-hard. We also obtain a lower bound on the running
time for solving the problem using the Exponential Time Hypothesis.

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The categories of this paper are: chem-ph
5
The categories of this paper are: gr-qc
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The categories of this paper are: q-fin
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The categories of this paper are: chao-dyn
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The categories of this paper are: ao-sci
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The categories of this paper are: adap-org
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The categories of this paper are: patt-sol
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The categories of this paper are: acc-phys
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The categories of this paper are: econ
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The categories of this paper are: dg-ga
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The categories of this paper are: supr-con
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The categories of this paper are: quant-ph
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The categories of this paper are: solv-int
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The categories of this paper are: eess
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The categories of this paper are: funct-an
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The categories of this paper are: bayes-an