1,229 research outputs found
On the (Parameterized) Complexity of Almost Stable Marriage
In the Stable Marriage problem, when the preference lists are complete, all agents of the smaller side can be matched. However, this need not be true when preference lists are incomplete. In most real-life situations, where agents participate in the matching market voluntarily and submit their preferences, it is natural to assume that each agent wants to be matched to someone in his/her preference list as opposed to being unmatched. In light of the Rural Hospital Theorem, we have to relax the "no blocking pair" condition for stable matchings in order to match more agents. In this paper, we study the question of matching more agents with fewest possible blocking edges. In particular, the goal is to find a matching whose size exceeds that of a stable matching in the graph by at least t and has at most k blocking edges. We study this question in the realm of parameterized complexity with respect to several natural parameters, k,t,d, where d is the maximum length of a preference list. Unfortunately, the problem remains intractable even for the combined parameter k+t+d. Thus, we extend our study to the local search variant of this problem, in which we search for a matching that not only fulfills each of the above conditions but is "closest", in terms of its symmetric difference to the given stable matching, and obtain an FPT algorithm
Solving Hard Stable Matching Problems Involving Groups of Similar Agents
Many important stable matching problems are known to be NP-hard, even when
strong restrictions are placed on the input. In this paper we seek to identify
structural properties of instances of stable matching problems which will allow
us to design efficient algorithms using elementary techniques. We focus on the
setting in which all agents involved in some matching problem can be
partitioned into k different types, where the type of an agent determines his
or her preferences, and agents have preferences over types (which may be
refined by more detailed preferences within a single type). This situation
would arise in practice if agents form preferences solely based on some small
collection of agents' attributes. We also consider a generalisation in which
each agent may consider some small collection of other agents to be
exceptional, and rank these in a way that is not consistent with their types;
this could happen in practice if agents have prior contact with a small number
of candidates. We show that (for the case without exceptions), several
well-studied NP-hard stable matching problems including Max SMTI (that of
finding the maximum cardinality stable matching in an instance of stable
marriage with ties and incomplete lists) belong to the parameterised complexity
class FPT when parameterised by the number of different types of agents needed
to describe the instance. For Max SMTI this tractability result can be extended
to the setting in which each agent promotes at most one `exceptional' candidate
to the top of his/her list (when preferences within types are not refined), but
the problem remains NP-hard if preference lists can contain two or more
exceptions and the exceptional candidates can be placed anywhere in the
preference lists, even if the number of types is bounded by a constant.Comment: Results on SMTI appear in proceedings of WINE 2018; Section 6
contains work in progres
Stable Marriage with Multi-Modal Preferences
We introduce a generalized version of the famous Stable Marriage problem, now
based on multi-modal preference lists. The central twist herein is to allow
each agent to rank its potentially matching counterparts based on more than one
"evaluation mode" (e.g., more than one criterion); thus, each agent is equipped
with multiple preference lists, each ranking the counterparts in a possibly
different way. We introduce and study three natural concepts of stability,
investigate their mutual relations and focus on computational complexity
aspects with respect to computing stable matchings in these new scenarios.
Mostly encountering computational hardness (NP-hardness), we can also spot few
islands of tractability and make a surprising connection to the \textsc{Graph
Isomorphism} problem
Local search for stable marriage problems
The stable marriage (SM) problem has a wide variety of practical
applications, ranging from matching resident doctors to hospitals, to matching
students to schools, or more generally to any two-sided market. In the
classical formulation, n men and n women express their preferences (via a
strict total order) over the members of the other sex. Solving a SM problem
means finding a stable marriage where stability is an envy-free notion: no man
and woman who are not married to each other would both prefer each other to
their partners or to being single. We consider both the classical stable
marriage problem and one of its useful variations (denoted SMTI) where the men
and women express their preferences in the form of an incomplete preference
list with ties over a subset of the members of the other sex. Matchings are
permitted only with people who appear in these lists, an we try to find a
stable matching that marries as many people as possible. Whilst the SM problem
is polynomial to solve, the SMTI problem is NP-hard. We propose to tackle both
problems via a local search approach, which exploits properties of the problems
to reduce the size of the neighborhood and to make local moves efficiently. We
evaluate empirically our algorithm for SM problems by measuring its runtime
behaviour and its ability to sample the lattice of all possible stable
marriages. We evaluate our algorithm for SMTI problems in terms of both its
runtime behaviour and its ability to find a maximum cardinality stable
marriage.For SM problems, the number of steps of our algorithm grows only as
O(nlog(n)), and that it samples very well the set of all stable marriages. It
is thus a fair and efficient approach to generate stable marriages.Furthermore,
our approach for SMTI problems is able to solve large problems, quickly
returning stable matchings of large and often optimal size despite the
NP-hardness of this problem.Comment: 12 pages, Proc. COMSOC 2010 (Third International Workshop on
Computational Social Choice
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