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Stable Fractional Matchings [article]

Ioannis Caragiannis, Aris Filos-Ratsikas, Panagiotis Kanellopoulos, Rohit Vaish
2020 arXiv   pre-print
an optimal (i.e., welfare-maximizing) or nearly-optimal stable fractional matching.  ...  We study a generalization of the classical stable matching problem that allows for cardinal preferences (as opposed to ordinal) and fractional matchings (as opposed to integral).  ...  Stable Fractional Matching.  ... 
arXiv:1902.06698v2 fatcat:fodx5zag4fa6xe2bnq4sszemxy

On the many-to-one strongly stable fractional matching set [article]

Pablo A. Neme, Jorge Oviedo
2020 arXiv   pre-print
Also, we prove that a strongly stable fractional matching is represented as a convex combination of stable matchings that are ordered in the common preferences of all firms.  ...  For a many-to-one matching market where firms have strict and q-responsive preferences, we give a characterization of the set of strongly stable fractional matchings as the union of the convex hull of  ...  That is to say, a random matching is always a stable fractional matching, but some stable fractional matchings cannot be written as a lottery over stable matchings.  ... 
arXiv:1905.12500v2 fatcat:6pkk4cm3rzf3bpvncl3n7spf4e

Fractionally Log-Concave and Sector-Stable Polynomials: Counting Planar Matchings and More [article]

Yeganeh Alimohammadi, Nima Anari, Kirankumar Shiragur, Thuy-Duong Vuong
2021 arXiv   pre-print
While perfect matchings on planar graphs can be counted exactly in polynomial time, counting non-perfect matchings was shown by [Jer87] to be #P-hard, who also raised the question of whether efficient  ...  As a byproduct of our techniques, we show that polynomials avoiding roots in a sector of the complex plane must satisfy what we call fractional log-concavity; this extends a classic result established  ...  Fractionally Log-Concave Polynomials In this section, first, we show that any sector-stable polynomial is a fractionally log-concave polynomial as well.  ... 
arXiv:2102.02708v2 fatcat:arzydkij3ngzlnmmqrjiqtk4oi

On the Complexity of Stable Fractional Hypergraph Matching

Takashi Ishizuka, Naoyuki Kamiyama, Michael Wagner
2018 International Symposium on Algorithms and Computation  
Aharoni and Fleiner proved that there exists a stable fractional matching in every hypergraphic preference system.  ...  In this paper, we consider the complexity of the problem of finding a stable fractional matching in a hypergraphic preference system.  ...  Notice that since a stable fractional matching in P always exists, an -stable fractional matching in P always exists. The goal of this problem is to find an -stable fractional matching in P .  ... 
doi:10.4230/lipics.isaac.2018.11 dblp:conf/isaac/IshizukaK18 fatcat:4wq6xif54bgwxbojmi2ew67uym

The Geometry of Fractional Stable Matchings and Its Applications

Chung-Piaw Teo, Jay Sethuraman
1998 Mathematics of Operations Research  
We propose a new LP formulation for the stable roommates problem, which has a feasible solution if and only if the underlying roommates problem has a stable matching.  ...  For the stable marriage problem, we show that a related geometry allows us to express any fractional solution in the stable marriage polytope as a convex combination of stable marriage solutions.  ...  Fractional stable roommates polytope.  ... 
doi:10.1287/moor.23.4.874 fatcat:f36v5qpxebhulpkrt2vlckzzxe

Fractional solutions for capacitated NTU-games, with applications to stable matchings

Péter Biró, Tamás Fleiner
2016 Discrete Optimization  
First, we introduce the notion of fractional core for NTU-games, which is always nonempty by the Lemma.  ...  This problem is relevant in many practical applications, such as NRMP (National Resident Matching Program).  ...  Aharoni and Fleiner [1] showed that a fractional stable matching can be assumed to be an extremal point of the fractional matching polytope.  ... 
doi:10.1016/j.disopt.2015.02.002 fatcat:omycadr2ifbhnp7xtbfrr2yiy4

On the Coexistence of Stability and Incentive Compatibility in Fractional Matchings [article]

Shivika Narang, Y Narahari
2022 arXiv   pre-print
Our study leads to a class of matching instances that admit unique stable fractional matchings.  ...  With this as the backdrop, our paper studies the important topic of incentive compatibility of mechanisms to find stable fractional matchings.  ...  Definition 3 (Stable Fractional Matchings).  ... 
arXiv:2001.05652v2 fatcat:reddz2x7tbg25nericmkt5sob4

Stable matchings and linear programming

Hernán Abeledo, Yosef Blum
1996 Linear Algebra and its Applications  
We establish here additional properties of fractional stable matchings and use linear programming to obtain an alternative polynomial algorithm for solving stable matching problems. 1.  ...  This paper continues the work of Abeledo and Rothblum, who study nonbipartite stable matching problems from a polyhedral perspective.  ...  Solutions of (21, (3), and (4) are called fractional stable matchings of (G; P). The fractional stable matching polytope is the set of all fractional stable matchings and will be denoted FSM(G; P).  ... 
doi:10.1016/0024-3795(95)00052-6 fatcat:nr7prmqqdncpxmbfvz2hnvjufi

Random Matching under Priorities: Stability and No Envy Concepts [article]

Haris Aziz, Bettina Klaus
2017 arXiv   pre-print
When matchings can be random, there are a number of natural stability / fairness concepts that coincide with stability / no envy whenever matchings are deterministic.  ...  When matchings are deterministic, the standard stability concept also captures the fairness property of no (justified) envy.  ...  that a fractionally stable random matching is ex-post stable.  ... 
arXiv:1707.01231v1 fatcat:zf7lnn6cbnautpdobrpu4twapu

Random matching under priorities: stability and no envy concepts

Haris Aziz, Bettina Klaus
2019 Social Choice and Welfare  
Secondly, the framework of random matchings also helps to reason about fractional matchings that capture time sharing arrangements (Aziz, 2019; Roth, Rothblum, and Vande Vate, 1993; Teo and Sethuraman,  ...  Various stability concepts for random and fractional matchings have been introduced and studied in various papers, but the picture of how exactly they relate to each other and how their formulations change  ...  that a fractionally stable random matching is ex-post stable.  ... 
doi:10.1007/s00355-019-01181-x fatcat:3kjuegjzj5bkbhtp4bjpwt6jba

Friendship and Stable Matching [chapter]

Elliot Anshelevich, Onkar Bhardwaj, Martin Hoefer
2013 Lecture Notes in Computer Science  
We consider the existence, computation, and inefficiency of stable matchings from which no pair of players wants to deviate.  ...  Furthermore, a good stable matching achieving the price of stability bound always exists and can be reached in polynomial time.  ...  Most of our results can be extended (with minor modifications) to contribution games [8] as well, as they can be considered non-standard fractional versions of stable matching.  ... 
doi:10.1007/978-3-642-40450-4_5 fatcat:dqm5yerh5fbffp5hlpaox5rntm

Stable matchings and linear inequalities

Hernán G. Abeledo, Uriel G. Rothblum
1994 Discrete Applied Mathematics  
Here we extend the approach to the general stable matching problem in which the structure of matchable pairs need not be bipartite.  ...  The theory of linear inequalities and linear programming was recently applied to study the stable marriage problem which until then has been studied by mostly combinatorial methods.  ...  We now show that the median of three fractional stable matchings is also a fractional stable matching.  ... 
doi:10.1016/0166-218x(94)90130-9 fatcat:vofidiw5bnfwbk2bxeab2bhkaq

Fractionally total colouring Gn,p

Conor Meagher, Bruce Reed
2008 Discrete Applied Mathematics  
We study the fractional total chromatic number of G n,p as p varies from 0 to 1.  ...  We also present an algorithm that computes the fractional total chromatic number of a random graph in polynomial expected time.  ...  A total stable set is the union of a matching and a stable set such that no vertex of the stable set is an endpoint of an edge in the matching.  ... 
doi:10.1016/j.dam.2007.05.052 fatcat:onqxzu7gwbabfa7q4pxcnujw7a

Stabilizing Weighted Graphs

Zhuan Khye Koh, Laura Sanità, Michael Wagner
2018 International Colloquium on Automata, Languages and Programming  
An edge-weighted graph G = (V, E) is called stable if the value of a maximum-weight matching equals the value of a maximum-weight fractional matching.  ...  The algorithm is combinatorial and exploits new structural properties of basic fractional matchings, which are of independent interest.  ...  An edge-weighted graph G = (V, E) is called stable if the value ν(G) of a maximum-weight matching equals the value of a maximum-weight fractional matching, denoted as ν f (G).  ... 
doi:10.4230/lipics.icalp.2018.83 dblp:conf/icalp/KohS18 fatcat:yg2xxp5uorbajlj2grkt3doijq

Fractional Matchings under Preferences: Stability and Optimality [article]

Jiehua Chen and Sanjukta Roy and Manuel Sorge
2020 arXiv   pre-print
picture regarding the computational complexity of finding an optimal ordinally stable or cardinally stable matching.  ...  After having observed that ordinal stability always exists and implies cardinal stability, and that the set of ordinally stable matchings in a restricted case admits a lattice structure, we obtain a complete  ...  The acronyms CSM, OSM, and LSM stand for cardinally stable, ordinally stable, and linearly stable fractional matching, respectively. ⋄ Remark.  ... 
arXiv:2011.12259v1 fatcat:xmur7y45mzcfblxcph5yaroh4a
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