84 research outputs found
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
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
Online Duet between Metric Embeddings and Minimum-Weight Perfect Matchings
Low-distortional metric embeddings are a crucial component in the modern
algorithmic toolkit. In an online metric embedding, points arrive sequentially
and the goal is to embed them into a simple space irrevocably, while minimizing
the distortion. Our first result is a deterministic online embedding of a
general metric into Euclidean space with distortion (or,
if the metric has doubling
dimension ), solving a conjecture by Newman and Rabinovich (2020), and
quadratically improving the dependence on the aspect ratio from Indyk et
al.\ (2010). Our second result is a stochastic embedding of a metric space into
trees with expected distortion , generalizing previous
results (Indyk et al.\ (2010), Bartal et al.\ (2020)).
Next, we study the \emph{online minimum-weight perfect matching} problem,
where a sequence of metric points arrive in pairs, and one has to maintain
a perfect matching at all times. We allow recourse (as otherwise the order of
arrival determines the matching). The goal is to return a perfect matching that
approximates the \emph{minimum-weight} perfect matching at all times, while
minimizing the recourse. Our third result is a randomized algorithm with
competitive ratio and recourse against an
oblivious adversary, this result is obtained via our new stochastic online
embedding. Our fourth result is a deterministic algorithm against an adaptive
adversary, using recourse, that maintains a matching of weight at
most times the weight of the MST, i.e., a matching of lightness
. We complement our upper bounds with a strategy for an oblivious
adversary that, with recourse , establishes a lower bound of
for both competitive ratio and lightness.Comment: 53 pages, 8 figures, to be presented at the ACM-SIAM Symposium on
Discrete Algorithms (SODA24
Graph theoretic generalizations of clique: optimization and extensions
This dissertation considers graph theoretic generalizations of the maximum
clique problem. Models that were originally proposed in social network analysis literature, are investigated from a mathematical programming perspective for the first
time. A social network is usually represented by a graph, and cliques were the first
models of "tightly knit groups" in social networks, referred to as cohesive subgroups.
Cliques are idealized models and their overly restrictive nature motivated the development of clique relaxations that relax different aspects of a clique. Identifying large
cohesive subgroups in social networks has traditionally been used in criminal network
analysis to study organized crimes such as terrorism, narcotics and money laundering.
More recent applications are in clustering and data mining wireless networks, biological networks as well as graph models of databases and the internet. This research
has the potential to impact homeland security, bioinformatics, internet research and
telecommunication industry among others.
The focus of this dissertation is a degree-based relaxation called k-plex. A
distance-based relaxation called k-clique and a diameter-based relaxation called k-club are also investigated in this dissertation. We present the first systematic study
of the complexity aspects of these problems and application of mathematical programming techniques in solving them. Graph theoretic properties of the models are
identified and used in the development of theory and algorithms.
Optimization problems associated with the three models are formulated as binary integer programs and the properties of the associated polytopes are investigated. Facets and valid inequalities are identified based on combinatorial arguments.
A branch-and-cut framework is designed and implemented to solve the optimization
problems exactly. Specialized preprocessing techniques are developed that, in conjunction with the branch-and-cut algorithm, optimally solve the problems on real-life
power law graphs, which is a general class of graphs that include social and biological
networks. Computational experiments are performed to study the effectiveness of the
proposed solution procedures on benchmark instances and real-life instances.
The relationship of these models to the classical maximum clique problem is
studied, leading to several interesting observations including a new compact integer
programming formulation. We also prove new continuous non-linear formulations for
the classical maximum independent set problem which maximize continuous functions
over the unit hypercube, and characterize its local and global maxima. Finally, clustering and network design extensions of the clique relaxation models are explored
Cross-modal and synaesthetic perception in music and vision
This thesis is concerned with the cross-modal and synaesthetic perception of musical
and visual stimuli. Each of these types of perception has been researched separately,
and a hypothesis is presented here that accounts for both cross-modal matching and the development of synaesthesia. This hypothesis claims that sensory information can be evaluated in another modality by using a scale of comparison in that modality.
The first set of experiments examines normal subjects performing cross-modal matching with coloured circles and auditory stimuli that vary in complexity. It is shown that subjects use a variety of scales of comparison from both visual and auditory
modalities to form matches. As the stimuli increase in complexity, the individual
variation in cross-modal matching also increases. The second set of experiments examines matching performance using higher order stimuli, by having subjects evaluate fragments of melodies and complete melodies on affective and descriptive adjective scales. Melodies were also matched with landscape scenes to examine if subjects could form matches between two highly complex sets of stimuli. The final experiments examine synaesthetic associations with colour, evoked from music, letters, numbers, and other categorical information. Common features of synaesthesia from a population of synaesthetes are identified, and experiments performed to test the interference of the synaesthetic associations. Additional experiments are presented that explore the superior short-term memory of one synaesthete, and the role of his associations as a mnemonic device
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