726 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
New models for two variants of popular matching
We study the problem of matching a set of applicants to a set of posts, where each applicant has an ordinal preference list, which may contain ties, ranking a subset of posts. A matching M is popular if there exists no matching M0 where more applicants prefer M0 to M. Several notions of optimality are studied in the literature for the case of strictly ordered preference lists. In this paper we address the case involving ties and propose novel algorithmic and complexity results for this variant. Next, we focus on the NP-hard case where additional copies of posts can be added in the preference lists, called Popular Matching with Copies. We define new dominance rules for this problem and present several novel graph properties characterising the posts that should be copied with priority. We present a comprehensive set of experiments for the popular matching problem with copies to evaluate our dominance rules as well as the different branching strategies. Our experimental study emphasizes the importance of the dominance rules and characterises the key aspects of a good branching strategy
Neural Attentive Session-based Recommendation
Given e-commerce scenarios that user profiles are invisible, session-based
recommendation is proposed to generate recommendation results from short
sessions. Previous work only considers the user's sequential behavior in the
current session, whereas the user's main purpose in the current session is not
emphasized. In this paper, we propose a novel neural networks framework, i.e.,
Neural Attentive Recommendation Machine (NARM), to tackle this problem.
Specifically, we explore a hybrid encoder with an attention mechanism to model
the user's sequential behavior and capture the user's main purpose in the
current session, which are combined as a unified session representation later.
We then compute the recommendation scores for each candidate item with a
bi-linear matching scheme based on this unified session representation. We
train NARM by jointly learning the item and session representations as well as
their matchings. We carried out extensive experiments on two benchmark
datasets. Our experimental results show that NARM outperforms state-of-the-art
baselines on both datasets. Furthermore, we also find that NARM achieves a
significant improvement on long sessions, which demonstrates its advantages in
modeling the user's sequential behavior and main purpose simultaneously.Comment: Proceedings of the 2017 ACM on Conference on Information and
Knowledge Management. arXiv admin note: text overlap with arXiv:1511.06939,
arXiv:1606.08117 by other author
LP-Based Algorithms for Capacitated Facility Location
Linear programming has played a key role in the study of algorithms for
combinatorial optimization problems. In the field of approximation algorithms,
this is well illustrated by the uncapacitated facility location problem. A
variety of algorithmic methodologies, such as LP-rounding and primal-dual
method, have been applied to and evolved from algorithms for this problem.
Unfortunately, this collection of powerful algorithmic techniques had not yet
been applicable to the more general capacitated facility location problem. In
fact, all of the known algorithms with good performance guarantees were based
on a single technique, local search, and no linear programming relaxation was
known to efficiently approximate the problem.
In this paper, we present a linear programming relaxation with constant
integrality gap for capacitated facility location. We demonstrate that the
fundamental theories of multi-commodity flows and matchings provide key
insights that lead to the strong relaxation. Our algorithmic proof of
integrality gap is obtained by finally accessing the rich toolbox of LP-based
methodologies: we present a constant factor approximation algorithm based on
LP-rounding.Comment: 25 pages, 6 figures; minor revision
Pseudo-random graphs
Random graphs have proven to be one of the most important and fruitful
concepts in modern Combinatorics and Theoretical Computer Science. Besides
being a fascinating study subject for their own sake, they serve as essential
instruments in proving an enormous number of combinatorial statements, making
their role quite hard to overestimate. Their tremendous success serves as a
natural motivation for the following very general and deep informal questions:
what are the essential properties of random graphs? How can one tell when a
given graph behaves like a random graph? How to create deterministically graphs
that look random-like? This leads us to a concept of pseudo-random graphs and
the aim of this survey is to provide a systematic treatment of this concept.Comment: 50 page
Investigation of Matching Problems using Constraint Programming and Optimisation Methods
This thesis focuses on matching under ordinal preferences, i.e. problems where agents may be required to list other agents that they find acceptable in order of preference. In particular, we focus on two main cases: the popular matching and the kidney exchange problem. These problems are important in practice and in this thesis we develop novel algorithms and techniques to solve them as combinatorial optimisation problems. The first part of the thesis focuses on one-sided matching on a bipartite graph, specifically the popular matching. When the participants express their preferences in an ordinal order, one might want to guarantee that no two applicants are inclined to form a coalition in order to maximise their welfare, thus finding a stable matching is needed. Popularity is a concept that offers an attractive trade- off between these two notions. In particular, we examine the popular matching in the context of constraint programming using global constraints. We discuss the possibility to find a popular matching even for the instances that does not admit one.
The second part of the thesis focuses on non-bipartite graphs, i.e. the kidney exchange problem. Kidney transplant is the most effective treatment to cure end-stage renal disease, affecting one in every thousand European citizen. Motivated by the observation that the kidney exchange is inherently a stochastic online problem, first, we give a stochastic online method, which provides an expected value estimation that is correct within the limit of sampling errors. Second, we show that by taking into consideration a probabilistic model of future arrivals and drop-offs, we can get reduce sampling scenarios, and we can even construct a sampling-free probabilistic model, called the Abstract Exchange Graph (AEG). A final contribution of this thesis is related to finding robust solutions when uncertainty occurs. Uncertainty is inherent to most real world problems
Truthful Facility Assignment with Resource Augmentation: An Exact Analysis of Serial Dictatorship
We study the truthful facility assignment problem, where a set of agents with
private most-preferred points on a metric space are assigned to facilities that
lie on the metric space, under capacity constraints on the facilities. The goal
is to produce such an assignment that minimizes the social cost, i.e., the
total distance between the most-preferred points of the agents and their
corresponding facilities in the assignment, under the constraint of
truthfulness, which ensures that agents do not misreport their most-preferred
points.
We propose a resource augmentation framework, where a truthful mechanism is
evaluated by its worst-case performance on an instance with enhanced facility
capacities against the optimal mechanism on the same instance with the original
capacities. We study a very well-known mechanism, Serial Dictatorship, and
provide an exact analysis of its performance. Although Serial Dictatorship is a
purely combinatorial mechanism, our analysis uses linear programming; a linear
program expresses its greedy nature as well as the structure of the input, and
finds the input instance that enforces the mechanism have its worst-case
performance. Bounding the objective of the linear program using duality
arguments allows us to compute tight bounds on the approximation ratio. Among
other results, we prove that Serial Dictatorship has approximation ratio
when the capacities are multiplied by any integer . Our
results suggest that even a limited augmentation of the resources can have
wondrous effects on the performance of the mechanism and in particular, the
approximation ratio goes to 1 as the augmentation factor becomes large. We
complement our results with bounds on the approximation ratio of Random Serial
Dictatorship, the randomized version of Serial Dictatorship, when there is no
resource augmentation
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