3,660 research outputs found
A Combinatorial Algorithm to Establish a Fair Border
A finite algorithm is given for the following problem: a piece of land bordered by n countries is to be divided equally among these n countries in such a way that each country's share is connected and adjacent to its original border
Recognizing Graph Theoretic Properties with Polynomial Ideals
Many hard combinatorial problems can be modeled by a system of polynomial
equations. N. Alon coined the term polynomial method to describe the use of
nonlinear polynomials when solving combinatorial problems. We continue the
exploration of the polynomial method and show how the algorithmic theory of
polynomial ideals can be used to detect k-colorability, unique Hamiltonicity,
and automorphism rigidity of graphs. Our techniques are diverse and involve
Nullstellensatz certificates, linear algebra over finite fields, Groebner
bases, toric algebra, convex programming, and real algebraic geometry.Comment: 20 pages, 3 figure
Finding Fair and Efficient Allocations
We study the problem of allocating a set of indivisible goods among a set of
agents in a fair and efficient manner. An allocation is said to be fair if it
is envy-free up to one good (EF1), which means that each agent prefers its own
bundle over the bundle of any other agent up to the removal of one good. In
addition, an allocation is deemed efficient if it satisfies Pareto optimality
(PO). While each of these well-studied properties is easy to achieve
separately, achieving them together is far from obvious. Recently, Caragiannis
et al. (2016) established the surprising result that when agents have additive
valuations for the goods, there always exists an allocation that simultaneously
satisfies these two seemingly incompatible properties. Specifically, they
showed that an allocation that maximizes the Nash social welfare (NSW)
objective is both EF1 and PO. However, the problem of maximizing NSW is
NP-hard. As a result, this approach does not provide an efficient algorithm for
finding a fair and efficient allocation.
In this paper, we bypass this barrier, and develop a pseudopolynomial time
algorithm for finding allocations that are EF1 and PO; in particular, when the
valuations are bounded, our algorithm finds such an allocation in polynomial
time. Furthermore, we establish a stronger existence result compared to
Caragiannis et al. (2016): For additive valuations, there always exists an
allocation that is EF1 and fractionally PO.
Another contribution of our work is to show that our algorithm provides a
polynomial-time 1.45-approximation to the NSW objective. This improves upon the
best known approximation ratio for this problem (namely, the 2-approximation
algorithm of Cole et al. (2017)). Unlike many of the existing approaches, our
algorithm is completely combinatorial.Comment: 40 pages. Updated versio
Designing Fair Ranking Schemes
Items from a database are often ranked based on a combination of multiple
criteria. A user may have the flexibility to accept combinations that weigh
these criteria differently, within limits. On the other hand, this choice of
weights can greatly affect the fairness of the produced ranking. In this paper,
we develop a system that helps users choose criterion weights that lead to
greater fairness.
We consider ranking functions that compute the score of each item as a
weighted sum of (numeric) attribute values, and then sort items on their score.
Each ranking function can be expressed as a vector of weights, or as a point in
a multi-dimensional space. For a broad range of fairness criteria, we show how
to efficiently identify regions in this space that satisfy these criteria.
Using this identification method, our system is able to tell users whether
their proposed ranking function satisfies the desired fairness criteria and, if
it does not, to suggest the smallest modification that does. We develop
user-controllable approximation that and indexing techniques that are applied
during preprocessing, and support sub-second response times during the online
phase. Our extensive experiments on real datasets demonstrate that our methods
are able to find solutions that satisfy fairness criteria effectively and
efficiently
Optimizing the location of weather monitoring stations using estimation uncertainty
In this article, we address the problem of planning a network of weather monitoring stations observing average air temperature (AAT). Assuming the network planning scenario as a location problem, an optimization model and an operative methodology are proposed. The model uses the geostatistical uncertainty of estimation and the indicator formalism to consider in the location process a variable demand surface, depending on the spatial arrangement of the stations. This surface is also used to express a spatial representativeness value for each element in the network. It is then possible to locate such a network using optimization techniques, such as the used methods of simulated annealing (SA) and construction heuristics. This new approach was applied in the optimization of the Portuguese network of weather stations monitoring the AAT variable. In this case study, scenarios of reduction in the number of stations were generated and analysed: the uncertainty of estimation was computed, interpreted and applied to model the varying demand surface that is used in the optimization process. Along with the determination of spatial representativeness value of individual stations, SA was used to detect redundancies on the existing network and establish the base for its expansion. Using a greedy algorithm, a new network for monitoring average temperature in the selected study area is proposed and its effectiveness is compared with the current distribution of stations. For this proposed network distribution maps of the uncertainty of estimation and the temperature distribution were created. Copyright (c) 2011 Royal Meteorological Societyinfo:eu-repo/semantics/publishedVersio
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