24 research outputs found
School Choice as a One-Sided Matching Problem: Cardinal Utilities and Optimization
The school choice problem concerns the design and implementation of matching mechanisms that produce school assignments for students within a given public school district. Previously considered criteria for evaluating proposed mechanisms such as stability, strategyproofness and Pareto efficiency do not always translate into desirable student assignments. In this note, we explore a class of one-sided, cardinal utility maximizing matching mechanisms focused exclusively on student preferences. We adapt a well-known combinatorial optimization technique (the Hungarian algorithm) as the kernel of this class of matching mechanisms. We find that, while such mechanisms can be adapted to meet desirable criteria not met by any previously employed mechanism in the school choice literature, they are not strategyproof. We discuss the practical implications and limitations of our approach at the end of the article
Decomposition of Trees and Paths via Correlation
We study the problem of decomposing (clustering) a tree with respect to costs
attributed to pairs of nodes, so as to minimize the sum of costs for those
pairs of nodes that are in the same component (cluster). For the general case
and for the special case of the tree being a star, we show that the problem is
NP-hard. For the special case of the tree being a path, this problem is known
to be polynomial time solvable. We characterize several classes of facets of
the combinatorial polytope associated with a formulation of this clustering
problem in terms of lifted multicuts. In particular, our results yield a
complete totally dual integral (TDI) description of the lifted multicut
polytope for paths, which establishes a connection to the combinatorial
properties of alternative formulations such as set partitioning.Comment: v2 is a complete revisio
Adapting a Kidney Exchange Algorithm to Align with Human Values
The efficient and fair allocation of limited resources is a classical problem
in economics and computer science. In kidney exchanges, a central market maker
allocates living kidney donors to patients in need of an organ. Patients and
donors in kidney exchanges are prioritized using ad-hoc weights decided on by
committee and then fed into an allocation algorithm that determines who gets
what--and who does not. In this paper, we provide an end-to-end methodology for
estimating weights of individual participant profiles in a kidney exchange. We
first elicit from human subjects a list of patient attributes they consider
acceptable for the purpose of prioritizing patients (e.g., medical
characteristics, lifestyle choices, and so on). Then, we ask subjects
comparison queries between patient profiles and estimate weights in a
principled way from their responses. We show how to use these weights in kidney
exchange market clearing algorithms. We then evaluate the impact of the weights
in simulations and find that the precise numerical values of the weights we
computed matter little, other than the ordering of profiles that they imply.
However, compared to not prioritizing patients at all, there is a significant
effect, with certain classes of patients being (de)prioritized based on the
human-elicited value judgments
A polynomial-time algorithm for optimization of quadratic pseudo-boolean functions
We develop a polynomial-time algorithm to minimize pseudo-Boolean functions. The computational complexity is O(n â–ˇ(15/2)), although very conservative, it is su_cient to prove that this minimization problem is in the class P. A direct application of the algorithm is the 3-SAT problem, which is also guaranteed to be in the class P with a computational complexity of order O(n â–ˇ(45/2)). The algorithm was implemented in MATLAB and checked by generating one million matrices of arbitrary dimension up to 24 with random entries in the range [-50; 50]. All the experiments were successful
Certifying Solvers for Clique and Maximum Common (Connected) Subgraph Problems
An algorithm is said to be certifying if it outputs, together with a solution to the problem it solves, a proof that this solution is correct. We explain how state of the art maximum clique, maximum weighted clique, maximal clique enumeration and maximum common (connected) induced subgraph algorithms can be turned into certifying solvers by using pseudo-Boolean models and cutting planes proofs, and demonstrate that this approach can also handle reductions between problems. The generality of our results suggests that this method is ready for widespread adoption in solvers for combinatorial graph problems