Repairing non-monotone ordinal data sets by changing class labels

__Abstract__ Ordinal data sets often contain a certain amount of non-monotone noise. This paper proposes three algorithms for removing these non-monotonicities by relabeling the noisy instances. The first one is a naive algorithm. The second one is a refinement of this naive algorithm which minimizes the difference between the old and the new label. The third one is optimal in the sense that the number of unchanged instances is maximized. The last algorithm is a refinement of the second. In addition, the runtime complexities are discussed

Adding feasibility constraints to a ranking rule under a monotonicity constraint

We propose a new point of view in the long-standing problem where several voters have expressed a linear order relation (or ranking) over a set of candidates. For a ranking a > b > c to represent a group's opinion, it would be logical that the strength with which a > c is supported should not be less than the strength with which either a > b or b > c is supported. This intuitive property can be considered a monotonicity constraint, and has been addressed before. We extend previous approaches in the following way: as the voters are expressing linear orders, we can take the number of candidates between two candidates to be a measure of the degree to which one candidate is preferred to the other. In this way, intensity of support is both counted as the number of voters who indicate a > c is true, as well as the distance between a and c in these voters' rankings. The resulting distributions serve as input for a natural ranking rule that is based on stochastic monotonicity and stochastic dominance. Adapting the previous methodology turns out to be non-trivial once we add some natural feasibility constraints

Adaptive Priority Mechanisms

How should authorities that care about match quality and diversity allocate resources when they are uncertain about the market? We introduce adaptive priority mechanisms (APM) that prioritize agents based on both their scores and characteristics. We derive an APM that is optimal and show that the ubiquitous priority and quota mechanisms are optimal if and only if the authority is risk-neutral or extremely risk-averse over diversity, respectively. With many authorities, each authority using the optimal APM is dominant and implements the unique stable matching. Using Chicago Public Schools data, we find that the gains from adopting APM may be considerable

Rank-based linkage I: triplet comparisons and oriented simplicial complexes

Rank-based linkage is a new tool for summarizing a collection $S$ of objects according to their relationships. These objects are not mapped to vectors, and ``similarity'' between objects need be neither numerical nor symmetrical. All an object needs to do is rank nearby objects by similarity to itself, using a Comparator which is transitive, but need not be consistent with any metric on the whole set. Call this a ranking system on $S$. Rank-based linkage is applied to the $K$-nearest neighbor digraph derived from a ranking system. Computations occur on a 2-dimensional abstract oriented simplicial complex whose faces are among the points, edges, and triangles of the line graph of the undirected $K$-nearest neighbor graph on $S$. In $|S| K^2$ steps it builds an edge-weighted linkage graph $(S, \mathcal{L}, \sigma)$ where $\sigma(\{x, y\})$ is called the in-sway between objects $x$ and $y$. Take $\mathcal{L}_t$ to be the links whose in-sway is at least $t$, and partition $S$ into components of the graph $(S, \mathcal{L}_t)$, for varying $t$. Rank-based linkage is a functor from a category of out-ordered digraphs to a category of partitioned sets, with the practical consequence that augmenting the set of objects in a rank-respectful way gives a fresh clustering which does not ``rip apart`` the previous one. The same holds for single linkage clustering in the metric space context, but not for typical optimization-based methods. Open combinatorial problems are presented in the last section.Comment: 37 pages, 12 figure

Sparsifying, Shrinking and Splicing for Minimum Path Cover in Parameterized Linear Time

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