An Analysis Framework for Metric Voting based on LP Duality

Distortion-based analysis has established itself as a fruitful framework for comparing voting mechanisms. m voters and n candidates are jointly embedded in an (unknown) metric space, and the voters submit rankings of candidates by non-decreasing distance from themselves. Based on the submitted rankings, the social choice rule chooses a winning candidate; the quality of the winner is the sum of the (unknown) distances to the voters. The rule's choice will in general be suboptimal, and the worst-case ratio between the cost of its chosen candidate and the optimal candidate is called the rule's distortion. It was shown in prior work that every deterministic rule has distortion at least 3, while the Copeland rule and related rules guarantee worst-case distortion at most 5; a very recent result gave a rule with distortion $2+\sqrt{5} \approx 4.236$. We provide a framework based on LP-duality and flow interpretations of the dual which provides a simpler and more unified way for proving upper bounds on the distortion of social choice rules. We illustrate the utility of this approach with three examples. First, we give a fairly simple proof of a strong generalization of the upper bound of 5 on the distortion of Copeland, to social choice rules with short paths from the winning candidate to the optimal candidate in generalized weak preference graphs. A special case of this result recovers the recent $2+\sqrt{5}$ guarantee. Second, using this generalized bound, we show that the Ranked Pairs and Schulze rules have distortion $\Theta(\sqrt(n))$. Finally, our framework naturally suggests a combinatorial rule that is a strong candidate for achieving distortion 3, which had also been proposed in recent work. We prove that the distortion bound of 3 would follow from any of three combinatorial conjectures we formulate.Comment: 23 pages An abbreviated version appears in Proceedings of AAAI 202

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 $g/(g-2)$ when the capacities are multiplied by any integer $g \geq 3$. 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

Three Puzzles on Mathematics, Computation, and Games

In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure

An oil painters recognition method based on cluster multiple kernel learning algorithm

A lot of image processing research works focus on natural images, such as in classification, clustering, and the research on the recognition of artworks (such as oil paintings), from feature extraction to classifier design, is relatively few. This paper focuses on oil painter recognition and tries to find the mobile application to recognize the painter. This paper proposes a cluster multiple kernel learning algorithm, which extracts oil painting features from three aspects: color, texture, and spatial layout, and generates multiple candidate kernels with different kernel functions. With the results of clustering numerous candidate kernels, we selected the sub-kernels with better classification performance, and use the traditional multiple kernel learning algorithm to carry out the multi-feature fusion classification. The algorithm achieves a better result on the Painting91 than using traditional multiple kernel learning directly

Algorithm Design for Ordinal Settings

Social choice theory is concerned with aggregating the preferences of agents into a single outcome. While it is natural to assume that agents have cardinal utilities, in many contexts, we can only assume access to the agents’ ordinal preferences, or rankings over the outcomes. As ordinal preferences are not as expressive as cardinal utilities, a loss of efficiency is unavoidable. Procaccia and Rosenschein (2006) introduced the notion of distortion to quantify this worst-case efficiency loss for a given social choice function. We primarily study distortion in the context of election, or equivalently clustering problems, where we are given a set of agents and candidates in a metric space; each agent has a preference ranking over the set of candidates and we wish to elect a committee of k candidates that minimizes the total social cost incurred by the agents. In the single-winner setting (when k = 1), we give a novel LP-duality based analysis framework that makes it easier to analyze the distortion of existing social choice functions, and extends readily to randomized social choice functions. Using this framework, we show that it is possible to give simpler proofs of known results. We also show how to efficiently compute an optimal randomized social choice function for any given instance. We utilize the latter result to obtain an instance for which any randomized social choice function has distortion at least 2.063164. This disproves the long-standing conjecture that there exists a randomized social choice function that has a worst-case distortion of at most 2. When k is at least 2, it is not possible to compute an O(1)-distortion committee using purely ordinal information. We develop two O(1)-distortion mechanisms for this problem: one having a polylog(n) (per agent) query complexity, where n is the number of agents; and the other having O(k) query complexity (i.e., no dependence on n). We also study a much more general setting called minimum-norm k-clustering recently proposed in the clustering literature, where the objective is some monotone, symmetric norm of the the agents' costs, and we wish to find a committee of k candidates to minimize this objective. When the norm is the sum of the p largest costs, which is called the p-centrum problem in the clustering literature, we give low-distortion mechanisms by adapting our mechanisms for k-median. En route, we give a simple adaptive-sampling algorithm for this problem. Finally, we show how to leverage this adaptive-sampling idea to also obtain a constant-factor bicriteria approximation algorithm for minimum-norm k-clustering (in its full generality)