We provide the first large-scale data collection of real-world approval-based committee elections. These elections have been conducted on the Polkadot blockchain as part of their Nominated Proof-of-Stake mechanism and contain around one thousand candidates and tens of thousands of (weighted) voters each. We conduct an in-depth study of application-relevant questions, including a quantitative and qualitative analysis of the outcomes returned by different voting rules. Besides considering proportionality measures that are standard in the multiwinner voting literature, we pay particular attention to less-studied measures of overrepresentation, as these are closely related to the security of the Polkadot network. We also analyze how different design decisions such as the committee size affect the examined measures.

The concept of proportional representation in approval-based committee elections has appeared in the social choice literature for over a century and is typically understood as avoiding the underrepresentation of minorities. However, we argue that the security of some distributed systems critically depends on the opposite goal of preventing the overrepresentation of any minority, a goal not previously formalized that leads us to an optimization objective known as maximin support. After providing a thorough analysis of the computational complexity of this objective, we propose a new efficient election rule that simultaneously achieves a) a constant-factor approximation guarantee for it, and b) the property of proportional justified representation (PJR) - one of the strongest forms of proportional representation. However, the most striking feature of the new rule is that one can verify in linear time that the winning committee satisfies the two aforementioned guarantees, even when the algorithm is executed by an untrusted party who only communicates the output. As a result, the rule can be adapted into a verifiable computing scheme. Moreover, its verification procedure easily admits parallel processing for further efficiency.

Diversity maximization is an important geometric optimization problem with many applications in recommender systems, machine learning or search engines among others. A typical diversification problem is as follows: Given a finite metric space $(X,d)$ and a parameter $k \in \mathbb{N}$, find a subset of $k$ elements of $X$ that has maximum diversity. There are many functions that measure diversity. One of the most popular measures, called remote-clique, is the sum of the pairwise distances of the chosen elements. In this paper, we present novel results on three widely used diversity measures: Remote-clique, remote-star and remote-bipartition. Our main result are polynomial time approximation schemes for these three diversification problems under the assumption that the metric space is doubling. This setting has been discussed in the recent literature. The existence of such a PTAS however was left open. Our results also hold in the setting where the distances are raised to a fixed power $q\geq 1$, giving rise to more variants of diversity functions, similar in spirit to the variations of clustering problems depending on the power applied to the distances. Finally, we provide a proof of NP-hardness for remote-clique with squared distances in doubling metric spaces.

Mixed-integer mathematical programs are among the most commonly used models for a wide set of problems in Operations Research and related fields. However, there is still very little known about what can be expressed by small mixed-integer programs. In particular, prior to this work, it was open whether some classical problems, like the minimum odd-cut problem, can be expressed by a compact mixed-integer program with few (even constantly many) integer variables. This is in stark contrast to linear formulations, where recent breakthroughs in the field of extended formulations have shown that many polytopes associated to classical combinatorial optimization problems do not even admit approximate extended formulations of sub-exponential size. We provide a general framework for lifting inapproximability results of extended formulations to the setting of mixed-integer extended formulations, and obtain almost tight lower bounds on the number of integer variables needed to describe a variety of classical combinatorial optimization problems. Among the implications we obtain, we show that any mixed-integer extended formulation of sub-exponential size for the matching polytope, cut polytope, travelling salesman polytope or dominant of the odd-cut polytope, needs Ω(n / log n) many integer variables, where n is the number of vertices of the underlying graph. Conversely, the above-mentioned polyhedra admit polynomial-size mixed-integer formulations with only O(n) or O(n log n) (for the traveling salesman polytope) many integer variables. Our results build upon a new decomposition technique that, for any convex set C, allows for approximating any mixed-integer description of C by the intersection of C with the union of a small number of affine subspaces.

$2$-level polytopes naturally appear in several areas of pure and applied mathematics, including combinatorial optimization, polyhedral combinatorics, communication complexity, and statistics. In this paper, we present a study of some $2$-level polytopes arising in combinatorial settings. Our first contribution is proving that $f_0(P)f_{d-1}(P)\leq d2^{d+1}$ for a large collection of families of such polytopes $P$. Here $f_0(P)$ (resp., $f_{d-1}(P)$) is the number of vertices (resp., facets) of $P$, and $d$ is its dimension. Whether this holds for all 2-level polytopes was asked in [A. Bohn, Y. Faenza, S. Fiorini, V. Fisikopoulos, M. Macchia, and K. Pashkovich, in Algorithms--ESA 2015, Springer, Berlin, 2015, pp. 191--202], and experimental results from [S. Fiorini, V. Fisikopoulos, and M. Macchia, in Combinatorial Optimization, Springer, Cham, 2016, pp. 285--296] showed it true for $d\leq 7$. The key to most of our proofs is a deeper understanding of the relations among those polytopes and their underlying combinatorial structure. This leads to a number of results that we believe to be of independent interest: a trade-off formula for the number of cliques and stable sets in a graph, a description of stable matching polytopes as affine projections of certain order polytopes, and a linear-size description of the base polytope of matroids that are 2-level in terms of cuts of an associated tree.

We provide simple and fast polynomial-time approximation schemes (PTASs) for several variants of the max-sum diversification problem which, in its most basic form, is as follows: given n points p1,…,pn ∊ ℝq and an integer k, select k points such that the average Euclidean distance between these points is maximized. This problem is commonly applied in web search and information retrieval in order to select a diverse set of representative points from the input. In this context, it has recently received a lot of attention. We present new techniques to analyze natural local- search algorithms. This leads to a for distances of negative type, even subject to a general matroid constraint of rank k, in time O(nk2 log k), when assuming that distance evaluations and calls to the independence oracle are constant time. Negative-type distances include as special cases Euclidean and Manhattan distances, among other natural distances. Our result easily transforms into a PTAS. It improves on the only previously known PTAS for this setting, which relies on convex optimization techniques in an n-dimensional space and is impractical for large data sets. In contrast, our procedure has an (optimal) linear dependence on n. Using generalized exchange properties of matroid intersection, we show that a PTAS can be obtained for matroid- intersection constraints as well. Moreover, our techniques, being based on local search, are conceptually simple and allow for various extensions. In particular, we get asymptotically optimal O(1)-approximations when combining the classic dispersion function with a monotone submodular objective, which is a very common class of functions to measure diversity and relevance. This result leverages recent advances on local-search techniques based on proxy functions to obtain optimal approximations for monotone submodular function maximization subject to a matroid constraint.