arXiv subjects

Alfonso Cevallos

Publications and source records attributed to Alfonso Cevallos.

9 recordsLinked to original sources

arXiv2025

The Merkle Mountain Belt

Merkle structures are widely used as commitment schemes: they allow a prover to publish a compact commitment to an ordered list $X$ of items, and then efficiently prove to a verifier that $x_i\in X$ is the $i$-th item in it. We compare different Merkle structures and their corresponding properties as commitment schemes in the context of blockchain applications. Our primary goal is to speed up light client protocols so that, e.g., a user can verify a transaction efficiently from their smartphone. For instance, the Merkle Mountain Range (MMR) yields a succinct scheme: a light client synchronizing for the first time can do so with a complexity sublinear in $|X|$. On the other hand, the Merkle chain, traditionally used to commit to block headers, is not succinct, but it is incremental - a light client resynchronizing frequently can do so with constant complexity - and optimally additive - the structure can be updated in constant time when a new item is appended to list $X$. We introduce new Merkle structures, most notably the Merkle Mountain Belt (MMB), the first to be simultaneously succinct, incremental and optimally additive. A variant called UMMB is also asynchronous: a light client may continue to interact with the network even when out of sync with the public commitment. Our Merkle structures are slightly unbalanced, so that items recently appended to $X$ receive shorter membership proofs than older items. This feature reduces a light client's expected costs, in applications where queries are biased towards recently generated data.

cs.DS↗

arXiv2023

Approval-Based Committee Voting in Practice: A Case Study of (Over-)Representation in the Polkadot Blockchain

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.

cs.GT↗

arXiv2021

A verifiably secure and proportional committee election rule

The property 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 is directly linked to the opposite goal of avoiding 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. Our work is motivated by an application on blockchain networks that implement Nominated Proof-of-Stake, where the community elects a committee of validators to participate in the consensus protocol, and where preventing overrepresentation protects the network against attacks by an adversarial minority. Our election rule enables a validator selection protocol with formal guarantees on security and proportionality, and its adaptation as a verifiable computing scheme with a parallelized verification proves to be key for its successful implementation given the computationally limited nature of the blockchain architecture.

cs.DS↗

arXiv2020

Overview of Polkadot and its Design Considerations

In this paper we describe the design components of the heterogenous multi-chain protocol Polkadot and explain how these components help Polkadot address some of the existing shortcomings of blockchain technologies. At present, a vast number of blockchain projects have been introduced and employed with various features that are not necessarily designed to work with each other. This makes it difficult for users to utilise a large number of applications on different blockchain projects. Moreover, with the increase in number of projects the security that each one is providing individually becomes weaker. Polkadot aims to provide a scalable and interoperable framework for multiple chains with pooled security that is achieved by the collection of components described in this paper.

cs.CR↗

arXiv2018

Diversity maximization in doubling metrics

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.

cs.DM↗

arXiv2017

On 2-level polytopes arising in combinatorial settings

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 v(P)*f(P) is upper bounded by d*2^(d+1), for a large collection of families of such polytopes P. Here v(P) (resp. f(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 [Bohn et al., ESA 2015], and experimental results from [Fiorini et al., ISCO 2016] showed it true up to dimension 7. The key to most of our proofs is a deeper understanding of the relations among those polytopes and their underlying combinatorial structures. 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.

math.CO↗

arXiv2017

Lifting Linear Extension Complexity Bounds to the Mixed-Integer Setting

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, traveling 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.

cs.DM↗

arXiv2016

Local Search for Max-Sum Diversification

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 p_1,...,p_n in R^d and an integer k, select k points such that the average Euclidean distance between these points is maximized. This problem commonly appears in information retrieval and web-search in order to select a diverse set of 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 (1-O(1/k))-approximation for distances of negative type, even subject to any matroid constraint of rank k, in time O(n k^2 log k), when assuming that distance evaluations and calls to the independence oracle are constant time. Negative type distances include as special cases Euclidean distances and many further natural distances. Our result easily transforms into a PTAS and 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.

cs.DS↗

arXiv2015

Max-sum diversity via convex programming

Diversity maximization is an important concept in information retrieval, computational geometry and operations research. Usually, it is a variant of the following problem: Given a ground set, constraints, and a function $f(\cdot)$ that measures diversity of a subset, the task is to select a feasible subset $S$ such that $f(S)$ is maximized. The \emph{sum-dispersion} function $f(S) = \sum_{x,y \in S} d(x,y)$, which is the sum of the pairwise distances in $S$, is in this context a prominent diversification measure. The corresponding diversity maximization is the \emph{max-sum} or \emph{sum-sum diversification}. Many recent results deal with the design of constant-factor approximation algorithms of diversification problems involving sum-dispersion function under a matroid constraint. In this paper, we present a PTAS for the max-sum diversification problem under a matroid constraint for distances $d(\cdot,\cdot)$ of \emph{negative type}. Distances of negative type are, for example, metric distances stemming from the $\ell_2$ and $\ell_1$ norm, as well as the cosine or spherical, or Jaccard distance which are popular similarity metrics in web and image search.

cs.DS↗