In an instance of the weighted Nash Social Welfare problem, we are given a set of m indivisible items, G, and n agents, A, where each agent i ∈ A has a valuation vij ≥ 0 for each item j ∈ G. In addition, every agent i has a non-negative weight wi such that the weights collectively sum up to 1. The goal is to find an assignment σ : G → A that maximizes . When all the weights equal to , the problem reduces to the classical Nash Social Welfare problem, which has recently received much attention. In this work, we present a -approximation algorithm for the weighted Nash Social Welfare problem, where denotes the KL-divergence between the distribution w and the uniform distribution on [n].

Beyond self-report data, we lack reliable and non-intrusive methods for identifying flow. However, taking a step back and acknowledging that flow occurs during periods of focus gives us the opportunity to make progress towards measuring flow by isolating focused work. Here, we take a mixed-methods approach to design a logs-based metric that leverages machine learning and a comprehensive collection of logs data to identify periods of related actions (indicating focus), and validate this metric against self-reported time in focus or flow using diary data and quarterly survey data. Our results indicate that we can determine when software engineers at a large technology company experience focused work which includes instances of flow. This metric speaks to engineering work, but can be leveraged in other domains to non-disruptively measure when people experience focus. Future research can build upon this work to identify signals associated with other facets of flow.

Determinant maximization problem gives a general framework that models problems arising in as diverse fields as statistics [1], convex geometry [2], fair allocations [3], combinatorics [4], spectral graph theory [5], network design, and random processes [6]. In an instance of a determinant maximization problem, we are given a collection of vectors $U=\{v_{1},\cdots,\ v_{n}\}\subset \mathbb{R}^{d}$, and a goal is to pick a subset $S\subseteq U$ of given vectors to maximize the determinant of the matrix $\displaystyle \sum_{i\in S}v_{i}v_{i}^{\text{T}}$. Often, the set S of picked vectors must satisfy additional combinatorial constraints such as cardinality constraint $(|S|\leq k)$ or matroid constraint $(S$ is a basis of a matroid defined on the vectors). In this paper, we give a polynomial-time deterministic algorithm that returns a $r^{O(r)}$-approximation for any matroid of rank $r \leq d$. This improves previous results that give $e^{O(r^{2})}$-approximation algorithms relying on $e^{O(r)}$-approximate estimation algorithms [4], [7] –[9] for any r$\leq$d. All previous results use convex relaxations and their relationship to stable polynomials and strongly $\log$-concave polynomials or non-convex relaxations for the problem [10]. In contrast, our algorithm builds on combinatorial algorithms for matroid intersection, which iteratively improve any solution by finding an alternating negative cycle in the exchange graph defined by the matroids. While the $\det(.)$ function is not linear, we show that taking appropriate linear approximations at each iteration suffice to give the improved approximation algorithm.

Abstract We investigate a sheaf-theoretic interpretation of stratification learning from geometric and topological perspectives. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al. (Proceedings of the 23rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1355–1370, ACM, New York, 2012), and the cohomology stratification algorithm given in Nanda (Found. Comput. Math. 20(2), 195–222, 2020). Additionally, we give examples of stratifications based on the geometric techniques of Breiding et al. (Rev. Mat. Complut. 31(3), 545–593, 2018), illustrating how the sheaf-theoretic approach can be used to study stratifications from both topological and geometric perspectives. This approach also points toward future applications of sheaf theory in the study of topological data analysis by illustrating the utility of the language of sheaf theory in generalizing existing algorithms.

In this paper, we investigate a sheaf-theoretic interpretation of stratification learning. Motivated by the work of Alexandroff (1937) and McCord (1978), we aim to redirect efforts in the computational topology of triangulated compact polyhedra to the much more computable realm of sheaves on partially ordered sets. Our main result is the construction of stratification learning algorithms framed in terms of a sheaf on a partially ordered set with the Alexandroff topology. We prove that the resulting decomposition is the unique minimal stratification for which the strata are homogeneous and the given sheaf is constructible. In particular, when we choose to work with the local homology sheaf, our algorithm gives an alternative to the local homology transfer algorithm given in Bendich et al. (2012), and the cohomology stratification algorithm given in Nanda (2017). We envision that our sheaf-theoretic algorithm could give rise to a larger class of stratification beyond homology-based stratification. This approach also points toward future applications of sheaf theory in the study of topological data analysis by illustrating the utility of the language of sheaf theory in generalizing existing algorithms.

This paper presents a new state space reduction technique that applies to model checking of software. The new technique, precision abstraction, borrows ideas from dataflow analysis to identify procedures that can be analyzed context-insensitively without affecting the accuracy of the verification of a given property. These context-insensitive procedures can then be represented with fewer states than would be needed context-sensitive analysis. Preliminary results indicate that the number of transitions in the analysis prescribed by our approach is at least 155 times fewer than the exhaustive analysis a model checker would otherwise perform.