Fair and Efficient Allocations under Subadditive Valuations

We study the problem of allocating a set of indivisible goods among agents with subadditive valuations in a fair and efficient manner. Envy-Freeness up to any good (EFX) is the most compelling notion of fairness in the context of indivisible goods. Although the existence of EFX is not known beyond the simple case of two agents with subadditive valuations, some good approximations of EFX are known to exist, namely $\tfrac{1}{2}$-EFX allocation and EFX allocations with bounded charity. Nash welfare (the geometric mean of agents' valuations) is one of the most commonly used measures of efficiency. In case of additive valuations, an allocation that maximizes Nash welfare also satisfies fairness properties like Envy-Free up to one good (EF1). Although there is substantial work on approximating Nash welfare when agents have additive valuations, very little is known when agents have subadditive valuations. In this paper, we design a polynomial-time algorithm that outputs an allocation that satisfies either of the two approximations of EFX as well as achieves an $\mathcal{O}(n)$ approximation to the Nash welfare. Our result also improves the current best-known approximation of $\mathcal{O}(n \log n)$ and $\mathcal{O}(m)$ to Nash welfare when agents have submodular and subadditive valuations, respectively. Furthermore, our technique also gives an $\mathcal{O}(n)$ approximation to a family of welfare measures, $p$-mean of valuations for $p\in (-\infty, 1]$, thereby also matching asymptotically the current best known approximation ratio for special cases like $p =-\infty$ while also retaining the fairness properties

Fair and Efficient Allocations under Subadditive Valuations

We study the problem of allocating a set of indivisible goods among agents with subadditive valuations in a fair and efficient manner. Envy-Freeness up to any good (EFX) is the most compelling notion of fairness in the context of indivisible goods. Although the existence of EFX is not known beyond the simple case of two agents with subadditive valuations, some good approximations of EFX are known to exist, namely $\tfrac{1}{2}$-EFX allocation and EFX allocations with bounded charity. Nash welfare (the geometric mean of agents' valuations) is one of the most commonly used measures of efficiency. In case of additive valuations, an allocation that maximizes Nash welfare also satisfies fairness properties like Envy-Free up to one good (EF1). Although there is substantial work on approximating Nash welfare when agents have additive valuations, very little is known when agents have subadditive valuations. In this paper, we design a polynomial-time algorithm that outputs an allocation that satisfies either of the two approximations of EFX as well as achieves an $\mathcal{O}(n)$ approximation to the Nash welfare. Our result also improves the current best-known approximation of $\mathcal{O}(n \log n)$ and $\mathcal{O}(m)$ to Nash welfare when agents have submodular and subadditive valuations, respectively. Furthermore, our technique also gives an $\mathcal{O}(n)$ approximation to a family of welfare measures, $p$-mean of valuations for $p\in (-\infty, 1]$, thereby also matching asymptotically the current best known approximation ratio for special cases like $p =-\infty$ while also retaining the fairness properties

Evaluation of female factors in infertility by diagnostic laparohysteroscopy in a tertiary health care centre

Background: The inability to conceive is one of the most distressing conditions for a couple. It not only makes the female incomplete but also the social taboos attached are phenomenal. The problem of infertility as long as the recorded history of mankind. Fertility in our culture stands for reproductivity, growth and continuity. Reproduction is one of the basic essential for the survival of a species.  Diagnostic laparoscopy & hysteroscopy have emerged as an accurate method of assessing, evaluating and treating infertility. Direct visualization of the abdominal and pelvic organs in laparohysteroscopy allows a definitive diagnosis to be made in many conditions where clinical examination and less invasive techniques such as ultrasound and hysterosalpingography fail to identify the problem.Methods: A prospective study was conducted in Department of Obstetrics and Gynaecology, AGMC& GBP Hospital Agartala. 50 infertile women suspected with pelvic (tubal, peritoneal, adnexal) and intrauterine (uterine polyp, septa, submucous fibroid, intrauterine adhesions) pathologies were included in the study for further evaluation and correlation of clinical findings with Laparohysteroscopy observations.Results: Out of 50 cases, 27 (54%) patients had primary infertility. While laparoscopy detected abnormalities in 60% of the cases, significant hysteroscopy findings were noted in 66% of cases. The most common laparoscopic abnormality was tubal (22%) ovarian and peritoneal (16%) in primary and secondary infertile patients respectively. On hysteroscopy, endometrial polyp (30%) was found as the commonest abnormality in both the groups.Conclusions: Laparoscopy and hysteroscopy are both diagnostic and therapeutic procedures. If pathology is discovered, it can often be treated simultaneously. Laparoscopy combined with hysteroscopy is the sole technique to have a direct view of the female reproductive tract and to find out the various causes of infertility

EFX exists for three agents

We study the problem of distributing a set of indivisible items among agents with additive valuations in a $\mathit{fair}$ manner. The fairness notion under consideration is Envy-freeness up to any item (EFX). Despite significant efforts by many researchers for several years, the existence of EFX allocations has not been settled beyond the simple case of two agents. In this paper, we show constructively that an EFX allocation always exists for three agents. Furthermore, we falsify the conjecture by Caragiannis et al. by showing an instance with three agents for which there is a partial EFX allocation (some items are not allocated) with higher Nash welfare than that of any complete EFX allocation

Data Augmentation for Low-Resource Keyphrase Generation

Keyphrase generation is the task of summarizing the contents of any given article into a few salient phrases (or keyphrases). Existing works for the task mostly rely on large-scale annotated datasets, which are not easy to acquire. Very few works address the problem of keyphrase generation in low-resource settings, but they still rely on a lot of additional unlabeled data for pretraining and on automatic methods for pseudo-annotations. In this paper, we present data augmentation strategies specifically to address keyphrase generation in purely resource-constrained domains. We design techniques that use the full text of the articles to improve both present and absent keyphrase generation. We test our approach comprehensively on three datasets and show that the data augmentation strategies consistently improve the state-of-the-art performance. We release our source code at https://github.com/kgarg8/kpgen-lowres-data-aug.Comment: 9 pages, 8 tables, To appear at the Findings of the Proceedings of the 61st Annual Meeting of the Association for Computational Linguistics, Toronto, Canad

Tighter Estimates for ϵ-nets for Disks

International audienceThe geometric hitting set problem is one of the basic geometric combinatorial optimization problems: given a set P of points, and a set D of geometric objects in the plane, the goal is to compute a small-sized subset of P that hits all objects in D. In 1994, Bronniman and Goodrich [5] made an important connection of this problem to the size of fundamental combinatorial structures called ϵ-nets, showing that small-sized ϵ-nets imply approximation algorithms with correspondingly small approximation ratios. Very recently, Agarwal and Pan [2] showed that their scheme can be implemented in near-linear time for disks in the plane. Altogether this gives O(1)-factor approximation algorithms in O(n) time for hitting sets for disks in the plane. This constant factor depends on the sizes of ϵ-nets for disks; unfortunately, the current state-of-the-art bounds are large – at least 24/ϵ and most likely larger than 40/ϵ. Thus the approximation factor of the Agarwal and Pan algorithm ends up being more than 40. The best lower-bound is 2/ϵ, which follows from the Pach-Woeginger construction [32] for halfplanes in two dimensions. Thus there is a large gap between the best-known upper and lower bounds. Besides being of independent interest, finding precise bounds is important since this immediately implies an improved linear-time algorithm for the hitting-set problem. The main goal of this paper is to improve the upper-bound to 13.4/ϵ for disks in the plane. The proof is constructive, giving a simple algorithm that uses only Delaunay triangulations. We have implemented the algorithm, which is available as a public open-source module. Experimental results show that the sizes of-nets for a variety of data-sets is lower, around 9/ϵ

On Fair Division of Indivisible Items

We consider the task of assigning indivisible goods to a set of agents in a fair manner. Our notion of fairness is Nash social welfare, i.e., the goal is to maximize the geometric mean of the utilities of the agents. Each good comes in multiple items or copies, and the utility of an agent diminishes as it receives more items of the same good. The utility of a bundle of items for an agent is the sum of the utilities of the items in the bundle. Each agent has a utility cap beyond which he does not value additional items. We give a polynomial time approximation algorithm that maximizes Nash social welfare up to a factor of $e^{1/e} \approx 1.445$

On Fair Division for Indivisible Items

We consider the task of assigning indivisible goods to a set of agents in a fair manner. Our notion of fairness is Nash social welfare, i.e., the goal is to maximize the geometric mean of the utilities of the agents. Each good comes in multiple items or copies, and the utility of an agent diminishes as it receives more items of the same good. The utility of a bundle of items for an agent is the sum of the utilities of the items in the bundle. Each agent has a utility cap beyond which he does not value additional items. We give a polynomial time approximation algorithm that maximizes Nash social welfare up to a factor of e^{1/{e}} ~~ 1.445. The computed allocation is Pareto-optimal and approximates envy-freeness up to one item up to a factor of 2 + epsilon