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We study the fair and efficient allocation of a set of indivisible goods among agents, where each good has several copies, and each agent has an additively separable concave valuation function with a threshold. These valuations capture the property of diminishing marginal returns, and they are more general than the well-studied case of additive valuations. We present a polynomial-time algorithm that approximates the optimal Nash social welfare (NSW) up to a factor of e1/e ≈ 1.445. This matches with the state-of-the-art approximation factor for additive valuations. The computed allocation also satisfies the popular fairness guarantee of envy-freeness up to one good (EF1) up to a factor of 2 + ε. For instances without thresholds, it is also approximately Pareto-optimal. For instances satisfying a large market property, we show an improved approximation factor. Lastly, we show that the upper bounds on the optimal NSW introduced in Cole and Gkatzelis (2018) and Barman et al. (2018) have the same value. 

Abstract Deep neural networks (DNNs) have achieved state-of-the-art performance in many important domains, including medical diagnosis, security, and autonomous driving. In domains where safety is highly critical, an erroneous decision can result in serious consequences. While a perfect prediction accuracy is not always achievable, recent work on Bayesian deep networks shows that it is possible to know when DNNs are more likely to make mistakes. Knowing what DNNs do not know is desirable to increase the safety of deep learning technology in sensitive applications; Bayesian neural networks attempt to address this challenge. Traditional approaches are computationally intractable and do not scale well to large, complex neural network architectures. In this paper, we develop a theoretical framework to approximate Bayesian inference for DNNs by imposing a Bernoulli distribution on the model weights. This method called Monte Carlo DropConnect (MC-DropConnect) gives us a tool to represent the model uncertainty with little change in the overall model structure or computational cost. We extensively validate the proposed algorithm on multiple network architectures and datasets for classification and semantic segmentation tasks. We also propose new metrics to quantify uncertainty estimates. This enables an objective comparison between MC-DropConnect and prior approaches. Our empirical results demonstrate that the proposed framework yields significant improvement in both prediction accuracy and uncertainty estimation quality compared to the state of the art. 

Vehicle routing problems are a broad class of combinatorial optimization problems that can be formulated as the problem of finding a tour in a weighted graph that optimizes some function of the visited vertices. For instance, a canonical and extensively studied vehicle routing problem is the orienteering problem where the goal is to find a tour that maximizes the number of vertices visited by a given deadline. In this paper, we consider the computational tractability of a well-known generalization of the orienteering problem called the Orient-MTW problem. The input to Orient-MTW consists of a weighted graph G(V, E) where for each vertex v ∊ V we are given a set of time instants Tv ⊆ [T], and a source vertex s. A tour starting at s is said to visit a vertex v if it transits through v at any time in the set Tv. The goal is to find a tour starting at the source vertex that maximizes the number of vertices visited. It is known that this problem admits a quasi-polynomial time O(log OPT)-approximation ratio where OPT is the optimal solution value but until now no hardness better than an APX-hardness was known for this problem. Our main result is an -hardness for this problem that holds even when the underlying graph G is an undirected tree. This is the first super-constant hardness result for the Orient-MTW problem. The starting point for our result is the hardness of the SetCover problem which is known to hold on instances with a special structure. We exploit this special structure of the hard SetCover instances to first obtain a new proof of the APX-hardness result for Orient-MTW that holds even on trees of depth 2. We then recursively amplify this constant factor hardness to an -hardness, while keeping the resulting topology to be a tree. Our amplified hardness proof crucially utilizes a delicate concavity property which shows that in our encoding of SetCover instances as instances of the Orient-MTW problem, whenever the optimal cost for SetCover instance is large, any tour, no matter how it allocates its time across different sub-trees, can not visit too many vertices overall. We believe that this reduction template may also prove useful in showing hardness of other vehicle routing problems. 

Computer-aided diagnosis (CAD) systems must constantly cope with the perpetual changes in data distribution caused by different sensing technologies, imaging protocols, and patient populations. Adapting these systems to new domains often requires significant amounts of labeled data for re-training. This process is labor-intensive and time-consuming. We propose a memory-augmented capsule network for the rapid adaptation of CAD models to new domains. It consists of a capsule network that is meant to extract feature embeddings from some high-dimensional input, and a memory-augmented task network meant to exploit its stored knowledge from the target domains. Our network is able to efficiently adapt to unseen domains using only a few annotated samples. We evaluate our method using a large-scale public lung nodule dataset (LUNA), coupled with our own collected lung nodules and incidental lung nodules datasets. When trained on the LUNA dataset, our network requires only 30 additional samples from our collected lung nodule and incidental lung nodule datasets to achieve clinically relevant performance (0.925 and 0.891 area under receiving operating characteristic curves (AUROC), respectively). This result is equivalent to using two orders of magnitude less labeled training data while achieving the same performance. We further evaluate our method by introducing heavy noise, artifacts, and adversarial attacks. Under these severe conditions, our network’s AUROC remains above 0.7 while the performance of state-of-the-art approaches reduce to chance level 

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.