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We introduce and study the problem of balanced districting, where given an undirected graph with vertices carrying two types of weights (different population, resource types, etc) the goal is to maximize the total weights covered in vertex disjoint districts such that each district is a star or (in general) a connected induced subgraph with the two weights to be balanced. This problem is strongly motivated by political redistricting, where contiguity, population balance, and compactness are essential. We provide hardness and approximation algorithms for this problem. In particular, we show NP-hardness for an approximation better than n^{1/2-δ} for any constant δ > 0 in general graphs even when the districts are star graphs, as well as NP-hardness on complete graphs, tree graphs, planar graphs and other restricted settings. On the other hand, we develop an algorithm for balanced star districting that gives an O(√n)-approximation on any graph (which is basically tight considering matching hardness of approximation results), an O(log n) approximation on planar graphs with extensions to minor-free graphs. Our algorithm uses a modified Whack-a-Mole algorithm [Bhattacharya, Kiss, and Saranurak, SODA 2023] to find a sparse solution of a fractional packing linear program (despite exponentially many variables) which requires a new design of a separation oracle specific for our balanced districting problem. To turn the fractional solution to a feasible integer solution, we adopt the randomized rounding algorithm by [Chan and Har-Peled, SoCG 2009]. To get a good approximation ratio of the rounding procedure, a crucial element in the analysis is the balanced scattering separators for planar graphs and minor-free graphs - separators that can be partitioned into a small number of k-hop independent sets for some constant k - which may find independent interest in solving other packing style problems. Further, our algorithm is versatile - the very same algorithm can be analyzed in different ways on various graph classes, which leads to class-dependent approximation ratios. We also provide a FPTAS algorithm for complete graphs and tree graphs, as well as greedy algorithms and approximation ratios when the district cardinality is bounded, the graph has bounded degree or the weights are binary. We refer the readers to the full version of the paper for complete set of results and proofs. 

People with blindness and low vision (pBLV) encounter substantial challenges when it comes to comprehensive scene recognition and precise object identification in unfamiliar environments. Additionally, due to the vision loss, pBLV have difficulty in accessing and identifying potential tripping hazards independently. Previous assistive technologies for the visually impaired often struggle in real-world scenarios due to the need for constant training and lack of robustness, which limits their effectiveness, especially in dynamic and unfamiliar environments, where accurate and efficient perception is crucial. Therefore, we frame our research question in this paper as: How can we assist pBLV in recognizing scenes, identifying objects, and detecting potential tripping hazards in unfamiliar environments, where existing assistive technologies often falter due to their lack of robustness? We hypothesize that by leveraging large pretrained foundation models and prompt engineering, we can create a system that effectively addresses the challenges faced by pBLV in unfamiliar environments. Motivated by the prevalence of large pretrained foundation models, particularly in assistive robotics applications, due to their accurate perception and robust contextual understanding in real-world scenarios induced by extensive pretraining, we present a pioneering approach that leverages foundation models to enhance visual perception for pBLV, offering detailed and comprehensive descriptions of the surrounding environment and providing warnings about potential risks. Specifically, our method begins by leveraging a large-image tagging model (i.e., Recognize Anything Model (RAM)) to identify all common objects present in the captured images. The recognition results and user query are then integrated into a prompt, tailored specifically for pBLV, using prompt engineering. By combining the prompt and input image, a vision-language foundation model (i.e., InstructBLIP) generates detailed and comprehensive descriptions of the environment and identifies potential risks in the environment by analyzing environmental objects and scenic landmarks, relevant to the prompt. We evaluate our approach through experiments conducted on both indoor and outdoor datasets. Our results demonstrate that our method can recognize objects accurately and provide insightful descriptions and analysis of the environment for pBLV. 

Peer prediction refers to a collection of mechanisms for eliciting information from human agents when direct verification of the obtained information is unavailable. They are designed to have a game-theoretic equilibrium where everyone reveals their private information truthfully. This result holds under the assumption that agents are Bayesian and they each adopt a fixed strategy across all tasks. Human agents however are observed in many domains to exhibit learning behavior in sequential settings. In this paper, we explore the dynamics of sequential peer prediction mechanisms when participants are learning agents. We first show that the notion of no regret alone for the agents’ learning algorithms cannot guaran- tee convergence to the truthful strategy. We then focus on a family of learning algorithms where strategy updates only depend on agents’ cumulative rewards and prove that agents’ strategies in the popular Correlated Agreement (CA) mechanism converge to truthful reporting when they use algorithms from this family. This fam- ily of algorithms is not necessarily no-regret, but includes several familiar no-regret learning algorithms (e.g multiplicative weight update and Follow the Perturbed Leader) as special cases. Simulation of several algorithms in this family as well as the ε-greedy algorithm, which is outside of this family, shows convergence to the truthful strategy in the CA mechanism. 

We propose new differential privacy solutions for when external invariants and integer constraints are simultaneously enforced on the data product. These requirements arise in real world applications of private data curation, including the public release of the 2020 U.S. Decennial Census. They pose a great challenge to the production of provably private data products with adequate statistical usability. We propose integer subspace differential privacy to rigorously articulate the privacy guarantee when data products maintain both the invariants and integer characteristics, and demonstrate the composition and post-processing properties of our proposal. To address the challenge of sampling from a potentially highly restricted discrete space, we devise a pair of unbiased additive mechanisms, the generalized Laplace and the generalized Gaussian mechanisms, by solving the Diophantine equations as defined by the constraints. The proposed mechanisms have good accuracy, with errors exhibiting sub-exponential and sub-Gaussian tail probabilities respectively. To implement our proposal, we design an MCMC algorithm and supply empirical convergence assessment using estimated upper bounds on the total variation distance via L-lag coupling. We demonstrate the efficacy of our proposal with applications to a synthetic problem with intersecting invariants, a sensitive contingency table with known margins, and the 2010 Census county-level demonstration data with mandated fixed state population totals.