We prove the following asymptotically tight lower bound for $k$-color discrepancy: For any $k \geq 2$, there exists a hypergraph with $n$ hyperedges such that its $k$-color discrepancy is at least $Ω(\sqrt{n})$. This improves on the previously known lower bound of $Ω(\sqrt{n/\log k})$ due to Caragiannis et al. (arXiv:2502.10516). As an application, we show that our result implies improved lower bounds for group fair division.
In the Densest $k$-Subgraph problem, given an undirected graph $G$ and an integer $k$, the goal is to find a subgraph of $G$ on $k$ vertices that contains maximum number of edges. Even though the state-of-the-art algorithm for the problem achieves only $O(n^{1/4 + \varepsilon})$ approximation ratio (Bhaskara et al., 2010), previous attempts at proving hardness of approximation, including those under average case assumptions, fail to achieve a polynomial ratio; the best ratios ruled out under any worst case assumption and any average case assumption are only any constant (Raghavendra and Steurer, 2010) and $2^{Ω(\log^{2/3} n)}$ (Alon et al., 2011) respectively. In this work, we propose the Almost-Exponential Time Hypothesis (Almost-ETH), which postulates that no algorithm can solve 3SAT on $n$ variables in $O(2^{n^{1 - γ}})$ time for any constant $γ> 0$. Assuming Almost-ETH, we show that, for any constant $\varepsilon > 0$, there is no polynomial-time algorithm that approximates {\sc D$k$S} to within $2^{(\log n)^{1 - \varepsilon}}$ factor of the optimum. In addition, our result has perfect completeness, meaning that, assuming Almost-ETH, no polynomial-time algorithm can even distinguish between the case in which $G$ contains a $k$-clique and the case in which every induced $k$-subgraph of $G$ has density at most $2^{-(\log n)^{1 - \varepsilon}}$.
Benade et al. (EC 2024) introduced a semi-random model for non-additive fair division in which each agent starts with an arbitrary base valuation over bundles of items and independently permutes the item names uniformly at random. While they proved that envy-free allocations exist with high probability as the number of items grows for $n = 2$ agents, they left open whether the same holds for $n > 2$ agents. We resolve this open question in the affirmative: for any fixed number of agents $n$ and any number of items $m$ divisible by $n$, an envy-free allocation exists with probability $1 - o(1)$ as $m \to \infty$. Our proof combines the second moment method with a connection to the (small-set) expansion of generalized Johnson graphs, which may be of independent interest.
Daogao Liu, Edith Cohen, Badih Ghazi, Peter Kairouz, Pritish Kamath, Alexander Knop, Ravi Kumar, Pasin Manurangsi 等 11 位作者
We introduce $Urania$, a novel framework for generating insights about LLM chatbot interactions with rigorous differential privacy (DP) guarantees. The framework employs a private clustering mechanism and innovative keyword extraction methods, including frequency-based, TF-IDF-based, and LLM-guided approaches. By leveraging DP tools such as clustering, partition selection, and histogram-based summarization, $Urania$ provides end-to-end privacy protection. Our evaluation assesses lexical and semantic content preservation, pair similarity, and LLM-based metrics, benchmarking against a non-private Clio-inspired pipeline (Tamkin et al., 2024). Moreover, we develop a simple empirical privacy evaluation that demonstrates the enhanced robustness of our DP pipeline. The results show the framework's ability to extract meaningful conversational insights while maintaining stringent user privacy, effectively balancing data utility with privacy preservation.