Marek Eliáš, Gautam Kamath, Janardhan Kulkarni, George Z. Li, Jingcheng Liu, Sepideh Mahabadi, Pan Peng, Jakub Tarnawski 等 10 位作者
We give a polynomial-time $(\varepsilon, δ)$-edge-differentially private algorithm that releases a synthetic graph approximately preserving all cut values of a nonnegative weighted input graph $G$. Our algorithm outputs a nonnegative weighted graph $H$ such that, with probability at least $1-β$, simultaneously for all $S\subseteq V$, \[ \left|w_G(S,V\setminus S)-w_H(S,V\setminus S)\right| \leq γw_G(S,V\setminus S) + O\left(\frac{n}{γ\varepsilon} \log^7 \frac{n}{δβγ\varepsilon}\right). \] For fixed $γ, \varepsilon$ and $δ= β= n^{-10}$, the additive error is $n \operatorname{polylog} n$, improving upon the previous efficient bound of $n^{13/12 + o(1)}$. Furthermore, we establish an $Ω(n/γ\varepsilon)$ lower bound for the relevant parameter regime, showing that our result is tight up to polylogarithmic factors. Our algorithm uses a log-det-regularized SDP to generate fractional queries for multiplicative-weights updates and certify accuracy for all cuts. The privacy analysis charges privacy loss only fractionally per mechanism call, based on the amount of mass on edge coordinates retired from subsequent iterations. Our synthetic graph release yields improved error bounds for several downstream tasks in private graph analysis including correlation clustering, Max-Cut, Max-$k$-Cut, constrained cut problems and hierarchical clustering. Most notably, for unweighted complete-graph correlation clustering, we achieve the optimal $\widetilde O(n)$ additive error while reducing the best previously known approximation ratio for this additive error from several thousand to $1.387$. We also provide a formalization of our results in Lean.
Gary Hoppenworth, George Z. Li, Sepideh Mahabadi, Jakub Tarnawski
In the semi-streaming model, the edges of an $n$-vertex graph arrive in a stream, and an algorithm makes multiple passes over that stream while using only $\widetilde{O}(n)$ space. The number of passes needed to compute single-source reachability in directed graphs is a long-standing open question in streaming algorithms, going back to Henzinger, Raghavan, and Rajagopalan (1998). The best known semi-streaming algorithm uses $n^{1/2+o(1)}$ passes due to Jambulapati, Liu, and Sidford (2019), while the best lower bound is $Ω(\log n/\log\log n)$ passes due to Guruswami and Onak (2016). We give streaming algorithms for computing single-source reachability and shortest paths on $n$-vertex directed graphs using $\widetilde{O}(n)$ space and $n^{1/3+o(1)}$ passes. Our algorithms are based on implicit hopsets, which are large hopsets that can be represented implicitly in $\widetilde{O}(n)$ space. We introduce this notion and give new constructions for implicit hopsets with small hopbound. Along the way, we obtain new work-depth tradeoffs for parallel shortest paths in directed graphs, improving the recent result of Ashvinkumar, Bernstein, Probst Gutenberg, and Saranurak (2026).
Sepideh Mahabadi, Prasanna Ramakrishnan, Jakub Tarnawski
We consider the problem of allocating indivisible goods to agents with additive valuations, under two classic notions of fairness and efficiency: envy-freeness up to any good (EFX) and Pareto optimality. While some of the earliest work on EFX showed that Pareto optimal EFX allocations may not exist if agents can have zero value for some goods, it has remained open whether the same is true when all values are positive. We present a simple instance in which every good has positive value to every agent, but no Pareto optimal EFX allocation exists. We then generalize to a family of instances showing that Pareto optimal $α$-EFX allocations may not exist for any $α> 2/3$.