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The field of learning-augmented algorithms seeks to use ML techniques on past instances of a problem to inform an algorithm designed for a future instance. In this paper, we introduce a novel model for learning-augmented algorithms inspired by online learning. In this model, we are given a sequence of instances of a problem and the goal of the learning-augmented algorithm is to use prior instances to propose a solution to a future instance of the problem. The performance of the algorithm is measured by its average performance across all the instances, where the performance on a single instance is the ratio between the cost of the algorithm’s solution and that of an optimal solution for that instance. We apply this framework to the classic k-median clustering problem, and give an efficient learning algorithm that can approximately match the average performance of the best fixed k-median solution in hindsight across all the instances. We also experimentally evaluate our algorithm and show that its empirical performance is close to optimal, and also that it automatically adapts the solution to a dynamically changing sequence. 

The goal in the stochastic vertex cover problem is to obtain an approximately minimum vertex cover for a graph G⋆ that is realized by sampling each edge independently with some probability p∈ (0, 1] in a base graph G = (V, E). The algorithm is given the base graph G and the probability p as inputs, but its only access to the realized graph G⋆ is through queries on individual edges in G that reveal the existence (or not) of the queried edge in G⋆. In this paper, we resolve the central open question for this problem: to find a (1+ε)-approximate vertex cover using only Oε(n/p) edge queries. Prior to our work, there were two incomparable state-of-the-art results for this problem: a (3/2+ε)-approximation using Oε(n/p) queries (Derakhshan, Durvasula, and Haghtalab, 2023) and a (1+ε)-approximation using Oε((n/p)· RS(n)) queries (Derakhshan, Saneian, and Xun, 2025), where RS(n) is known to be at least 2Ω(logn/loglogn) and could be as large as n/2Θ(log* n). Our improved upper bound of Oε(n/p) matches the known lower bound of Ω(n/p) for any constant-factor approximation algorithm for this problem (Behnezhad, Blum, and Derakhshan, 2022). A key tool in our result is a new concentration bound for the size of minimum vertex cover on random graphs, which might be of independent interest. 

The vertex connectivity of an m-edge n-vertex undirected graph is the smallest number of vertices whose removal disconnects the graph or leaves only a singleton vertex. In 1974, Aho Hopcroft and Ullman asked if vertex connectivity can be computed in linear time. Despite the substantial effort in the past five decades, the best-known running time is O-tilde(mn) by Henzinger-Rao-Gabow (FOCS 1996). Indeed, no algorithm with an o(mn) running time is known even if we assume a linear-time max-flow algorithm. In this article, we give an affirmative answer to this long-standing open problem (up to a sub-polynomial factor). We present a randomized reduction from the vertex connectivity problem to the max-flow problem which incurs only a poly-logarithmic overhead in runtime. Using this reduction, we can solve vertex connectivity in almost linear time by using the celebrated almost-linear-time max-flow algorithms by Chen-Kyng-Liu-Peng-Probst Gutenberg-Sachdeva (FOCS 2022) and Brand-Chen-Kyng-Liu-Peng-Probst Gutenberg-Sachdeva-Sidford (FOCS 2023). Using our new techniques, we also obtain an algorithm for directed vertex connectivity with a running time of n^(2+o(1)) time which improves the best-known bound of O-tilde(mn) by Henzinger-Rao-Gabow (FOCS 1996). 

Advances in Neural Information Processing Systems 37