Abstract. Given an [Formula: see text]-vertex undirected graph [Formula: see text], and a parameter [Formula: see text], a path-reporting distance oracle (PRDO) is a data structure of size [Formula: see text] that given a query [Formula: see text] returns an [Formula: see text]-approximate shortest [Formula: see text]-[Formula: see text] path [Formula: see text] in [Formula: see text] within time [Formula: see text]. Here [Formula: see text], [Formula: see text], and [Formula: see text] are arbitrary (hopefully slowly growing) functions. A distance oracle that only returns an approximate estimate [Formula: see text] of the distance [Formula: see text] between the queried vertices is called a non-path-reporting distance oracle. A landmark PRDO [M. Thorup and U. Zwick, Proceedings of the 33 rd ACM Symposium on Theory of Computing, 2001, pp. 183–192; C. Wulff-Nilsen, Proceedings of the 24 th Annual ACM-SIAM Symposium on Discrete Algorithms, 2013, pp. 539–549] has [Formula: see text], [Formula: see text], and [Formula: see text]. The size of this oracle is [Formula: see text] for all [Formula: see text]. PRDOs with [Formula: see text], [Formula: see text], and [Formula: see text], for [Formula: see text], were devised in [M. Elkin and S. Pettie, ACM Trans. Algorithms, 12 (2016), 50; O. Neiman and I. Shabat, 19 th Scandinavian Symposium and Workshops on Algorithm Theory, Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, Germany, 2024, 36]. These oracles can be much sparser than [Formula: see text], but their stretch is polynomially larger than the optimal bound of [Formula: see text]. On the other hand, a long line of non-path-reporting distance oracles culminated in a celebrated result by Chechik [ Proceedings of the 47 th Annual ACM on Symposium on Theory of Computing, ACM, New York, 2015, pp. 1–10], in which [Formula: see text], [Formula: see text], and [Formula: see text]. In this paper we make dramatic progress in bridging the gap between path-reporting and non-path-reporting distance oracles. In particular, we devise a PRDO with size [Formula: see text], stretch [Formula: see text], and query time [Formula: see text]. As [Formula: see text] for [Formula: see text], its size is always at most [Formula: see text], and its query time is [Formula: see text]. Moreover, for [Formula: see text], we have [Formula: see text], i.e., [Formula: see text], [Formula: see text], and [Formula: see text]. For [Formula: see text], our oracle has size [Formula: see text], stretch [Formula: see text], and query time [Formula: see text]. These trade-offs exhibit polynomial improvement in stretch over the PRDOs of Elkin and Pettie and of Neiman and Shabat. For [Formula: see text], our trade-offs also strictly improve the long-standing bounds of Thorup and Zwick and of Wulff-Nilsen. Our results on PRDOs are based on novel constructions of approximate distance preservers, which we devise in this paper. Specifically, we show that for any [Formula: see text], any [Formula: see text], and any graph [Formula: see text] and a collection [Formula: see text] of [Formula: see text] vertex pairs, there exists a [Formula: see text]-approximate preserver for [Formula: see text] with [Formula: see text] edges, where [Formula: see text]. These new preservers are significantly sparser than the previous state-of-the-art approximate preservers due to Kogan and Parter [63 rd Annual Symposium on Foundations of Computer Science, IEEE, 2022, pp. 766–777].
Abstract. We study the power of query access for the fundamental task of agnostic learning under the Gaussian distribution. In the agnostic model, no assumptions are made on the labels of the examples, and the goal is to compute a hypothesis that is competitive with the best-fit function in a known class; i.e., it achieves error [Formula: see text], where [Formula: see text] is the error of the best function in the class. We focus on a general family of multi-Index models (MIMs), which are [Formula: see text]-variate functions that depend only on a few relevant directions, i.e., have the form [Formula: see text] for an unknown link function [Formula: see text] and a [Formula: see text] matrix [Formula: see text]. MIMs cover a wide range of commonly studied function classes, including real-valued function classes, such as constant-depth neural networks with ReLU activations, and Boolean concept classes, such as intersections of halfspaces. Our main result shows that query access gives significant runtime improvements over random examples for agnostically learning both real-valued and Boolean-valued MIMs. Under standard regularity assumptions for the link function (namely, bounded variation or surface area), we give an agnostic query learner for MIMs with running time [Formula: see text]. In contrast, algorithms that rely only on random labeled examples inherently require [Formula: see text] samples and runtime, even for the basic problem of agnostically learning a single ReLU or a halfspace. As special cases of our general approach, we obtain the following results: [Formula: see text] For the class of depth-[Formula: see text], width-[Formula: see text] ReLU networks on [Formula: see text], our agnostic query learner runs in time [Formula: see text]. This bound qualitatively matches the runtime of an algorithm by Chen, Klivans, and Meka [ Learning deep ReLU networks is fixed-parameter tractable, 2022] for the realizable PAC setting with random examples. [Formula: see text] For the class of arbitrary intersections of [Formula: see text] halfspaces on [Formula: see text], our agnostic query learner runs in time [Formula: see text]. Prior to our work, no improvement over the agnostic PAC model complexity (without queries) was known, even for the case of a single halfspace. In both these settings, we provide evidence that the [Formula: see text] runtime dependence is required for proper query learners, even for agnostically learning a single ReLU or halfspace. Our algorithmic result establishes a strong computational separation between the agnostic PAC and the agnostic PAC + Query models under the Gaussian distribution for a range of natural function classes. Prior to our work, no such separation was known for any natural concept class, even for the case of a single halfspace, for which it was an open problem posed by Feldman [ On the power of membership queries in agnostic learning, 2008]. Our results are enabled by a general dimension-reduction technique that leverages query access to estimate gradients of (a smoothed version of) the underlying label function.
Let G = (V, E, \ell ) be an n-node m-edge weighted undirected graph, where \ell : E-(0, oo) is a real length function defined on its edges, and let g denote the girth of G, i.e., the length of a shortest cycle. We present an algorithm that, for any input, integer k \geq 1, in O(kn1+1/k log n + m(k + log n)) expected time finds a cycle of length at most 4k3 g. This algorithm nearly matches an O(n1+1/k log n)-time algorithm of Kadria et al. [Algorithmic trade-offs for girth approximation in undirected graphs, in Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2022, pp. 1471--1492] which applied to unweighted graphs of girth 3. For weighted graphs, this result also improves upon the previous state-of-the-art algorithm that in O((n1+1/k log n + m) log(nM)) time, where \ell : E-[1, M] is an integral length function, finds a cycle of length at most 2kg of Kadria et al. [Algorithmic trade-offs for girth approximation in undirected graphs, in Proceedings of the 2022 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), SIAM, 2022, pp. 1471--1492]. For k = 1, this result improves upon the result of Roditty and Tov [ACM Trans. Algorithms, 9 (2013), pp. 15:1-15:13].
We provide new deterministic algorithms for the edge coloring problem, which is one of the classic and highly studied distributed local symmetry breaking problems. As our main result, we show that a (2\Delta - 1)-edge coloring can be computed in time poly log \Delta + O(log* n) in the uted Computing (PODC), 2020, pp. 289--298], who gave an algorithm with a quasi-polylogarithmic dependency on \Delta . We further show that in the CONGEST model, an (8 + \varepsilon )\Delta -edge coloring can be computed in poly log \Delta + O(log* n) rounds. The best previous O(\Delta )-edge coloring algorithm that can be implemented in the CONGEST model is by Barenboim and Elkin [J. ACM, 58 (2011), 23] and it computes a 2O(1/\varepsilon )\Delta -edge coloring in time O(\Delta \varepsilon + log* n) for any \varepsilon \in (0, 1].
