arXiv subjects

Sepideh Mahabadi

Publications and source records attributed to Sepideh Mahabadi.

At least 19 recordsLinked to original sources

arXiv2026

Semi-Streaming Reachability and Shortest Paths in $n^{1/3+o(1)}$ Passes

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).

cs.DS↗

arXiv2026

Pareto Inefficiency of EFX

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 $α> 2/3$.

cs.GT↗

arXiv2026

A Better Lower Bound for Sublinear Metric Steiner Tree

In the metric Steiner tree problem, we are given an $n$-point metric space $(V,w)$ and a set of terminals $T\subseteq V$, and we are interested in the smallest total weight $\mathsf{st}(V,T,w)$ of a tree that contains every terminal and may also use points of $V\setminus T$. We work in the distance-oracle model: the algorithm knows $V$ and $T$ in advance and learns $w$ only by querying a single entry $w(u,v)$ at a time. The goal is to estimate $\mathsf{st}(V,T,w)$ with the minimum number of queries. A factor of $2$ is achievable with nearly linear number of queries, so the interesting question is the query complexity of algorithms with an approximation ratio strictly below $2$. We prove that for every fixed $η\in(0,1)$, every randomized $(2-η)$-approximation algorithm must make more than $\widetildeΩ(n^{7/5})$ queries, improving upon the previously known bound of $\widetildeΩ(n^{6/5})$. Given that the best known upper bound shows that $\widetilde O(n^{5/3})$ queries suffice, the exponent of this problem now lies between $7/5$ and $5/3$, up to logarithmic factors.

cs.DS↗

arXiv2026

Random Fourier Features in Near-Linear Time

Random Fourier features approximate a shift-invariant kernel in $\mathbb{R}^d$, such as the Gaussian kernel $\exp(-\tfrac12\|x-y\|_2^2)$, using independent samples from its spectral measure to construct an unbiased estimator for the kernel value with subgaussian concentration. However, generating $m$ such features requires $O(md)$ time. A long line of work on structured random features has accelerated this computation for the Gaussian kernel and other radial kernels, albeit with weaker concentration guarantees. We close this gap by showing a random feature map that can be computed in near-linear time $O((d+m)\log d)$ and yields near-subgaussian concentration. We obtain it through a new and tighter analysis of the Fastfood feature map of Le, Sarlós and Smola (ICML 2013), using Brascamp-Lieb inequalities and the Schoenberg representation of radial kernels.

cs.LG↗

arXiv2026

Improved Sublinear Algorithms for Maximal Independent Set and Metric Steiner Forest

In this work we consider the Maximal Independent Set (MIS) problem and the metric Steiner Forest problem in the sublinear time setting, under the adjacency/distance matrix query model. First, we give an algorithm that estimates the size of an MIS up to a multiplicative factor of $(1+\eps)$ using $\tO(n^{4/3}/\eps^2)$ queries. This improves the best previous algorithm by Mahabadi, Roghani, Tarnawski, and Vakilian (SODA 2026), which had a query complexity of $\tO(n^{3/2}/\eps^2)$. Via a reduction from that work, this would automatically imply the same improvement for the problem of estimating the metric Steiner Forest cost up to an $O(\log n)$ factor. However, as our second contribution, we consider the Steiner Forest problem directly and provide an algorithm with $\tO(n)$ query complexity that is very simple and does not proceed via MIS.

cs.DS↗

arXiv2026

Local Search for Fair Max-Min Diversification

Given $n$ points in a metric space, Max-Min diversification asks for a subset of $k$ points maximizing the minimum pairwise distance between the selected points. This is arguably the most fundamental notion of diversity with applications across a wide range of domains. We consider this problem under partition constraints, previously studied as Fair Max-Min Diversification (FMMD). Here, each point has a color in $[m]$, and a feasible solution must contain exactly $k_i$ points of color $i$, where $k_1,\ldots,k_m$ are prescribed parameters satisfying $\sum_i k_i=k$. We give the first constant factor approximation for the problem using local search, that runs in time $f(m)\cdot \operatorname{poly}(n)$, in which all constraints are satisfied exactly. All previously known algorithms either provided an $\widetilde Θ(m)$ approximation factor, had running times exponential in the solution size $k$, or satisfied the fairness constraints only approximately or in expectation. We further generalize our result to the problem where each point may belong to an arbitrary subset of colors. Given lower and upper bounds $\ell_i$ and $u_i$ for every color $i$, the goal is to find $k$ points whose color counts satisfy all these bounds while maximizing their diversity.

cs.DS↗

arXiv2026

Hardness of Online Directed Steiner Network

In the Directed Steiner Network (DSN) problem we are given a directed graph and a set of demands $(s_i,t_i)$, and asked to find a cheap subgraph connecting each terminal pair. In its online version, the demands arrive online and must be served by buying edges irrevocably. DSN is a fundamental hard problem in network design, heavily studied in both the offline and the online setting. Offline, it has a superpolylogarithmic hardness of approximation. However, offline hardness says nothing about online algorithms, which are computationally unrestricted. It has been an open question whether uncertainty itself (needing to commit to a solution without knowing future demands) rules out polylogarithmic-competitive online algorithms. In this work, we show the first such unconditional, information-theoretic hardness. Namely, we give an $\exp\!\bigl(Ω(\sqrt{\log n})\bigr)$ bound on the competitive ratio, which holds even for randomized algorithms against an oblivious adversary, and on unit-cost DAGs. Our proof uses a novel connection between online network design and algebraic coding theory. We encode requests using a hidden low-degree polynomial, whose past evaluations reveal nothing about future ones. We then use list-recovery bounds to show that an algorithm cannot make cheaply reusable decisions without knowing those future evaluations.

cs.DS↗

arXiv2026

The Expiration Streaming Model: Diameter, $k$-Center, Counting, Sampling, and Friends

An important thread in the study of data-stream algorithms focuses on settings where stream items are active only for a limited time. We introduce a new expiration model, where each item arrives with its own arbitrary expiration time. The special case where items expire in the order that they arrive, which we call consistent expirations, contains the classical sliding-window model of Datar, Gionis, Indyk, and Motwani [SICOMP 2002] and its timestamp-based variant of Braverman and Ostrovsky [FOCS 2007]. Our first set of results explores the expiration streaming model and presents algorithms for several fundamental problems, including approximate counting, uniform sampling, and weighted sampling by efficiently tracking active items without explicitly storing them all. Naturally, these algorithms have many immediate applications, e.g., to range counting. Our second and main set of results for the expiration model designs algorithms for the diameter and k-center problems, where items are points in a metric space. Our results significantly extend those known for the special case of sliding-window streams by Cohen-Addad, Schwiegelshohn, and Sohler [ICALP 2016], and obtain a strictly better approximation factor for the diameter in the important special case of high-dimensional Euclidean metrics. We develop new decomposition and coordination techniques along with a geometric dominance framework to filter out redundant points based on both temporal and spatial proximity.

cs.DS↗

arXiv2026

Online Steiner Forest with Recourse

In the online Steiner forest problem we are given a graph $G$, and a sequence of terminal pairs $(u_i,v_i)$ which arrive in an online fashion. We are asked to maintain a low-cost subgraph in which each $u_i$ is connected to $v_i$ for all the pairs that have arrived so far. If we are not allowed to delete edges from our solution, then the best possible competitive ratio is $Θ(\log n)$. In this work, we initiate the study of low-recourse algorithms for online Steiner forest. We give an algorithm that maintains a constant-competitive solution and has an amortized recourse of $O(\log n)$, i.e., inserts and deletes $O(\log n)$ edges per demand on average.

cs.DS↗

arXiv2026

Improved Algorithms for Fair Matroid Submodular Maximization

Submodular maximization subject to matroid constraints is a central problem with many applications in machine learning. As algorithms are increasingly used in decision-making over datapoints with sensitive attributes such as gender or race, it is becoming crucial to enforce fairness to avoid bias and discrimination. Recent work has addressed the challenge of developing efficient approximation algorithms for fair matroid submodular maximization. However, the best algorithms known so far are only guaranteed to satisfy a relaxed version of the fairness constraints that loses a factor 2, i.e., the problem may ask for $\ell$ elements with a given attribute, but the algorithm is only guaranteed to find $\lfloor \ell/2 \rfloor$. In particular, there is no provable guarantee when $\ell=1$, which corresponds to a key special case of perfect matching constraints. In this work, we achieve a new trade-off via an algorithm that gets arbitrarily close to full fairness. Namely, for any constant $\varepsilon>0$, we give a constant-factor approximation to fair monotone matroid submodular maximization that in expectation loses only a factor $(1-\varepsilon)$ in the lower-bound fairness constraint. Our empirical evaluation on a standard suite of real-world datasets -- including clustering, recommendation, and coverage tasks -- demonstrates the practical effectiveness of our methods.

cs.DS↗

arXiv2025

Sublinear Metric Steiner Forest via Maximal Independent Set

In this work we consider the Metric Steiner Forest problem in the sublinear time model. Given a set $V$ of $n$ points in a metric space where distances are provided by means of query access to an $n\times n$ distance matrix, along with a set of $k$ terminal pairs $(s_1,t_1), \dots, (s_k,t_k)\in V\times V$, the goal is to find a minimum-weight subset of edges that connects each terminal pair. Although sublinear time algorithms have been studied for estimating the weight of a minimum spanning tree in both general and metric settings, as well as for the metric Steiner Tree problem, no sublinear time algorithm was known for the metric Steiner Forest problem. Here, we give an $O(\log k)$-approximation algorithm for the problem that runs in time $\widetilde{O}(n^{3/2})$. Along the way, we provide the first sublinear-time algorithm for estimating the size of a Maximal Independent Set (MIS). Our algorithm runs in time $\widetilde{O}(n^{3/2}/\varepsilon^2)$ under the adjacency matrix oracle model and obtains a purely multiplicative $(1+\varepsilon)$-approximation. Previously, sublinear-time algorithms for MIS were only known for bounded-degree graphs.

cs.DS↗

arXiv2025

Composable Coresets for Constrained Determinant Maximization and Beyond

We study algorithms for construction of composable coresets for the task of Determinant Maximization under partition constraint. Given a point set $V\subset \mathbb{R}^d$ that is partitioned into $s$ groups $V_1,\cdots, V_s$, and integers $k_1,...,k_s$, where $k=\sum_i k_i$, the goal is to pick $k_i$ points from group $V_i$ such that the overall determinant of the picked $k$ points is maximized. Determinant Maximization and its constrained variants have gained a lot of interest for modeling diversity, and have found applications in the context of data summarization. When the cardinality $k$ of the selected set is greater than the dimension $d$, we show a peeling algorithm that gives us a composable coreset of size $kd$ with a provably optimal approximation factor of $d^{O(d)}.$ When $k\leq d$, we show a simple coreset construction with optimal size and approximation factor. As a further application of our technique, we get a composable coreset for determinant maximization under the more general laminar matroid constraints, and a composable coreset for unconstrained determinant maximization in a previously unresolved regime. Our results generalize to all strongly Rayleigh distributions and to several other experimental design problems. As an application, we improve the runtime of the practical local-search based algorithm of [Anari-Vuong--COLT'22] for determinantal maximization under partition constraint from $O(n^{2^s}k^{2^s})$ to $O(n k^{2^s})$, making it only linear on the number of points $n$.

cs.DS↗

arXiv2025

A 0.51-Approximation of Maximum Matching in Sublinear $n^{1.5}$ Time

We study the problem of estimating the size of a maximum matching in sublinear time. The problem has been studied extensively in the literature and various algorithms and lower bounds are known for it. Our result is a $0.5109$-approximation algorithm with a running time of $\tilde{O}(n\sqrt{n})$. All previous algorithms either provide only a marginal improvement (e.g., $2^{-280}$) over the $0.5$-approximation that arises from estimating a \emph{maximal} matching, or have a running time that is nearly $n^2$. Our approach is also arguably much simpler than other algorithms beating $0.5$-approximation.

cs.DS↗

arXiv2025

Guessing Efficiently for Constrained Subspace Approximation

In this paper we study constrained subspace approximation problem. Given a set of $n$ points $\{a_1,\ldots,a_n\}$ in $\mathbb{R}^d$, the goal of the {\em subspace approximation} problem is to find a $k$ dimensional subspace that best approximates the input points. More precisely, for a given $p\geq 1$, we aim to minimize the $p$th power of the $\ell_p$ norm of the error vector $(\|a_1-\bm{P}a_1\|,\ldots,\|a_n-\bm{P}a_n\|)$, where $\bm{P}$ denotes the projection matrix onto the subspace and the norms are Euclidean. In \emph{constrained} subspace approximation (CSA), we additionally have constraints on the projection matrix $\bm{P}$. In its most general form, we require $\bm{P}$ to belong to a given subset $\mathcal{S}$ that is described explicitly or implicitly. We introduce a general framework for constrained subspace approximation. Our approach, that we term coreset-guess-solve, yields either $(1+\varepsilon)$-multiplicative or $\varepsilon$-additive approximations for a variety of constraints. We show that it provides new algorithms for partition-constrained subspace approximation with applications to {\it fair} subspace approximation, $k$-means clustering, and projected non-negative matrix factorization, among others. Specifically, while we reconstruct the best known bounds for $k$-means clustering in Euclidean spaces, we improve the known results for the remainder of the problems.

cs.DS↗

arXiv2025

Streaming Algorithms for Network Design

We consider the Survivable Network Design problem (SNDP) in the single-pass insertion-only streaming model. The input to SNDP is an edge-weighted graph $G = (V, E)$ and an integer connectivity requirement $r(uv)$ for each $u, v \in V$. The objective is to find a min-weight subgraph $H \subseteq G$ s.t., for every pair of $u, v \in V$, $u$ and $v$ are $r(uv)$-edge/vertex-connected. Recent work by Jin et al. [JKMV24] obtained approximation algorithms for edge-connectivity augmentation, and via that, also derived algorithms for edge-connectivity SNDP (EC-SNDP). We consider vertex-connectivity setting (VC-SNDP) and obtain several results for it as well as improved results for EC-SNDP. * We provide a general framework for solving connectivity problems in streaming; this is based on a connection to fault-tolerant spanners. For VC-SNDP, we provide an $O(tk)$-approximation in $\tilde O(k^{1-1/t}n^{1 + 1/t})$ space, where $k$ is the maximum connectivity requirement, assuming an exact algorithm at the end of the stream. Using a refined LP-based analysis, we provide an $O(βt)$-approximation where $β$ is the integrality gap of the natural cut-based LP relaxation. When applied to the EC-SNDP, our framework provides an $O(t)$-approximation in $\tilde O(k^{1/2-1/(2t)}n^{1 + 1/t} + kn)$ space, improving the $O(t \log k)$-approximation of [JKMV24] using $\tilde O(kn^{1+1/t})$ space; this also extends to element-connectivity SNDP. * We consider vertex connectivity-augmentation in the link-arrival model. The input is a $k$-vertex-connected subgraph $G$, and the weighted links $L$ arrive in the stream; the goal is to store the min-weight set of links s.t. $G \cup L$ is $(k+1)$-vertex-connected. We obtain $O(1)$ approximations in near-linear space for $k = 1, 2$. Our result for $k=2$ is based on SPQR tree, a novel application for this well-known representation of $2$-connected graphs.

cs.DS↗

arXiv2025

Graph-Based Algorithms for Diverse Similarity Search

Nearest neighbor search is a fundamental data structure problem with many applications in machine learning, computer vision, recommendation systems and other fields. Although the main objective of the data structure is to quickly report data points that are closest to a given query, it has long been noted (Carbonell and Goldstein, 1998) that without additional constraints the reported answers can be redundant and/or duplicative. This issue is typically addressed in two stages: in the first stage, the algorithm retrieves a (large) number $r$ of points closest to the query, while in the second stage, the $r$ points are post-processed and a small subset is selected to maximize the desired diversity objective. Although popular, this method suffers from a fundamental efficiency bottleneck, as the set of points retrieved in the first stage often needs to be much larger than the final output. In this paper we present provably efficient algorithms for approximate nearest neighbor search with diversity constraints that bypass this two stage process. Our algorithms are based on popular graph-based methods, which allows us to "piggy-back" on the existing efficient implementations. These are the first graph-based algorithms for nearest neighbor search with diversity constraints. For data sets with low intrinsic dimension, our data structures report a diverse set of $k$ points approximately closest to the query, in time that only depends on $k$ and $\log Δ$, where $Δ$ is the ratio of the diameter to the closest pair distance in the data set. This bound is qualitatively similar to the best known bounds for standard (non-diverse) graph-based algorithms. Our experiments show that the search time of our algorithms is substantially lower than that using the standard two-stage approach.

cs.DS↗

arXiv2024

Sublinear Metric Steiner Tree via Improved Bounds for Set Cover

We study the metric Steiner tree problem in the sublinear query model. In this problem, for a set of $n$ points $V$ in a metric space given to us by means of query access to an $n\times n$ matrix $w$, and a set of terminals $T\subseteq V$, the goal is to find the minimum-weight subset of the edges that connects all the terminal vertices. Recently, Chen, Khanna and Tan [SODA'23] gave an algorithm that uses $\widetilde{O}(n^{13/7})$ queries and outputs a $(2-η)$-estimate of the metric Steiner tree weight, where $η>0$ is a universal constant. A key component in their algorithm is a sublinear algorithm for a particular set cover problem where, given a set system $(U, F)$, the goal is to provide a multiplicative-additive estimate for $|U|-\textsf{SC}(U, F)$. Here $U$ is the set of elements, $F$ is the collection of sets, and $\textsf{SC}(U, F)$ denotes the optimal set cover size of $(U, F)$. In particular, their algorithm returns a $(1/4, \varepsilon\cdot|U|)$-multiplicative-additive estimate for this set cover problem using $\widetilde{O}(|F|^{7/4})$ membership oracle queries (querying whether a set $S$ contains an $e$), where $\varepsilon$ is a fixed constant. In this work, we improve the query complexity of $(2-η)$-estimating the metric Steiner tree weight to $\widetilde{O}(n^{5/3})$ by showing a $(1/2, \varepsilon \cdot |U|)$-estimate for the above set cover problem using $\widetilde{O}(|F|^{5/3})$ membership queries. To design our set cover algorithm, we estimate the size of a random greedy maximal matching for an auxiliary multigraph that the algorithm constructs implicitly, without access to its adjacency list or matrix.

cs.DS↗

arXiv2024

Efficiently Computing Similarities to Private Datasets

Many methods in differentially private model training rely on computing the similarity between a query point (such as public or synthetic data) and private data. We abstract out this common subroutine and study the following fundamental algorithmic problem: Given a similarity function $f$ and a large high-dimensional private dataset $X \subset \mathbb{R}^d$, output a differentially private (DP) data structure which approximates $\sum_{x \in X} f(x,y)$ for any query $y$. We consider the cases where $f$ is a kernel function, such as $f(x,y) = e^{-\|x-y\|_2^2/σ^2}$ (also known as DP kernel density estimation), or a distance function such as $f(x,y) = \|x-y\|_2$, among others. Our theoretical results improve upon prior work and give better privacy-utility trade-offs as well as faster query times for a wide range of kernels and distance functions. The unifying approach behind our results is leveraging `low-dimensional structures' present in the specific functions $f$ that we study, using tools such as provable dimensionality reduction, approximation theory, and one-dimensional decomposition of the functions. Our algorithms empirically exhibit improved query times and accuracy over prior state of the art. We also present an application to DP classification. Our experiments demonstrate that the simple methodology of classifying based on average similarity is orders of magnitude faster than prior DP-SGD based approaches for comparable accuracy.

cs.CR↗