arXiv subjects

Ermiya Farokhnejad

Publications and source records attributed to Ermiya Farokhnejad.

9 recordsLinked to original sources

arXiv2026

Connectivity Oracles Under Vertex Failures via a Simple and Fast Low-Degree Steiner Forest Decomposition

We study the low-degree Steiner forest decomposition. Given a graph $G=(V,E)$ and a terminal set $U\subseteq V$, the standard decomposition returns a set $X\subseteq V$ of size at most $|U|/2$ and a forest $T\subseteq G-X$ of maximum degree $Δ$ such that, for every connected component $C$ of $G-X$, some connected component of $T$ contains all terminals in $U\cap V(C)$. This is the central decomposition behind several connectivity oracles under vertex failures [DP20, LS22, LW24]. The state-of-the-art algorithms either take $O(mn\log n)$ time with degree bound $4$ [DP20], or take $m^{1+o(1)}$ time with the weaker degree bound $O(\log^{2}n)$ [LW24]. We show that if $T$ is allowed to contain vertices of $X$, then a degree-$4$ decomposition can be computed by a very simple algorithm in $O(mα(n))$ time. Further, we show that this relaxed decomposition is equally useful for constructing connectivity oracles under vertex failures. As a consequence, we obtain a deterministic connectivity oracle under $d$ vertex failures with $\tilde{O}(m)$ space, $\tilde{O}(md_\star)$ preprocessing time ($d_\star$ is an upper bound on the number of failed vertices), $\tilde{O}(d^{2})$ update time, and $O(d)$ query time. Up to polylogarithmic factors, this oracle strictly improves all known oracles; in particular, it removes the $n^{o(1)}$ factors from the preprocessing and update times of [LS22, LW24].

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arXiv2026

Minimum Degree Spanning Tree: $(1+ε,1)$-Approximation in Near-Linear Time

The minimum degree spanning tree problem is a classic NP-hard problem whose optimal approximation guarantee was established since the early 1990s: Fürer and Raghavachari [FR92] gave an $\tilde O(mn)$-time algorithm that computes a spanning tree with maximum degree $Δ^\star+1$, where $Δ^\star$ denotes the optimum value. Whether similarly strong guarantees can be achieved in near-linear time has remained open for over three decades. We give the first near-linear-time algorithm that computes a spanning tree with maximum degree $\lceil (1+ε)Δ^\star\rceil+1$ in $\tilde O(m/ε^2)$ time. Prior near-linear-time algorithms either achieved the weaker bound $\lceil (1+ε)Δ^\star\rceil + O(\log n/ε^2)$ [DHZ20] or required dense graphs with $m\ge n^{7/4}$ [CQT21,BFW26]. Using the same framework, our algorithm can also compute a spanning tree with maximum degree $Δ^\star+1$ in $\tilde O(mn^{2/3})$ time, improving upon the recent $\tilde O(mn^{3/4})$-time algorithm of [BFW26]. These two results strictly improve all previous construction algorithms for the minimum degree spanning tree problem.

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arXiv2026

Distributed Dominating Set With Optimal Rounds and Message Size in Bounded Arboricity Graphs

We study the distributed minimum dominating set problem on graphs of arboricity $α$. Dory, Ghaffari, and Ilchi [PODC'22] showed that any algorithm achieving a constant or poly-logarithmic approximation factor needs at least $Ω(\logΔ/\log\logΔ)$ rounds in graphs of maximum degree $Δ$ and arboricity $α$, even when $α=2$ and even when the message sizes are unbounded. Although there is a variety of algorithms with a near-optimal round complexity of $O(\logΔ)$, it is natural to ask: What is the best approximation factor in the optimal round complexity of $O(\logΔ/\log\logΔ)$? We make progress in answering this question by describing a deterministic algorithm that obtains a $O\left( α\log Δ/ \log\log Δ\right)$ approximation without prior knowledge of $α$ with optimal round complexity of $O\left( \log Δ/ \log\log Δ\right)$ and optimal message size of $1$ bit per round. Among all of the previous results, the only algorithm that achieves the optimal round complexity of $O\left( \log Δ/ \log\log Δ\right)$ without prior knowledge of $α$ is due to Lenzen and Wattenhofer [DISC'10] that obtains a $O(α\log^{1+\varepsilon}Δ/ (\varepsilon\log\log Δ))$ approximation in $O(\logΔ/(\varepsilon\log\logΔ))$ rounds and $O(\log(\varepsilon^{-1}\logΔ))$ message size. Our algorithm simplifies and improves upon this result. The only downside of our algorithm compared to the algorithm of Lenzen and Wattenhofer is that it needs prior knowledge of $Δ$. The previous state-of-the-art algorithm by Dory, Ghaffari, and Ilchi [PODC'22] has a dependency on $\log n$ in the round complexity for unknown $α$, which is far from optimal.

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arXiv2026

Fully Dynamic Euclidean k-Means

We consider the Euclidean $k$-means clustering problem in a dynamic setting, where we have to explicitly maintain a solution (a set of $k$ centers) $S \subseteq \mathbb{R}^d$ subject to point insertions/deletions in $\mathbb{R}^d$. We present a dynamic algorithm for Euclidean $k$-means with $\mathrm{poly}(1/ε)$-approximation ratio, $\tilde{O}(k^ε)$ update time, and $\tilde{O}(1)$ recourse, for any $ε\in (0,1)$, even when $d$ and $k$ are both part of the input. This is the first algorithm to achieve a constant ratio with $o(k)$ update time for this problem, whereas the previous $O(1)$-approximation runs in $\tilde O(k)$ update time [Bhattacharya, Costa, Farokhnejad; STOC'25]. In fact, previous algorithms cannot go beyond $O(k)$ update time precisely because they are designed for general metrics where an $Ω(k)$ lower bound is known. We break this $O(k)$ barrier by devising new fundamental data structures to utilize Euclidean properties: a structure that (implicitly) maintains a clustering subject to both center and data point updates, and a range query structure that can evaluate a mergeable function over any metric ball range given as a query. To obtain these structures, we devise the first consistent hashing scheme [Czumaj, Jiang, Krauthgamer, Vesel{ý}, Yang; FOCS'22] that achieves $\tilde O(n^ε)$ running time per point evaluation with competitive parameters. Our final algorithm exploits the framework of [Bhattacharya, Costa, Farokhnejad; STOC'25] for general metrics. The key change is to redesign several critical subroutines so that they reduce to our new Euclidean data structures, replacing the general-metric implementations that are unlikely to run efficiently even when Euclidean properties are provided.

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arXiv2026

Deterministic $k$-Median Clustering in Near-Optimal Time

The metric $k$-median problem is a textbook clustering problem. As input, we are given a metric space $V$ of size $n$ and an integer $k$, and our task is to find a subset $S \subseteq V$ of at most $k$ `centers' that minimizes the total distance from each point in $V$ to its nearest center in $S$. Mettu and Plaxton [UAI'02] gave a randomized algorithm for $k$-median that computes a $O(1)$-approximation in $\tilde O(nk)$ time. They also showed that any algorithm for this problem with a bounded approximation ratio must have a running time of $Ω(nk)$. Thus, the running time of their algorithm is optimal up to polylogarithmic factors. For deterministic $k$-median, Guha et al.~[FOCS'00] gave an algorithm that computes a $\text{poly}(\log (n/k))$-approximation in $\tilde O(nk)$ time, where the degree of the polynomial in the approximation is unspecified. To the best of our knowledge, this remains the state-of-the-art approximation of any deterministic $k$-median algorithm with this running time. This leads us to the following natural question: What is the best approximation of a deterministic $k$-median algorithm with near-optimal running time? We make progress in answering this question by giving a deterministic algorithm that computes a $O(\log(n/k))$-approximation in $\tilde O(nk)$ time. We also provide a lower bound showing that any deterministic algorithm with this running time must have an approximation ratio of $Ω(\log n/(\log k + \log \log n))$, establishing a gap between the randomized and deterministic settings for $k$-median.

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arXiv2026

Additive One Approximation for Minimum Degree Spanning Tree: Breaking the $O(mn)$ Time Barrier

We consider the ``minimum degree spanning tree'' problem. As input, we receive an undirected, connected graph $G=(V, E)$ with $n$ nodes and $m$ edges, and our task is to find a spanning tree $T$ of $G$ that minimizes $\max_{u \in V} \text{deg}_T(u)$, where $\text{deg}_T(u)$ denotes the degree of $u \in V$ in $T$. The problem is known to be NP-hard. In the early 1990s, an influential work by Fürer and Raghavachari presented a local search algorithm that runs in $\tilde{O}(mn)$ time, and returns a spanning tree with maximum degree at most $Δ^\star+1$, where $Δ^\star$ is the optimal objective. This remained the state-of-the-art runtime bound for computing an additive one approximation, until now. We break this $O(mn)$ runtime barrier dating back to three decades, by providing a deterministic algorithm that returns an additive one approximate optimal spanning tree in $\tilde{O}(mn^{3/4})$ time. This constitutes a substantive progress towards answering an open question that has been repeatedly posed in the literature [Pettie'2016, Duan and Pettie'2020, Saranurak'2024]. Our algorithm is based on a novel application of the blocking flow paradigm.

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arXiv2025

Almost Optimal Fully Dynamic $k$-Center Clustering with Recourse

In this paper, we consider the \emph{metric $k$-center} problem in the fully dynamic setting, where we are given a metric space $(V,d)$ evolving via a sequence of point insertions and deletions and our task is to maintain a subset $S \subseteq V$ of at most $k$ points that minimizes the objective $\max_{x \in V} \min_{y \in S}d(x, y)$. We want to design our algorithm so that we minimize its \emph{approximation ratio}, \emph{recourse} (the number of changes it makes to the solution $S$), and \emph{update time} (the time it takes to handle an update). We give a simple algorithm for dynamic $k$-center that maintains a $O(1)$-approximate solution with $O(1)$ amortized recourse and $\tilde O(k)$ amortized update time, \emph{obtaining near-optimal approximation, recourse, and update time simultaneously}. We obtain our result by combining a variant of the dynamic $k$-center algorithm of Bateni et al.~[SODA'23] with the dynamic sparsifier of Bhattacharya et al.~[NeurIPS'23].

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arXiv2025

Improved Approximation Algorithms for (1,2)-TSP and Max-TSP Using Path Covers in the Semi-Streaming Model

We investigate semi-streaming algorithms for the Traveling Salesman Problem (TSP). Specifically, we focus on a variant known as the $(1,2)$-TSP, where the distances between any two vertices are either one or two. Our primary emphasis is on the closely related Maximum Path Cover Problem, which aims to find a collection of vertex-disjoint paths that cover the maximum number of edges in a graph. We propose an algorithm that, for any $ε> 0$, achieves a $(\frac{2}{3}-ε)$-approximation of the maximum path cover size for an $n$-vertex graph, using $\text{poly}(\frac{1}ε)$ passes. This result improves upon the previous $\frac{1}{2}$-approximation by Behnezhad et al. [ICALP 2024] in the semi-streaming model. Building on this result, we design a semi-streaming algorithm that constructs a tour for an instance of $(1,2)$-TSP with an approximation factor of $(\frac{4}{3} + ε)$, improving upon the previous $\frac{3}{2}$-approximation actor algorithm by Behnezhad et al. [ICALP 2024] (Although it is not explicitly stated in the paper that their algorithm works in the semi-streaming model, it is easy to verify). Furthermore, we extend our approach to develop an approximation algorithm for the Maximum TSP (Max-TSP), where the goal is to find a Hamiltonian cycle with the maximum possible weight in a given weighted graph $G$. Our algorithm provides a $(\frac{7}{12} - ε)$-approximation for Max-TSP in $\text{poly}(\frac{1}ε)$ passes, improving on the previously known $(\frac{1}{2}-ε)$-approximation obtained via maximum weight matching in the semi-streaming model.

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arXiv2024

Fully Dynamic $k$-Median with Near-Optimal Update Time and Recourse

In metric $k$-clustering, we are given as input a set of $n$ points in a general metric space, and we have to pick $k$ centers and cluster the input points around these chosen centers, so as to minimize an appropriate objective function. In recent years, significant effort has been devoted to the study of metric $k$-clustering problems in a dynamic setting, where the input keeps changing via updates (point insertions/deletions), and we have to maintain a good clustering throughout these updates. The performance of such a dynamic algorithm is measured in terms of three parameters: (i) Approximation ratio, which signifies the quality of the maintained solution, (ii) Recourse, which signifies how stable the maintained solution is, and (iii) Update time, which signifies the efficiency of the algorithm. We consider the metric $k$-median problem, where the objective is the sum of the distances of the points to their nearest centers. We design the first dynamic algorithm for this problem with near-optimal guarantees across all three performance measures (up to a constant factor in approximation ratio, and polylogarithmic factors in recourse and update time). Specifically, we obtain a $O(1)$-approximation algorithm for dynamic metric $k$-median with $\tilde{O}(1)$ recourse and $\tilde{O}(k)$ update time. Prior to our work, the state-of-the-art here was the recent result of [Bhattacharya et al., FOCS'24], who obtained $O(ε^{-1})$-approximation ratio with $\tilde{O}(k^ε)$ recourse and $\tilde{O}(k^{1+ε})$ update time. We achieve our results by carefully synthesizing the concept of robust centers introduced in [Fichtenberger et al., SODA'21] along with the randomized local search subroutine from [Bhattacharya et al., FOCS'24], in addition to several key technical insights of our own.

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