arXiv subjects

Sayan Bhattacharya

Publications and source records attributed to Sayan Bhattacharya.

At least 19 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 $\Delta$ 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 $O(m\alpha(n))$ 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: $(1+\epsilon,1)$-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: F\"urer and Raghavachari [FR92] gave an $\tilde O(mn)$-time algorithm that computes a spanning tree with maximum degree $\Delta^\star+1$, where $\Delta^\star$ 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 $\lceil (1+\epsilon)\Delta^\star\rceil+1$ in $\tilde O(m/\epsilon^2)$ time. Prior near-linear-time algorithms either achieved the weaker bound $\lceil (1+\epsilon)\Delta^\star\rceil + O(\log n/\epsilon^2)$ [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 $\Delta^\star+1$ 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

Reentrant localization transition, quantum butterfly and robust edge modes in aperiodic zig-zag ladder

Low dimensional tight-binding lattices in presence of quasiperiodic disorder generally exhibits localization transition. The system supports diffusive modes upto a limiting strength of disorder and all the eigenstates become localized beyond that critical strength thereby quenching the kinetic signature of the wavepacket. However, moving away from this situation, we demonstrate that with minimal long-range off-diagonal modulation, the eigenspectrum again may offer delocalization of electronic states for some subtle combination of kinetic parameters of the Hamiltonian leading to a second quantum phase change. The localization transitions are associated with the obvious presence of single-particle mobility edges. Multifractal energy landscape also shows quantum butterfly pattern with the in-gap robust edge modes. The re-emerging localization transition is manifested through the evaluation of inverse participation ratio, eigenspectrum and a pertinent quantum dynamical study.

cond-mat.other↗

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

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

Localization, transport, flux induced extended modes and mobility edge in a self-similar corral substrate

We address that a single-band tight-binding Hamiltonian defined on a self-similar corral substrate can give rise to a set of non-diffusive localized modes that follow the same hierarchical distribution. As the lattice, the spatial extent of quantum prison containing a cluster of atomic sites is dependent on the generation of fractal structure. Apart from the quantum imprisonment of the excitation, a magnetic flux threading each elementary plaquette is shown to destroy the boundedness and generate an absolutely continuous sub-band populated by resonant eigen functions. Flux induced engineering of quantum states is corroborated through the evaluation of inverse participation ratio and quantum transport. Moreover, the robustness of the extended states has been checked in presence of diagonal disorder and off-diagonal anisotropy. Flux modulated single-particle mobility edge is characterized through mutlifractal analysis. Quantum interference is the essential issue, reported here, that manipulates the kinematics of the excitation and this is manifested by the workout of persistent current.

cond-mat.mes-hall↗

arXiv2025

Fully Dynamic Set Cover: Worst-Case Recourse and Update Time

In (fully) dynamic set cover, the goal is to maintain an approximately optimal solution to a dynamically evolving instance of set cover, where in each step either an element is added to or removed from the instance. The two main desiderata of a dynamic set cover algorithm are to minimize at each time-step, the recourse, which is the number of sets removed from or added to the solution, and the update time to compute the updated solution. This problem has been extensively studied over the last decade leading to many results that achieve ever-improving bounds on the recourse and update time, while maintaining a solution whose cost is comparable to that of offline approximation algorithms. In this paper, we give the first algorithms to simultaneously achieve non-trivial worst-case bounds for recourse and update time. Specifically, we give fully-dynamic set cover algorithms that simultaneously achieve $O(\log n)$ recourse and $f\cdot \textrm{poly}\log(n)$ update time in the worst-case, for both approximation regimes: $O(\log n)$ and $O(f)$ approximation. (Here, $n, f$ respectively denote the maximum number of elements and maximum frequency of an element across all instances.) Prior to our work, all results for this problem either settled for amortized bounds on recourse and update time, or obtained $f\cdot \textrm{poly}\log(n)$ update time in the worst-case but at the cost of $Ω(m)$ worst-case recourse. (Here, $m$ denotes the number of sets. Note that any algorithm has recourse at most $m$.)

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arXiv2025

Separations between Oblivious and Adaptive Adversaries for Natural Dynamic Graph Problems

We establish the first update-time separation between dynamic algorithms against oblivious adversaries and those against adaptive adversaries in natural dynamic graph problems, based on popular fine-grained complexity hypotheses. Specifically, under the combinatorial BMM hypothesis, we show that every combinatorial algorithm against an adaptive adversary for the incremental maximal independent set problem requires $n^{1-o(1)}$ amortized update time. Furthermore, assuming either the 3SUM or APSP hypotheses, every algorithm for the decremental maximal clique problem needs $Δ/n^{o(1)}$ amortized update time when the initial maximum degree is $Δ\le \sqrt{n}$. These lower bounds are matched by existing algorithms against adaptive adversaries. In contrast, both problems admit algorithms against oblivious adversaries that achieve $\operatorname{polylog}(n)$ amortized update time [Behnezhad, Derakhshan, Hajiaghayi, Stein, Sudan; FOCS '19] [Chechik, Zhang; FOCS '19]. Therefore, our separations are exponential. Previously known separations for dynamic algorithms were either engineered for contrived problems and relied on strong cryptographic assumptions [Beimel, Kaplan, Mansour, Nissim, Saranurak, Stemmer; STOC '22], or worked for problems whose inputs are not explicitly given but are accessed through oracle calls [Bateni, Esfandiari, Fichtenberger, Henzinger, Jayaram, Mirrokni, Wiese; SODA '23]. As a byproduct, we also provide a separation between incremental and decremental algorithms for the triangle detection problem: we show a decremental algorithm with $\tilde{O}(n^ω)$ total update time, while every incremental algorithm requires $n^{3-o(1)}$ total update time, assuming the OMv hypothesis. To our knowledge this is the first separation of this kind.

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arXiv2025

Vizing's Theorem in Deterministic Almost-Linear Time

Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $Δ$ can be edge colored using at most $Δ+ 1$ different colors. Vizing's original proof is easily translated into a deterministic $O(mn)$ time algorithm. This deterministic time bound was subsequently improved to $\tilde O(m \sqrt n)$ time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985]. A series of recent papers improved the time bound of $\tilde O(m\sqrt{n})$ using randomization, culminating in the randomized near-linear time $(Δ+1)$-coloring algorithm by [Assadi, Behnezhad, Bhattacharya, Costa, Solomon, and Zhang, 2025]. At the heart of all of these recent improvements, there is some form of a sublinear time algorithm. Unfortunately, sublinear time algorithms as a whole almost always require randomization. This raises a natural question: can the deterministic time complexity of the problem be reduced below the $\tilde O(m\sqrt{n})$ barrier? In this paper, we answer this question in the affirmative. We present a deterministic almost-linear time $(Δ+1)$-coloring algorithm, namely, an algorithm running in $m \cdot 2^{O(\sqrt{\log Δ})} \cdot \log n = m^{1+o(1)}$ time. Our main technical contribution is to entirely forego sublinear time algorithms. We do so by presenting a new deterministic color-type sparsification approach that runs in almost-linear (instead of sublinear) time, but can be used to color a much larger set of edges.

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arXiv2025

Vizing's Theorem in Near-Linear Time

Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $Δ$ can be edge colored using at most $Δ+ 1$ different colors [Vizing, 1964]. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in $O(mn)$ time. This was subsequently improved to $\tilde O(m\sqrt{n})$ time, independently by [Arjomandi, 1982] and by [Gabow et al., 1985]. Very recently, independently and concurrently, using randomization, this runtime bound was further improved to $\tilde{O}(n^2)$ by [Assadi, 2024] and $\tilde O(mn^{1/3})$ by [Bhattacharya, Carmon, Costa, Solomon and Zhang, 2024] (and subsequently to $\tilde O(mn^{1/4})$ time by [Bhattacharya, Costa, Solomon and Zhang, 2024]). In this paper, we present a randomized algorithm that computes a $(Δ+1)$-edge coloring in near-linear time -- in fact, only $O(m\logΔ)$ time -- with high probability, giving a near-optimal algorithm for this fundamental problem.

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arXiv2025

Deterministic Dynamic Maximal Matching in Sublinear Update Time

We give a fully dynamic deterministic algorithm for maintaining a maximal matching of an $n$-vertex graph in $\tilde{O}(n^{8/9})$ amortized update time. This breaks the long-standing $Ω(n)$-update-time barrier on dense graphs, achievable by trivially scanning all incident vertices of the updated edge, and affirmatively answers a major open question repeatedly asked in the literature [BGS15, BCHN18, Sol22]. We also present a faster randomized algorithm against an adaptive adversary with $\tilde{O}(n^{3/4})$ amortized update time. Our approach employs the edge degree constrained subgraph (EDCS), a central object for optimizing approximation ratio, in a completely novel way; we instead use it for maintaining a matching that matches all high degree vertices in sublinear update time so that it remains to handle low degree vertices rather straightforwardly. To optimize this approach, we employ tools never used in the dynamic matching literature prior to our work, including sublinear-time algorithms for matching high degree vertices, random walks on directed expanders, and the monotone Even-Shiloach tree for dynamic shortest paths.

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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].

cs.DS↗

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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arXiv2024

Even Faster $(Δ+ 1)$-Edge Coloring via Shorter Multi-Step Vizing Chains

Vizing's Theorem from 1964 states that any $n$-vertex $m$-edge graph with maximum degree $Δ$ can be {\em edge colored} using at most $Δ+ 1$ colors. For over 40 years, the state-of-the-art running time for computing such a coloring, obtained independently by Arjomandi [1982] and by Gabow, Nishizeki, Kariv, Leven and Terada~[1985], was $\tilde O(m\sqrt{n})$. Very recently, this time bound was improved in two independent works, by Bhattacharya, Carmon, Costa, Solomon and Zhang to $\tilde O(mn^{1/3})$, and by Assadi to $\tilde O(n^2)$. In this paper we present an algorithm that computes such a coloring in $\tilde O(mn^{1/4})$ time. Our key technical contribution is a subroutine for extending the coloring to one more edge within time $\tilde O(Δ^2 + \sqrt{Δn})$. The best previous time bound of any color extension subroutine is either the trivial $O(n)$, dominated by the length of a Vizing chain, or the bound $\tilde{O}(Δ^6)$ by Bernshteyn [2022], dominated by the length of {\em multi-step Vizing chains}, which is basically a concatenation of multiple (carefully chosen) Vizing chains. Our color extension subroutine produces significantly shorter multi-step Vizing chains than in previous works, for sufficiently large $Δ$.

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arXiv2024

Density-Sensitive Algorithms for $(Δ+ 1)$-Edge Coloring

Vizing's theorem asserts the existence of a $(Δ+1)$-edge coloring for any graph $G$, where $Δ= Δ(G)$ denotes the maximum degree of $G$. Several polynomial time $(Δ+1)$-edge coloring algorithms are known, and the state-of-the-art running time (up to polylogarithmic factors) is $\tilde{O}(\min\{m \cdot \sqrt{n}, m \cdot Δ\})$, by Gabow et al.\ from 1985, where $n$ and $m$ denote the number of vertices and edges in the graph, respectively. (The $\tilde{O}$ notation suppresses polylogarithmic factors.) Recently, Sinnamon shaved off a polylogarithmic factor from the time bound of Gabow et al. The {arboricity} $α= α(G)$ of a graph $G$ is the minimum number of edge-disjoint forests into which its edge set can be partitioned, and it is a measure of the graph's "uniform density". While $α\le Δ$ in any graph, many natural and real-world graphs exhibit a significant separation between $α$ and $Δ$. In this work we design a $(Δ+1)$-edge coloring algorithm with a running time of $\tilde{O}(\min\{m \cdot \sqrt{n}, m \cdot Δ\})\cdot \fracαΔ$, thus improving the longstanding time barrier by a factor of $\fracαΔ$. In particular, we achieve a near-linear runtime for bounded arboricity graphs (i.e., $α= \tilde{O}(1)$) as well as when $α= \tilde{O}(\fracΔ{\sqrt{n}})$. Our algorithm builds on Sinnamon's algorithm, and can be viewed as a density-sensitive refinement of it.

cs.DS↗

arXiv2024

Fully Dynamic $k$-Clustering with Fast Update Time and Small Recourse

In the dynamic metric $k$-median problem, we wish to maintain a set of $k$ centers $S \subseteq V$ in an input metric space $(V, d)$ that gets updated via point insertions/deletions, so as to minimize the objective $\sum_{x \in V} \min_{y \in S} d(x, y)$. The quality of a dynamic algorithm is measured in terms of its approximation ratio, "recourse" (the number of changes in $S$ per update) and "update time" (the time it takes to handle an update). The ultimate goal in this line of research is to obtain a dynamic $O(1)$ approximation algorithm with $\tilde{O}(1)$ recourse and $\tilde{O}(k)$ update time. Dynamic $k$-median is a canonical example of a class of problems known as dynamic $k$-clustering, that has received significant attention in recent years. To the best of our knowledge, however, previous papers either attempt to minimize the algorithm's recourse while ignoring its update time, or minimize the algorithm's update time while ignoring its recourse. For dynamic $k$-median, we come arbitrarily close to resolving the main open question on this topic, with the following results. (I) We develop a new framework of randomized local search that is suitable for adaptation in a dynamic setting. For every $ε> 0$, this gives us a dynamic $k$-median algorithm with $O(1/ε)$ approximation ratio, $\tilde{O}(k^ε)$ recourse and $\tilde{O}(k^{1+ε})$ update time. This framework also generalizes to dynamic $k$-clustering with $\ell^p$-norm objectives, giving similar bounds for the dynamic $k$-means and a new trade-off for dynamic $k$-center. (II) If it suffices to maintain only an estimate of the value of the optimal $k$-median objective, then we obtain a $O(1)$ approximation algorithm with $\tilde{O}(k)$ update time. We achieve this result via adapting the Lagrangian Relaxation framework to the dynamic setting.

cs.DS↗

arXiv2024

Faster $(Δ+ 1)$-Edge Coloring: Breaking the $m \sqrt{n}$ Time Barrier

Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $Δ$ can be {\em edge colored} using at most $Δ+ 1$ different colors [Diskret.~Analiz, '64]. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in $\tilde{O}(mn)$ time. This was subsequently improved to $\tilde O(m\sqrt{n})$, independently by Arjomandi [1982] and by Gabow et al.~[1985]. In this paper we present an algorithm that computes such an edge coloring in $\tilde O(mn^{1/3})$ time, giving the first polynomial improvement for this fundamental problem in over 40 years.

cs.DS↗

arXiv2024

Chasing Positive Bodies

We study the problem of chasing positive bodies in $\ell_1$: given a sequence of bodies $K_{t}=\{x^{t}\in\mathbb{R}_{+}^{n}\mid C^{t}x^{t}\geq 1,P^{t}x^{t}\leq 1\}$ revealed online, where $C^{t}$ and $P^{t}$ are nonnegative matrices, the goal is to (approximately) maintain a point $x_t \in K_t$ such that $\sum_t \|x_t - x_{t-1}\|_1$ is minimized. This captures the fully-dynamic low-recourse variant of any problem that can be expressed as a mixed packing-covering linear program and thus also the fractional version of many central problems in dynamic algorithms such as set cover, load balancing, hyperedge orientation, minimum spanning tree, and matching. We give an $O(\log d)$-competitive algorithm for this problem, where $d$ is the maximum row sparsity of any matrix $C^t$. This bypasses and improves exponentially over the lower bound of $\sqrt{n}$ known for general convex bodies. Our algorithm is based on iterated information projections, and, in contrast to general convex body chasing algorithms, is entirely memoryless. We also show how to round our solution dynamically to obtain the first fully dynamic algorithms with competitive recourse for all the stated problems above; i.e. their recourse is less than the recourse of every other algorithm on every update sequence, up to polylogarithmic factors. This is a significantly stronger notion than the notion of absolute recourse in the dynamic algorithms literature.

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