arXiv subjects

Sayan Bhattacharya

Publications and source records attributed to Sayan Bhattacharya.

At least 37 records · Page 2Linked to original sources

arXiv2024

Dynamic $(1+ε)$-Approximate Matching Size in Truly Sublinear Update Time

We show a fully dynamic algorithm for maintaining $(1+ε)$-approximate \emph{size} of maximum matching of the graph with $n$ vertices and $m$ edges using $m^{0.5-Ω_ε(1)}$ update time. This is the first polynomial improvement over the long-standing $O(n)$ update time, which can be trivially obtained by periodic recomputation. Thus, we resolve the value version of a major open question of the dynamic graph algorithms literature (see, e.g., [Gupta and Peng FOCS'13], [Bernstein and Stein SODA'16],[Behnezhad and Khanna SODA'22]). Our key technical component is the first sublinear algorithm for $(1,εn)$-approximate maximum matching with sublinear running time on dense graphs. All previous algorithms suffered a multiplicative approximation factor of at least $1.499$ or assumed that the graph has a very small maximum degree.

cs.DS↗

arXiv2024

Near-Optimal Dynamic Rounding of Fractional Matchings in Bipartite Graphs

We study dynamic $(1-ε)$-approximate rounding of fractional matchings -- a key ingredient in numerous breakthroughs in the dynamic graph algorithms literature. Our first contribution is a surprisingly simple deterministic rounding algorithm in bipartite graphs with amortized update time $O(ε^{-1} \log^2 (ε^{-1} \cdot n))$, matching an (unconditional) recourse lower bound of $Ω(ε^{-1})$ up to logarithmic factors. Moreover, this algorithm's update time improves provided the minimum (non-zero) weight in the fractional matching is lower bounded throughout. Combining this algorithm with novel dynamic \emph{partial rounding} algorithms to increase this minimum weight, we obtain several algorithms that improve this dependence on $n$. For example, we give a high-probability randomized algorithm with $\tilde{O}(ε^{-1}\cdot (\log\log n)^2)$-update time against adaptive adversaries. (We use Soft-Oh notation, $\tilde{O}$, to suppress polylogarithmic factors in the argument, i.e., $\tilde{O}(f)=O(f\cdot \mathrm{poly}(\log f))$.) Using our rounding algorithms, we also round known $(1-ε)$-decremental fractional bipartite matching algorithms with no asymptotic overhead, thus improving on state-of-the-art algorithms for the decremental bipartite matching problem. Further, we provide extensions of our results to general graphs and to maintaining almost-maximal matchings.

cs.DS↗

arXiv2024

Arboricity-Dependent Algorithms for Edge Coloring

The problem of edge coloring has been extensively studied over the years. Recently, this problem has received significant attention in the dynamic setting, where we are given a dynamic graph evolving via a sequence of edge insertions and deletions and our objective is to maintain an edge coloring of the graph. Currently, it is not known whether it is possible to maintain a $(Δ+ O(Δ^{1 - μ}))$-edge coloring in $\tilde{O}(1)$ update time, for any constant $μ> 0$, where $Δ$ is the maximum degree of the graph. In this paper, we show how to efficiently maintain a $(Δ+ O(α))$-edge coloring in $\tilde O(1)$ amortized update time, where $α$ is the arboricty of the graph. Thus, we answer this question in the affirmative for graphs of sufficiently small arboricity.

cs.DS↗

arXiv2023

Nibbling at Long Cycles: Dynamic (and Static) Edge Coloring in Optimal Time

We consider the problem of maintaining a $(1+ε)Δ$-edge coloring in a dynamic graph $G$ with $n$ nodes and maximum degree at most $Δ$. The state-of-the-art update time is $O_ε(\text{polylog}(n))$, by Duan, He and Zhang [SODA'19] and by Christiansen [STOC'23], and more precisely $O(\log^7 n/ε^2)$, where $Δ= Ω(\log^2 n / ε^2)$. The following natural question arises: What is the best possible update time of an algorithm for this task? More specifically, \textbf{ can we bring it all the way down to some constant} (for constant $ε$)? This question coincides with the \emph{static} time barrier for the problem: Even for $(2Δ-1)$-coloring, there is only a naive $O(m \log Δ)$-time algorithm. We answer this fundamental question in the affirmative, by presenting a dynamic $(1+ε)Δ$-edge coloring algorithm with $O(\log^4 (1/ε)/ε^9)$ update time, provided $Δ= Ω_ε(\text{polylog}(n))$. As a corollary, we also get the first linear time (for constant $ε$) \emph{static} algorithm for $(1+ε)Δ$-edge coloring; in particular, we achieve a running time of $O(m \log (1/ε)/ε^2)$. We obtain our results by carefully combining a variant of the \textsc{Nibble} algorithm from Bhattacharya, Grandoni and Wajc [SODA'21] with the subsampling technique of Kulkarni, Liu, Sah, Sawhney and Tarnawski [STOC'22].

cs.DS↗

arXiv2023

Fully Dynamic $k$-Clustering in $\tilde O(k)$ Update Time

We present a $O(1)$-approximate fully dynamic algorithm for the $k$-median and $k$-means problems on metric spaces with amortized update time $\tilde O(k)$ and worst-case query time $\tilde O(k^2)$. We complement our theoretical analysis with the first in-depth experimental study for the dynamic $k$-median problem on general metrics, focusing on comparing our dynamic algorithm to the current state-of-the-art by Henzinger and Kale [ESA'20]. Finally, we also provide a lower bound for dynamic $k$-median which shows that any $O(1)$-approximate algorithm with $\tilde O(\text{poly}(k))$ query time must have $\tilde Ω(k)$ amortized update time, even in the incremental setting.

cs.DS↗

arXiv2023

Coarse-Grained Complexity for Dynamic Algorithms

To date, the only way to argue polynomial lower bounds for dynamic algorithms is via fine-grained complexity arguments. These arguments rely on strong assumptions about specific problems such as the Strong Exponential Time Hypothesis (SETH) and the Online Matrix-Vector Multiplication Conjecture (OMv). While they have led to many exciting discoveries, dynamic algorithms still miss out some benefits and lessons from the traditional ``coarse-grained'' approach that relates together classes of problems such as P and NP. In this paper we initiate the study of coarse-grained complexity theory for dynamic algorithms. Below are among questions that this theory can answer. What if dynamic Orthogonal Vector (OV) is easy in the cell-probe model? A research program for proving polynomial unconditional lower bounds for dynamic OV in the cell-probe model is motivated by the fact that many conditional lower bounds can be shown via reductions from the dynamic OV problem. Since the cell-probe model is more powerful than word RAM and has historically allowed smaller upper bounds, it might turn out that dynamic OV is easy in the cell-probe model, making this research direction infeasible. Our theory implies that if this is the case, there will be very interesting algorithmic consequences: If dynamic OV can be maintained in polylogarithmic worst-case update time in the cell-probe model, then so are several important dynamic problems such as $k$-edge connectivity, $(1+ε)$-approximate mincut, $(1+ε)$-approximate matching, planar nearest neighbors, Chan's subset union and 3-vs-4 diameter. The same conclusion can be made when we replace dynamic OV by, e.g., subgraph connectivity, single source reachability, Chan's subset union, and 3-vs-4 diameter. Lower bounds for $k$-edge connectivity via dynamic OV? (see the full abstract in the pdf file).

cs.CC↗

arXiv2023

Dynamic Matching with Better-than-2 Approximation in Polylogarithmic Update Time

We present dynamic algorithms with polylogarithmic update time for estimating the size of the maximum matching of a graph undergoing edge insertions and deletions with approximation ratio strictly better than $2$. Specifically, we obtain a $1+\frac{1}{\sqrt{2}}+ε\approx 1.707+ε$ approximation in bipartite graphs and a $1.973+ε$ approximation in general graphs. We thus answer in the affirmative the major open question first posed in the influential work of Onak and Rubinfeld (STOC'10) and repeatedly asked in the dynamic graph algorithms literature. Our randomized algorithms also work against an adaptive adversary and guarantee worst-case polylog update time, both w.h.p. Our algorithms are based on simulating new two-pass streaming matching algorithms in the dynamic setting. Our key new idea is to invoke the recent sublinear-time matching algorithm of Behnezhad (FOCS'21) in a white-box manner to efficiently simulate the second pass of our streaming algorithms, while bypassing the well-known vertex-update barrier.

cs.DS↗

arXiv2023

Sublinear Algorithms for $(1.5+ε)$-Approximate Matching

We study sublinear time algorithms for estimating the size of maximum matching. After a long line of research, the problem was finally settled by Behnezhad [FOCS'22], in the regime where one is willing to pay an approximation factor of $2$. Very recently, Behnezhad et al.[SODA'23] improved the approximation factor to $(2-\frac{1}{2^{O(1/γ)}})$ using $n^{1+γ}$ time. This improvement over the factor $2$ is, however, minuscule and they asked if even $1.99$-approximation is possible in $n^{2-Ω(1)}$ time. We give a strong affirmative answer to this open problem by showing $(1.5+ε)$-approximation algorithms that run in $n^{2-Θ(ε^{2})}$ time. Our approach is conceptually simple and diverges from all previous sublinear-time matching algorithms: we show a sublinear time algorithm for computing a variant of the edge-degree constrained subgraph (EDCS), a concept that has previously been exploited in dynamic [Bernstein Stein ICALP'15, SODA'16], distributed [Assadi et al. SODA'19] and streaming [Bernstein ICALP'20] settings, but never before in the sublinear setting. Independent work: Behnezhad, Roghani and Rubinstein [BRR'23] independently showed sublinear algorithms similar to our Theorem 1.2 in both adjacency list and matrix models. Furthermore, in [BRR'23], they show additional results on strictly better-than-1.5 approximate matching algorithms in both upper and lower bound sides.

cs.DS↗

arXiv2022

Efficient and Stable Fully Dynamic Facility Location

We consider the classic facility location problem in fully dynamic data streams, where elements can be both inserted and deleted. In this problem, one is interested in maintaining a stable and high quality solution throughout the data stream while using only little time per update (insertion or deletion). We study the problem and provide the first algorithm that at the same time maintains a constant approximation and incurs polylogarithmic amortized recourse per update. We complement our theoretical results with an experimental analysis showing the practical efficiency of our method.

cs.DS↗

arXiv2022

Dynamic Algorithms for Packing-Covering LPs via Multiplicative Weight Updates

In the dynamic linear program (LP) problem, we are given an LP undergoing updates and we need to maintain an approximately optimal solution. Recently, significant attention (e.g., [Gupta et al. STOC'17; Arar et al. ICALP'18, Wajc STOC'20]) has been devoted to the study of special cases of dynamic packing and covering LPs, such as the dynamic fractional matching and set cover problems. But until now, there is no non-trivial dynamic algorithm for general packing and covering LPs. In this paper, we settle the complexity of dynamic packing and covering LPs, up to a polylogarithmic factor in update time. More precisely, in the partially dynamic setting (where updates can either only relax or only restrict the feasible region), we give near-optimal deterministic $ε$-approximation algorithms with polylogarithmic amortized update time. Then, we show that both partially dynamic updates and amortized update time are necessary; without any of these conditions, the trivial algorithm that recomputes the solution from scratch after every update is essentially the best possible, assuming SETH. To obtain our results, we initiate a systematic study of the multiplicative weights update (MWU) method in the dynamic setting. As by-products of our techniques, we also obtain the first online $(1+ε)$-competitive algorithms for both covering and packing LPs with polylogarithmic recourse, and the first streaming algorithms for covering and packing LPs with linear space and polylogarithmic passes.

cs.DS↗

arXiv2022

Simple Dynamic Spanners with Near-optimal Recourse against an Adaptive Adversary

Designing dynamic algorithms against an adaptive adversary whose performance match the ones assuming an oblivious adversary is a major research program in the field of dynamic graph algorithms. One of the prominent examples whose oblivious-vs-adaptive gap remains maximally large is the \emph{fully dynamic spanner} problem; there exist algorithms assuming an oblivious adversary with near-optimal size-stretch trade-off using only $\operatorname{polylog}(n)$ update time [Baswana, Khurana, and Sarkar TALG'12; Forster and Goranci STOC'19; Bernstein, Forster, and Henzinger SODA'20], while against an adaptive adversary, even when we allow infinite time and only count recourse (i.e. the number of edge changes per update in the maintained spanner), all previous algorithms with stretch at most $\log^{5}(n)$ require at least $Ω(n)$ amortized recourse [Ausiello, Franciosa, and Italiano ESA'05]. In this paper, we completely close this gap with respect to recourse by showing algorithms against an adaptive adversary with near-optimal size-stretch trade-off and recourse. More precisely, for any $k\ge1$, our algorithm maintains a $(2k-1)$-spanner of size $O(n^{1+1/k}\log n)$ with $O(\log n)$ amortized recourse, which is optimal in all parameters up to a $O(\log n)$ factor. As a step toward algorithms with small update time (not just recourse), we show another algorithm that maintains a $3$-spanner of size $\tilde O(n^{1.5})$ with $\operatorname{polylog}(n)$ amortized recourse \emph{and} simultaneously $\tilde O(\sqrt{n})$ worst-case update time.

cs.DS↗

arXiv2022

A New Dynamic Algorithm for Densest Subhypergraphs

Computing a dense subgraph is a fundamental problem in graph mining, with a diverse set of applications ranging from electronic commerce to community detection in social networks. In many of these applications, the underlying context is better modelled as a weighted hypergraph that keeps evolving with time. This motivates the problem of maintaining the densest subhypergraph of a weighted hypergraph in a {\em dynamic setting}, where the input keeps changing via a sequence of updates (hyperedge insertions/deletions). Previously, the only known algorithm for this problem was due to Hu et al. [HWC17]. This algorithm worked only on unweighted hypergraphs, and had an approximation ratio of $(1+ε)r^2$ and an update time of $O(\text{poly} (r, \log n))$, where $r$ denotes the maximum rank of the input across all the updates. We obtain a new algorithm for this problem, which works even when the input hypergraph is weighted. Our algorithm has a significantly improved (near-optimal) approximation ratio of $(1+ε)$ that is independent of $r$, and a similar update time of $O(\text{poly} (r, \log n))$. It is the first $(1+ε)$-approximation algorithm even for the special case of weighted simple graphs. To complement our theoretical analysis, we perform experiments with our dynamic algorithm on large-scale, real-world data-sets. Our algorithm significantly outperforms the state of the art [HWC17] both in terms of accuracy and efficiency.

cs.DS↗

arXiv2021

Deterministic Rounding of Dynamic Fractional Matchings

We present a framework for deterministically rounding a dynamic fractional matching. Applying our framework in a black-box manner on top of existing fractional matching algorithms, we derive the following new results: (1) The first deterministic algorithm for maintaining a $(2-δ)$-approximate maximum matching in a fully dynamic bipartite graph, in arbitrarily small polynomial update time. (2) The first deterministic algorithm for maintaining a $(1+δ)$-approximate maximum matching in a decremental bipartite graph, in polylogarithmic update time. (3) The first deterministic algorithm for maintaining a $(2+δ)$-approximate maximum matching in a fully dynamic general graph, in small polylogarithmic (specifically, $O(\log^4 n)$) update time. These results are respectively obtained by applying our framework on top of the fractional matching algorithms of Bhattacharya et al. [STOC'16], Bernstein et al. [FOCS'20], and Bhattacharya and Kulkarni [SODA'19]. Prior to our work, there were two known general-purpose rounding schemes for dynamic fractional matchings. Both these schemes, by Arar et al. [ICALP'18] and Wajc [STOC'20], were randomized. Our rounding scheme works by maintaining a good {\em matching-sparsifier} with bounded arboricity, and then applying the algorithm of Peleg and Solomon [SODA'16] to maintain a near-optimal matching in this low arboricity graph. To the best of our knowledge, this is the first dynamic matching algorithm that works on general graphs by using an algorithm for low-arboricity graphs as a black-box subroutine. This feature of our rounding scheme might be of independent interest.

cs.DS↗

arXiv2020

Online Edge Coloring Algorithms via the Nibble Method

Nearly thirty years ago, Bar-Noy, Motwani and Naor [IPL'92] conjectured that an online $(1+o(1))Δ$-edge-coloring algorithm exists for $n$-node graphs of maximum degree $Δ=ω(\log n)$. This conjecture remains open in general, though it was recently proven for bipartite graphs under \emph{one-sided vertex arrivals} by Cohen et al.~[FOCS'19]. In a similar vein, we study edge coloring under widely-studied relaxations of the online model. Our main result is in the \emph{random-order} online model. For this model, known results fall short of the Bar-Noy et al.~conjecture, either in the degree bound [Aggarwal et al.~FOCS'03], or number of colors used [Bahmani et al.~SODA'10]. We achieve the best of both worlds, thus resolving the Bar-Noy et al.~conjecture in the affirmative for this model. Our second result is in the adversarial online (and dynamic) model with \emph{recourse}. A recent algorithm of Duan et al.~[SODA'19] yields a $(1+ε)Δ$-edge-coloring with poly$(\log n/ε)$ recourse. We achieve the same with poly$(1/ε)$ recourse, thus removing all dependence on $n$. Underlying our results is one common offline algorithm, which we show how to implement in these two online models. Our algorithm, based on the Rödl Nibble Method, is an adaptation of the distributed algorithm of Dubhashi et al.~[TCS'98]. The Nibble Method has proven successful for distributed edge coloring. We display its usefulness in the context of online algorithms.

cs.DS↗

arXiv2020

Pressure induced emergence of visible luminescence in $Cs_3Bi_2Br_9$: Effect of structural distortion in optical behaviour

We report emergence of photoluminescence at room temperature in trigonal $Cs_3Bi_2Br_9$ at high pressures. Enhancement in intensity with pressure is found to be driven by increase in distortion of $BiBr_6$ octahedra and iso-structural transitions. Electronic band structure calculations show the sample in the high pressure phase to be an indirect band gap semiconductor. The luminescence peak profile show signatures related to the recombination of free and self trapped excitons, respectively. Blue shift of the both peaks till about 4.4 GPa are due to the exciton recombination before relaxation due to the decrease in exciton lifetime with scattering from phonons

cond-mat.mtrl-sci↗

arXiv2020

Dynamic Set Cover: Improved Amortized and Worst-Case Update Time

In the dynamic minimum set cover problem, a challenge is to minimize the update time while guaranteeing close to the optimal $\min(O(\log n), f)$ approximation factor. (Throughout, $m$, $n$, $f$, and $C$ are parameters denoting the maximum number of sets, number of elements, frequency, and the cost range.) In the high-frequency range, when $f=Ω(\log n)$, this was achieved by a deterministic $O(\log n)$-approximation algorithm with $O(f \log n)$ amortized update time [Gupta et al. STOC'17]. In the low-frequency range, the line of work by Gupta et al. [STOC'17], Abboud et al. [STOC'19], and Bhattacharya et al. [ICALP'15, IPCO'17, FOCS'19] led to a deterministic $(1+ε)f$-approximation algorithm with $O(f \log (Cn)/ε^2)$ amortized update time. In this paper we improve the latter update time and provide the first bounds that subsume (and sometimes improve) the state-of-the-art dynamic vertex cover algorithms. We obtain: 1. $(1+ε)f$-approximation ratio in $O(f\log^2 (Cn)/ε^3)$ worst-case update time: No non-trivial worst-case update time was previously known for dynamic set cover. Our bound subsumes and improves by a logarithmic factor the $O(\log^3 n/\text{poly}(ε))$ worst-case update time for unweighted dynamic vertex cover (i.e., when $f=2$ and $C=1$) by Bhattacharya et al. [SODA'17]. 2. $(1+ε)f$-approximation ratio in $O\left((f^2/ε^3)+(f/ε^2) \log C\right)$ amortized update time: This result improves the previous $O(f \log (Cn)/ε^2)$ update time bound for most values of $f$ in the low-frequency range, i.e. whenever $f=o(\log n)$. It is the first that is independent of $m$ and $n$. It subsumes the constant amortized update time of Bhattacharya and Kulkarni [SODA'19] for unweighted dynamic vertex cover (i.e., when $f = 2$ and $C = 1$).

cs.DS↗

arXiv2019

Fully Dynamic $(Δ+1)$-Coloring in Constant Update Time

The problem of (vertex) $(Δ+1)$-coloring a graph of maximum degree $Δ$ has been extremely well-studied over the years in various settings and models. Surprisingly, for the dynamic setting, almost nothing was known until recently. In SODA'18, Bhattacharya, Chakrabarty, Henzinger and Nanongkai devised a randomized data structure for maintaining a $(Δ+1)$-coloring with $O(\log Δ)$ expected amortized update time. In this paper, we present a $(Δ+1)$-coloring data structure that achieves a constant amortized update time and show that this time bound holds not only in expectation but also with high probability.

cs.DS↗

arXiv2019

A New Deterministic Algorithm for Dynamic Set Cover

We present a deterministic dynamic algorithm for maintaining a $(1+ε)f$-approximate minimum cost set cover with $O(f\log(Cn)/ε^2)$ amortized update time, when the input set system is undergoing element insertions and deletions. Here, $n$ denotes the number of elements, each element appears in at most $f$ sets, and the cost of each set lies in the range $[1/C, 1]$. Our result, together with that of Gupta et al. [STOC`17], implies that there is a deterministic algorithm for this problem with $O(f\log(Cn))$ amortized update time and $O(\min(\log n, f))$-approximation ratio, which nearly matches the polynomial-time hardness of approximation for minimum set cover in the static setting. Our update time is only $O(\log (Cn))$ away from a trivial lower bound. Prior to our work, the previous best approximation ratio guaranteed by deterministic algorithms was $O(f^2)$, which was due to Bhattacharya et al. [ICALP`15]. In contrast, the only result that guaranteed $O(f)$-approximation was obtained very recently by Abboud et al. [STOC`19], who designed a dynamic algorithm with $(1+ε)f$-approximation ratio and $O(f^2 \log n/ε)$ amortized update time. Besides the extra $O(f)$ factor in the update time compared to our and Gupta et al.'s results, the Abboud et al. algorithm is randomized, and works only when the adversary is oblivious and the sets are unweighted (each set has the same cost). We achieve our result via the primal-dual approach, by maintaining a fractional packing solution as a dual certificate. Unlike previous primal-dual algorithms that try to satisfy some local constraints for individual sets at all time, our algorithm basically waits until the dual solution changes significantly globally, and fixes the solution only where the fix is needed.

cs.DS↗