arXiv subjects

Peter Kiss

Publications and source records attributed to Peter Kiss.

17 recordsLinked to original sources

arXiv2026

Online Preemptive Matching Revisited

We study the online preemptive matching problem, in which the edges of a graph arrive sequentially and the algorithm must maintain a matching by accepting or rejecting arriving edges and possibly discarding previously accepted ones. We prove a new upper bound of $0.5661$ on the competitive ratio achievable for the problem. This bound applies to arbitrary randomized algorithms, bipartite graphs and if we allow the algorithm to output a fractional solution. Our result improves upon the strongest previously known upper bound of $2-\sqrt{2} \approx 0.585$, due to Huang et al. [SODA'19]. Previous hardness constructions relied on edge sequences described by vertex arrivals where each arriving vertex reveals its edges to yet unvaried vertices. Under such sequences, Huang et al. showed that there exists a non-preemptive online algorithm with competitive ratio $\sim0.567$ (or $2-\sqrt{2}$ for fractional solutions). Consequently, our hardness construction is the first result which shows hardness for instances where the optimal algorithm employs preemption.

cs.DS↗

arXiv2026

Dynamic Hierarchical $j$-Tree Decomposition and Its Applications

We develop a new algorithmic framework for designing approximation algorithms for cut-based optimization problems on capacitated undirected graphs that undergo edge insertions and deletions. Specifically, our framework dynamically maintains a variant of the hierarchical $j$-tree decomposition of [Madry FOCS'10], achieving a poly-logarithmic approximation factor to the graph's cut structure and supporting edge updates in $O(n^{\epsilon})$ amortized update time, for any arbitrarily small constant $\epsilon \in (0,1)$. Consequently, we obtain new trade-offs between approximation and update/query time for fundamental cut-based optimization problems in the fully dynamic setting, including all-pairs minimum cuts, sparsest cut, multi-way cut, and multi-cut. For the last three problems, these trade-offs give the first fully-dynamic algorithms achieving poly-logarithmic approximation in sub-linear time per operation. The main technical ingredient behind our dynamic hierarchy is a dynamic cut-sparsifier algorithm that can handle vertex splits with low recourse. This is achieved by white-boxing the dynamic cut sparsifier construction of [Abraham et al. FOCS'16], based on forest packing, together with new structural insights about the maintenance of these forests under vertex splits. Given the versatility of cut sparsification in both the static and dynamic graph algorithms literature, we believe this construction may be of independent interest.

cs.DS↗

arXiv2025

Fully Dynamic Algorithms for Chamfer Distance

We study the problem of computing Chamfer distance in the fully dynamic setting, where two set of points $A, B \subset \mathbb{R}^{d}$, each of size up to $n$, dynamically evolve through point insertions or deletions and the goal is to efficiently maintain an approximation to $\mathrm{dist}_{\mathrm{CH}}(A,B) = \sum_{a \in A} \min_{b \in B} \textrm{dist}(a,b)$, where $\textrm{dist}$ is a distance measure. Chamfer distance is a widely used dissimilarity metric for point clouds, with many practical applications that require repeated evaluation on dynamically changing datasets, e.g., when used as a loss function in machine learning. In this paper, we present the first dynamic algorithm for maintaining an approximation of the Chamfer distance under the $\ell_p$ norm for $p \in \{1,2 \}$. Our algorithm reduces to approximate nearest neighbor (ANN) search with little overhead. Plugging in standard ANN bounds, we obtain $(1+\epsilon)$-approximation in $\tilde{O}(\epsilon^{-d})$ update time and $O(1/\epsilon)$-approximation in $\tilde{O}(d n^{\epsilon^2} \epsilon^{-4})$ update time. We evaluate our method on real-world datasets and demonstrate that it performs competitively against natural baselines.

cs.DS↗

arXiv2025

Tree Embedding in High Dimensions: Dynamic and Massively Parallel

Tree embedding has been a fundamental method in algorithm design with wide applications. We focus on the efficiency of building tree embedding in various computational settings under high-dimensional Euclidean $\mathbb{R}^d$. We devise a new tree embedding construction framework that operates on an arbitrary metric decomposition with bounded diameter, offering a tradeoff between distortion and the locality of its algorithmic steps. This framework works for general metric spaces and may be of independent interest beyond the Euclidean setting. Using this framework, we obtain a dynamic algorithm that maintains an $O_\epsilon(\log n)$-distortion tree embedding with update time $\tilde O(n^\epsilon + d)$ subject to point insertions/deletions, and a massively parallel algorithm that achieves $O_\epsilon(\log n)$-distortion in $O(1)$ rounds and total space $\tilde O(n^{1 + \epsilon})$ (for constant $\epsilon \in (0, 1)$). These new tree embedding results allow for a wide range of applications. Notably, under a similar performance guarantee as in our tree embedding algorithms, i.e., $\tilde O(n^\epsilon + d)$ update time and $O(1)$ rounds, we obtain $O_\epsilon(\log n)$-approximate dynamic and MPC algorithms for $k$-median and earth-mover distance in $\mathbb{R}^d$.

cs.DS↗

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 $\Delta/n^{o(1)}$ amortized update time when the initial maximum degree is $\Delta \le \sqrt{n}$. 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 $\tilde{O}(n^{\omega})$ 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.

cs.DS↗

arXiv2025

Fully Dynamic Euclidean Bi-Chromatic Matching in Sublinear Update Time

We consider the Euclidean bi-chromatic matching problem in the dynamic setting, where the goal is to efficiently process point insertions and deletions while maintaining a high-quality solution. Computing the minimum cost bi-chromatic matching is one of the core problems in geometric optimization that has found many applications, most notably in estimating Wasserstein distance between two distributions. In this work, we present the first fully dynamic algorithm for Euclidean bi-chromatic matching with sub-linear update time. For any fixed $\varepsilon > 0$, our algorithm achieves $O(1/\varepsilon)$-approximation and handles updates in $O(n^{\varepsilon})$ time. Our experiments show that our algorithm enables effective monitoring of the distributional drift in the Wasserstein distance on real and synthetic data sets, while outperforming the runtime of baseline approximations by orders of magnitudes.

cs.DS↗

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 $\Omega(n)$-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.

cs.DS↗

arXiv2024

Improved Bounds for Fully Dynamic Matching via Ordered Ruzsa-Szemeredi Graphs

In a very recent breakthrough, Behnezhad and Ghafari [FOCS'24] developed a novel fully dynamic randomized algorithm for maintaining a $(1-\epsilon)$-approximation of maximum matching with amortized update time potentially much better than the trivial $O(n)$ update time. The runtime of the BG algorithm is parameterized via the following graph theoretical concept: * For any $n$, define $ORS(n)$ -- standing for Ordered RS Graph -- to be the largest number of edge-disjoint matchings $M_1,\ldots,M_t$ of size $\Theta(n)$ in an $n$-vertex graph such that for every $i \in [t]$, $M_i$ is an induced matching in the subgraph $M_{i} \cup M_{i+1} \cup \ldots \cup M_t$. Then, for any fixed $\epsilon > 0$, the BG algorithm runs in \[ O\left( \sqrt{n^{1+O(\epsilon)} \cdot ORS(n)} \right) \] amortized update time with high probability, even against an adaptive adversary. $ORS(n)$ is a close variant of a more well-known quantity regarding RS graphs (which require every matching to be induced regardless of the ordering). It is currently only known that $n^{o(1)} \leqslant ORS(n) \leqslant n^{1-o(1)}$, and closing this gap appears to be a notoriously challenging problem. In this work, we further strengthen the result of Behnezhad and Ghafari and push it to limit to obtain a randomized algorithm with amortized update time of \[ n^{o(1)} \cdot ORS(n) \] with high probability, even against an adaptive adversary. In the limit, i.e., if current lower bounds for $ORS(n) = n^{o(1)}$ are almost optimal, our algorithm achieves an $n^{o(1)}$ update time for $(1-\epsilon)$-approximation of maximum matching, almost fully resolving this fundamental question. In its current stage also, this fully reduces the algorithmic problem of designing dynamic matching algorithms to a purely combinatorial problem of upper bounding $ORS(n)$ with no algorithmic considerations.

cs.DS↗

arXiv2023

Near-Optimal Dynamic Rounding of Fractional Matchings in Bipartite Graphs

We study dynamic $(1-\epsilon)$-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 $O(\epsilon^{-1} \log^2 (\epsilon^{-1} \cdot n))$, matching an (unconditional) recourse lower bound of $\Omega(\epsilon^{-1})$ 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 $\tilde{O}(\epsilon^{-1}\cdot (\log\log n)^2)$-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 $(1-\epsilon)$-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↗

arXiv2023

Incremental $(1-\epsilon)$-approximate dynamic matching in $O(poly(1/\epsilon))$ update time

In the dynamic approximate maximum bipartite matching problem we are given bipartite graph $G$ undergoing updates and our goal is to maintain a matching of $G$ which is large compared the maximum matching size $\mu(G)$. We define a dynamic matching algorithm to be $\alpha$ (respectively $(\alpha, \beta)$)-approximate if it maintains matching $M$ such that at all times $|M | \geq \mu(G) \cdot \alpha$ (respectively $|M| \geq \mu(G) \cdot \alpha - \beta$). We present the first deterministic $(1-\epsilon )$-approximate dynamic matching algorithm with $O(poly(\epsilon ^{-1}))$ amortized update time for graphs undergoing edge insertions. Previous solutions either required super-constant [Gupta FSTTCS'14, Bhattacharya-Kiss-Saranurak SODA'23] or exponential in $1/\epsilon $ [Grandoni-Leonardi-Sankowski-Schwiegelshohn-Solomon SODA'19] update time. Our implementation is arguably simpler than the mentioned algorithms and its description is self contained. Moreover, we show that if we allow for additive $(1, \epsilon \cdot n)$-approximation our algorithm seamlessly extends to also handle vertex deletions, on top of edge insertions. This makes our algorithm one of the few small update time algorithms for $(1-\epsilon )$-approximate dynamic matching allowing for updates both increasing and decreasing the maximum matching size of $G$ in a fully dynamic manner.

cs.DS↗

arXiv2023

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

We show a fully dynamic algorithm for maintaining $(1+\epsilon)$-approximate \emph{size} of maximum matching of the graph with $n$ vertices and $m$ edges using $m^{0.5-\Omega_{\epsilon}(1)}$ 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 $(1,\epsilon n)$-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↗

arXiv2022

Sublinear Algorithms for $(1.5+\epsilon)$-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 $(2-\frac{1}{2^{O(1/\gamma)}})$ using $n^{1+\gamma}$ time. This improvement over the factor $2$ is, however, minuscule and they asked if even $1.99$-approximation is possible in $n^{2-\Omega(1)}$ time. We give a strong affirmative answer to this open problem by showing $(1.5+\epsilon)$-approximation algorithms that run in $n^{2-\Theta(\epsilon^{2})}$ 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

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 $1+\frac{1}{\sqrt{2}}+\epsilon\approx 1.707+\epsilon$ approximation in bipartite graphs and a $1.973+\epsilon$ 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↗

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 $\epsilon$-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 $(1+\epsilon)$-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

Multimodal E-Commerce Product Classification Using Hierarchical Fusion

In this work, we present a multi-modal model for commercial product classification, that combines features extracted by multiple neural network models from textual (CamemBERT and FlauBERT) and visual data (SE-ResNeXt-50), using simple fusion techniques. The proposed method significantly outperformed the unimodal models' performance and the reported performance of similar models on our specific task. We did experiments with multiple fusing techniques and found, that the best performing technique to combine the individual embedding of the unimodal network is based on combining concatenation and averaging the feature vectors. Each modality complemented the shortcomings of the other modalities, demonstrating that increasing the number of modalities can be an effective method for improving the performance of multi-label and multimodal classification problems.

cs.AI↗

arXiv2021

Deterministic Dynamic Matching In Worst-Case Update Time

We present deterministic algorithms for maintaining a $(3/2 + \epsilon)$ and $(2 + \epsilon)$-approximate maximum matching in a fully dynamic graph with worst-case update times $\hat{O}(\sqrt{n})$ and $\tilde{O}(1)$ respectively. The fastest known deterministic worst-case update time algorithms for achieving approximation ratio $(2 - \delta)$ (for any $\delta > 0$) and $(2 + \epsilon)$ were both shown by Roghani et al. [2021] with update times $O(n^{3/4})$ and $O_\epsilon(\sqrt{n})$ respectively. We close the gap between worst-case and amortized algorithms for the two approximation ratios as the best deterministic amortized update times for the problem are $O_\epsilon(\sqrt{n})$ and $\tilde{O}(1)$ which were shown in Bernstein and Stein [SODA'2021] and Bhattacharya and Kiss [ICALP'2021] respectively. In order to achieve both results we explicitly state a method implicitly used in Nanongkai and Saranurak [STOC'2017] and Bernstein et al. [arXiv'2020] which allows to transform dynamic algorithms capable of processing the input in batches to a dynamic algorithms with worst-case update time. \textbf{Independent Work:} Independently and concurrently to our work Grandoni et al. [arXiv'2021] has presented a fully dynamic algorithm for maintaining a $(3/2 + \epsilon)$-approximate maximum matching with deterministic worst-case update time $O_\epsilon(\sqrt{n})$.

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 $(2-\delta)$-approximate maximum matching in a fully dynamic bipartite graph, in arbitrarily small polynomial update time. (2) The first deterministic algorithm for maintaining a $(1+\delta)$-approximate maximum matching in a decremental bipartite graph, in polylogarithmic update time. (3) The first deterministic algorithm for maintaining a $(2+\delta)$-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↗