arXiv subjects

Aditi Dudeja

Publications and source records attributed to Aditi Dudeja.

11 recordsLinked to original sources

arXiv2026

Randomized Algorithms for Learning Partitions with Near Optimal Query Complexity in Constant Rounds

We study the round complexity of learning a hidden partition $\mathcal{P}$ of an $n$-element universe using PAIR queries: PAIR($x,y$) tells us whether $x$ and $y$ belong to the same part of the partition or not. While it is easy to learn using $n|\mathcal{P}|$ queries using a basic algorithm and this query complexity is optimal, this basic algorithm is highly sequential. Black, Mazumdar, and Saha [COLT 2025] recently gave tight deterministic round/query tradeoffs when the number of parts of $\mathcal{P}$ is known. In particular they prove $\Theta(\log\log n)$ rounds are sufficient and necessary to limit the number of queries to $n|\mathcal{P}|$. They leave proving a randomized lower bound as an open direction. We show that randomization dramatically changes the picture. When the number of parts $k = |\mathcal{P}|$ is known, we give a simple 3-round randomized algorithm using $O(nk\log n)$ queries with high probability, and prove that 2 rounds require $\Omega(n^{4/3}k^{2/3})$ queries -- the same as deterministic algorithms. We also study a more general setting where the number of parts is unknown. In this case, we give a 4-round randomized algorithm using $O(n|\mathcal P|\log^2 n)$ queries with high probability, and prove that 3-rounds cannot achieve near-optimal query complexity. Furthermore, we show an even bigger separation in this regime between randomized and deterministic algorithms: for the latter, $\Theta(\log n/\log\log n)$ rounds are necessary and sufficient to obtain near-optimal query complexity.

cs.DS↗

arXiv2026

Distributed Stochastic Graph Algorithms

We study stochastic graph optimization problems in a novel distributed setting. As in the standard centralized setting, a random subgraph $G^*$ of a known base graph $G$ is realized by including each edge $e$ independently with a known probability $p_e$, and we must solve an optimization problem on $G^*$ despite uncertainty about its edges. In the standard setting, to cope with this uncertainty, the algorithm can query any edge of $G$ to learn if the edge exists in $G^*$, and its complexity is the number of queried edges. The distributed setting incorporates uncertainty in a natural manner, by having each vertex know only about its own edges in $G^*$ (and only communicate over them), and the complexity is measured by the number of synchronous communication rounds. We establish that distributed stochastic algorithms can be drastically faster than their non-stochastic counterparts and overcome known lower bounds, by showing fast distributed approximation algorithms for maximum matching, minimum vertex cover, and minimum dominating set.

cs.DS↗

arXiv2026

The Careless Coupon Collector's Problem

We initiate the study of the Careless Coupon Collector's Problem (CCCP), a novel variation of the classical coupon collector, that we envision as a model for information systems such as web crawlers, dynamic caches, and fault-resilient networks. In CCCP, a collector attempts to gather $n$ distinct coupon types by obtaining one coupon type uniformly at random in each discrete round, however the collector is \textit{careless}: at the end of each round, each collected coupon type is independently lost with probability $p$. We analyze the number of rounds required to complete the collection as a function of $n$ and $p$. In particular, we show that it transitions from $Θ(n \ln n)$ when $p = o\big(\frac{\ln n}{n^2}\big)$ up to $Θ\big((\frac{np}{1-p})^n\big)$ when $p=ω\big(\frac{1}{n}\big)$ in multiple distinct phases. Interestingly, when $p=\frac{c}{n}$, the process remains in a metastable phase, where the fraction of collected coupon types is concentrated around $\frac{1}{1+c}$ with probability $1-o(1)$, for a time window of length $e^{Θ(n)}$. Finally, we give an algorithm that computes the expected completion time of CCCP in $O(n^2)$ time.

cs.DM↗

arXiv2026

Frontier Space-Time Algorithms Using Only Full Memory

We develop catalytic algorithms for fundamental problems in algorithm design that run in polynomial time, use only $\mathcal{O}(\log(n))$ workspace, and use sublinear catalytic space matching the best-known space bounds of non-catalytic algorithms running in polynomial time. First, we design a polynomial time algorithm for directed $s$-$t$ connectivity using $n \big/ 2^{Θ(\sqrt{\log n})}$ catalytic space, which matches the state-of-the-art time-space bounds in the non-catalytic setting [Barnes et al., 1998], and improves the catalytic space usage of the best known algorithm [Cook and Pyne, 2026]. Furthermore, using only $\mathcal{O}(\log(n))$ random bits we get a randomized algorithm whose running time nearly matches the fastest time bounds known for space-unrestricted algorithms. Second, we design polynomial time algorithms for the problems of computing Edit Distance, Longest Common Subsequence, and the Discrete Fréchet Distance, again using $n \big/ 2^{Θ(\sqrt{\log n})}$ catalytic space. This again matches non-catalytic time-space frontier for Edit Distance and Least Common Subsequence [Kiyomi et al., 2021].

cs.DS↗

arXiv2026

A Weighted-to-Unweighted Reduction for Matroid Intersection

Given two matroids $\mathcal{M}_1$ and $\mathcal{M}_2$ over the same ground set, the matroid intersection problem is to find the maximum cardinality common independent set. In the weighted version of the problem, the goal is to find a maximum weight common independent set. It has been a matter of interest to find efficient approximation algorithms for this problem in various settings. In many of these models, there is a gap between the best known results for the unweighted and weighted versions. In this work, we address the question of closing this gap. Our main result is a reduction which converts any $α$-approximate unweighted matroid intersection algorithm into an $α(1-\varepsilon)$-approximate weighted matroid intersection algorithm, while increasing the runtime of the algorithm by a $\log W$ factor, where $W$ is the aspect ratio. Our framework is versatile and translates to settings such as streaming and one-way communication complexity where matroid intersection is well-studied. As a by-product of our techniques, we derive new results for weighted matroid intersection in these models.

cs.DS↗

arXiv2024

Randomized Greedy Online Edge Coloring Succeeds for Dense and Randomly-Ordered Graphs

Vizing's theorem states that any graph of maximum degree $Δ$ can be properly edge colored with at most $Δ+1$ colors. In the online setting, it has been a matter of interest to find an algorithm that can properly edge color any graph on $n$ vertices with maximum degree $Δ= ω(\log n)$ using at most $(1+o(1))Δ$ colors. Here we study the naïve random greedy algorithm, which simply chooses a legal color uniformly at random for each edge upon arrival. We show that this algorithm can $(1+ε)Δ$-color the graph for arbitrary $ε$ in two contexts: first, if the edges arrive in a uniformly random order, and second, if the edges arrive in an adversarial order but the graph is sufficiently dense, i.e., $n = O(Δ)$. Prior to this work, the random greedy algorithm was only known to succeed in trees. Our second result is applicable even when the adversary is adaptive, and therefore implies the existence of a deterministic edge coloring algorithm which $(1+ε)Δ$ edge colors a dense graph. Prior to this, the best known deterministic algorithm for this problem was the simple greedy algorithm which utilized $2Δ-1$ colors.

cs.DS↗

arXiv2024

Matching Composition and Efficient Weight Reduction in Dynamic Matching

We consider the foundational problem of maintaining a $(1-\varepsilon)$-approximate maximum weight matching (MWM) in an $n$-node dynamic graph undergoing edge insertions and deletions. We provide a general reduction that reduces the problem on graphs with a weight range of $\mathrm{poly}(n)$ to $\mathrm{poly}(1/\varepsilon)$ at the cost of just an additive $\mathrm{poly}(1/\varepsilon)$ in update time. This improves upon the prior reduction of Gupta-Peng (FOCS 2013) which reduces the problem to a weight range of $\varepsilon^{-O(1/\varepsilon)}$ with a multiplicative cost of $O(\log n)$. When combined with a reduction of Bernstein-Dudeja-Langley (STOC 2021) this yields a reduction from dynamic $(1-\varepsilon)$-approximate MWM in bipartite graphs with a weight range of $\mathrm{poly}(n)$ to dynamic $(1-\varepsilon)$-approximate maximum cardinality matching in bipartite graphs at the cost of a multiplicative $\mathrm{poly}(1/\varepsilon)$ in update time, thereby resolving an open problem in [GP'13; BDL'21]. Additionally, we show that our approach is amenable to MWM problems in streaming, shared-memory work-depth, and massively parallel computation models. We also apply our techniques to obtain an efficient dynamic algorithm for rounding weighted fractional matchings in general graphs. Underlying our framework is a new structural result about MWM that we call the "matching composition lemma" and new dynamic matching subroutines that may be of independent interest.

cs.DS↗

arXiv2024

Evolution of locally dependent random graphs

In this paper we study $d$-dependent random graphs -- introduced by Brody and Sanchez -- which are the family of random graph distributions where each edge is present with probability $p$, and each edge is independent of all but at most $d$ other edges. For this random graph model, we analyze degree sequences, jumbledness, connectivity, and subgraph containment. Our results mirror those of the classical Erdős--Rényi random graph, which are recovered by specializing our problem to $d=0$, although we show that in many regards our setting is appreciably more nuanced. We survey what is known for this model and conclude with a variety of open questions.

math.CO↗

arXiv2024

A Note on Rounding Matchings in General Graphs

In this note, we revisit the rounding algorithm of Wajc. Wajc gave a fully-adaptive randomized algorithm that rounds a dynamic fractional matching in an unweighted bipartite graph to an integral matching of nearly the same value in $O(\text{poly}(\log n,\frac{1}{\varepsilon}))$ update time. We give show that the guarantees of this algorithm hold for general graphs as well. Additionally, we show useful properties of this subroutine which have applications in rounding weighted fractional matchings.

cs.DS↗

arXiv2023

Decremental Matching in General Weighted Graphs

In this paper, we consider the problem of maintaining a $(1-\varepsilon)$-approximate maximum weight matching in a dynamic graph $G$, while the adversary makes changes to the edges of the graph. In the fully dynamic setting, where both edge insertions and deletions are allowed, Gupta and Peng gave an algorithm for this problem with an update time of $\tilde{O}_{\varepsilon}(\sqrt{m})$. We study a natural relaxation of this problem, namely the decremental model, where the adversary is only allowed to delete edges. For the cardinality version of this problem in general (possibly, non-bipartite) graphs, Assadi, Bernstein, and Dudeja gave a decremental algorithm with update time $O_{\varepsilon}(\text{poly}(\log n))$. However, beating $\tilde{O}_{\varepsilon}(\sqrt{m})$ update time remained an open problem for the \emph{weighted} version in \emph{general graphs}. In this paper, we bridge the gap between unweighted and weighted general graphs for the decremental setting. We give a $O_{\varepsilon}(\text{poly}(\log n))$ update time algorithm that maintains a $(1-\varepsilon)$-approximate maximum weight matching under adversarial deletions. Like the decremental algorithm of Assadi, Bernstein, and Dudeja, our algorithm is randomized, but works against an adaptive adversary. It also matches the time bound for the cardinality version upto dependencies on $\varepsilon$ and a $\log R$ factor, where $R$ is the ratio between the maximum and minimum edge weight in $G$.

cs.DS↗

arXiv2022

Decremental Matching in General Graphs

We consider the problem of maintaining an approximate maximum integral matching in a dynamic graph $G$, while the adversary makes changes to the edges of the graph. The goal is to maintain a $(1+ε)$-approximate maximum matching for constant $ε>0$, while minimizing the update time. In the fully dynamic setting, where both edge insertion and deletions are allowed, Gupta and Peng (see \cite{GP13}) gave an algorithm for this problem with an update time of $O(\sqrt{m}/ε^2)$. Motivated by the fact that the $O_ε(\sqrt{m})$ barrier is hard to overcome (see Henzinger, Krinninger, Nanongkai, and Saranurak [HKNS15]); Kopelowitz, Pettie, and Porat [KPP16]), we study this problem in the \emph{decremental} model, where the adversary is only allowed to delete edges. Recently, Bernstein, Probst-Gutenberg, and Saranurak (see [BPT20]) gave an $O_ε(1)$ update time decremental algorithm for this problem in \emph{bipartite graphs}. However, beating $O(\sqrt{m})$ update time remained an open problem for \emph{general graphs}. In this paper, we bridge the gap between bipartite and general graphs, by giving an $O_ε(1)$ update time algorithm that maintains a $(1+ε)$-approximate maximum integral matching under adversarial deletions. Our algorithm is randomized, but works against an adaptive adversary. Together with the work of Grandoni, Leonardi, Sankowski, Schwiegelshohn, and Solomon [GLSSS19] who give an $O_ε(1)$ update time algorithm for general graphs in the \emph{incremental} (insertion-only) model, our result essentially completes the picture for partially dynamic matching.

cs.DS↗