arXiv subjects

Kheeran K. Naidu

Publications and source records attributed to Kheeran K. Naidu.

8 recordsLinked to original sources

arXiv2026

UltRAG: a Universal Simple Scalable Recipe for Knowledge Graph RAG

Large language models (LLMs) frequently generate confident yet factually incorrect content when used for language generation (a phenomenon often known as hallucination). Retrieval augmented generation (RAG) tries to reduce factual errors by identifying information in a knowledge corpus and putting it in the context window of the model. While this approach is well-established for document-structured data, it is non-trivial to adapt it for Knowledge Graphs (KGs), especially for queries that require multi-node/multi-hop reasoning on graphs. We introduce UltRAG, a training-free KG-RAG recipe that combines LLM query generation, a fully inductive neural query executor, and LLM arbitration. This off-the-shelf composition achieves state-of-the-art results on Knowledge Graph Question Answering (KGQA) tasks without retraining the LLM or executor, while enabling language models to interface with Wikidata-scale graphs (116M entities, 1.6B relations) at comparable or lower costs. Our ablation studies indicate that these gains come from the full system design rather than from any single component.

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arXiv2026

On the Assadi Liu Tarjan Auction Algorithm for Bipartite Matching: Simplification, Alternative Analysis, and Hard Instance

Assadi, Liu, and Tarjan [SOSA'21] gave an auction algorithm that outputs a $(1-ε)$-approximation to Maximum Matching in bipartite graphs. Their algorithm computes a sequence of $O(\frac{1}{ε^2})$ maximal matchings in subgraphs of the input graph and can be implemented in the multi-pass streaming setting with $O(\frac{1}{ε^2})$ passes in a straightforward manner, which constitutes the state-of-the-art pass/approximation trade-off result in the multi-pass streaming setting. Their analysis uses tools from combinatorial auctions and, at its heart, relies on a clever potential function argument. Their proof, however, provides only limited insight into the inner workings of the algorithm. In this paper, we revisit the ALT-algorithm and present the following contributions. Simplification: The ALT-algorithm is built upon a freezing mechanism where vertices on one side of the bipartition that have already been rematched $Θ(\frac{1}ε)$ times over the course of the algorithm remain matched to their current partner forever. We show that this mechanism is in fact unnecessary, i.e., no special treatment of such vertices is needed. Alternative Analysis: We give an alternative analysis of the algorithm that is based on augmenting paths. Our analysis allows for a reinterpretation as one that follows the traditional approach of searching for and eliminating augmenting paths. Our analysis also copes with the removal of the freezing mechanism in a natural way, whereas the analysis of Assadi et al. strictly depends on its use. Hard Instance: We provide the first hard instance on which the algorithm requires $Ω(\frac{1}{ε^2})$ iterations/maximal matching computations. The instance is a simple path graph, where we exhibit a cyclic behaviour that prevents fast progress.

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arXiv2026

Unit Interval Selection in Random Order Streams

We consider the \textsf{Unit Interval Selection} problem in the one-pass random order streaming model. Here, an algorithm is presented a sequence of $n$ unit-length intervals on the line that arrive in uniform random order, and the objective is to output a largest set of disjoint intervals using space linear in the size of an optimal solution. Previous work only considered adversarially ordered streams and established that, in this space constraint, a $(2/3)$-approximation can be achieved, and this is also best possible, i.e. any improvement requires space $Ω(n)$ [Emek et al., TALG'16]. In this work, we show that an improved expected approximation factor can be achieved if the input stream is in uniform random order, with the expectation taken over the stream order. Specifically, we give a one-pass streaming algorithm with expected approximation factor $0.7401$ using space $O(|OPT|)$, where $OPT$ denotes an optimal solution. We also show that algorithms with expected approximation factor above $8/9$ require space $Ω(n)$, and algorithms that compute a better than $2/3$-approximation with probability above $2/3$ also require $Ω(n)$ space. On a technical note, we design an algorithm for the restricted domain $[0,Δ)$, for some constant $Δ$, and use standard techniques to obtain an algorithm for unrestricted domains. For the restricted domain $[0,Δ)$, we run $O(Δ)$ recursive instances of our algorithm, with each instance targeting the situation where a specific interval from $OPT$ arrives first. We establish the interesting property that our algorithm performs worst when the input stream is precisely a set of independent intervals. We then analyse the algorithm on these instances. Our lower bound is proved via communication complexity arguments, similar in spirit to the robust communication lower bounds by [Chakrabarti et al., Theory Comput. 2016].

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arXiv2024

Settling the Pass Complexity of Approximate Matchings in Dynamic Graph Streams

A semi-streaming algorithm in dynamic graph streams processes any $n$-vertex graph by making one or multiple passes over a stream of insertions and deletions to edges of the graph and using $O(n \cdot \mbox{polylog}(n))$ space. Semi-streaming algorithms for dynamic streams were first obtained in the seminal work of Ahn, Guha, and McGregor in 2012, alongside the introduction of the graph sketching technique, which remains the de facto way of designing algorithms in this model and a highly popular technique for designing graph algorithms in general. We settle the pass complexity of approximating maximum matchings in dynamic streams via semi-streaming algorithms by improving the state-of-the-art in both upper and lower bounds. We present a randomized sketching based semi-streaming algorithm for $O(1)$-approximation of maximum matching in dynamic streams using $O(\log\log{n})$ passes. The approximation ratio of this algorithm can be improved to $(1+ε)$ for any fixed $ε> 0$ even on weighted graphs using standard techniques. This exponentially improves upon several $O(\log{n})$ pass algorithms developed for this problem since the introduction of the dynamic graph streaming model. In addition, we prove that any semi-streaming algorithm (not only sketching based) for $O(1)$-approximation of maximum matching in dynamic streams requires $Ω(\log\log{n})$ passes. This presents the first multi-pass lower bound for this problem, which is already also optimal, settling a longstanding open question in this area.

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arXiv2023

$\mathcal{O}(\log\log{n})$ Passes is Optimal for Semi-Streaming Maximal Independent Set

In the semi-streaming model for processing massive graphs, an algorithm makes multiple passes over the edges of a given $n$-vertex graph and is tasked with computing the solution to a problem using $O(n \cdot \text{polylog}(n))$ space. Semi-streaming algorithms for Maximal Independent Set (MIS) that run in $O(\log\log{n})$ passes have been known for almost a decade, however, the best lower bounds can only rule out single-pass algorithms. We close this large gap by proving that the current algorithms are optimal: Any semi-streaming algorithm for finding an MIS with constant probability of success requires $Ω(\log\log{n})$ passes. This settles the complexity of this fundamental problem in the semi-streaming model, and constitutes one of the first optimal multi-pass lower bounds in this model. We establish our result by proving an optimal round vs communication tradeoff for the (multi-party) communication complexity of MIS. The key ingredient of this result is a new technique, called hierarchical embedding, for performing round elimination: we show how to pack many but small hard $(r-1)$-round instances of the problem into a single $r$-round instance, in a way that enforces any $r$-round protocol to effectively solve all these $(r-1)$-round instances also. These embeddings are obtained via a novel application of results from extremal graph theory -- in particular dense graphs with many disjoint unique shortest paths -- together with a newly designed graph product, and are analyzed via information-theoretic tools such as direct-sum and message compression arguments.

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arXiv2023

Improved Weighted Matching in the Sliding Window Model

We consider the Maximum-weight Matching (MWM) problem in the streaming sliding window model of computation. In this model, the input consists of a sequence of weighted edges on a given vertex set $V$ of size $n$. The objective is to maintain an approximation of a maximum-weight matching in the graph spanned by the $L$ most recent edges, for some integer $L$, using as little space as possible. Prior to our work, the state-of-the-art results were a $(3.5+\varepsilon)$-approximation algorithm for MWM by Biabani et al. [ISAAC'21] and a $(3+\varepsilon)$-approximation for (unweighted) Maximum Matching (MM) by Crouch et al. [ESA'13]. Both algorithms use space $\tilde{O}(n)$. We give the following results: 1. We give a $(2+\varepsilon)$-approximation algorithm for MWM with space $\tilde{O}(\sqrt{nL})$. Under the reasonable assumption that the graphs spanned by the edges in each sliding window are simple, our algorithm uses space $\tilde{O}(n \sqrt{n})$. 2. In the $\tilde{O}(n)$ space regime, we give a $(3+\varepsilon)$-approximation algorithm for MWM, thereby closing the gap between the best-known approximation ratio for MWM and MM. Similar to Biabani et al.'s MWM algorithm, both our algorithms execute multiple instances of the $(2+\varepsilon)$-approximation $\tilde{O}(n)$-space streaming algorithm for MWM by Paz and Schwartzman [SODA'17] on different portions of the stream. Our improvements are obtained by selecting these substreams differently. Furthermore, our $(2+\varepsilon)$-approximation algorithm runs the Paz-Schwartzman algorithm in reverse direction over some parts of the stream, and in forward direction over other parts, which allows for an improved approximation guarantee at the cost of increased space requirements.

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arXiv2022

Space Optimal Vertex Cover in Dynamic Streams

We optimally resolve the space complexity for the problem of finding an $α$-approximate minimum vertex cover ($α$MVC) in dynamic graph streams. We give a randomised algorithm for $α$MVC which uses $O(n^2/α^2)$ bits of space matching Dark and Konrad's lower bound [CCC 2020] up to constant factors. By computing a random greedy matching, we identify `easy' instances of the problem which can trivially be solved by returning the entire vertex set. The remaining `hard' instances, then have sparse induced subgraphs which we exploit to get our space savings and solve $α$MVC. Achieving this type of optimality result is crucial for providing a complete understanding of a problem, and it has been gaining interest within the dynamic graph streaming community. For connectivity, Nelson and Yu [SODA 2019] improved the lower bound showing that $Ω(n \log^3 n)$ bits of space is necessary while Ahn, Guha, and McGregor [SODA 2012] have shown that $O(n \log^3 n)$ bits is sufficient. For finding an $α$-approximate maximum matching, the upper bound was improved by Assadi and Shah [ITCS 2022] showing that $O(n^2/α^3)$ bits is sufficient while Dark and Konrad [CCC 2020] have shown that $Ω(n^2/α^3)$ bits is necessary. The space complexity, however, remains unresolved for many other dynamic graph streaming problems where further improvements can still be made. \end{abstract}

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arXiv2021

On Two-Pass Streaming Algorithms for Maximum Bipartite Matching

We study two-pass streaming algorithms for Maximum Bipartite Matching (MBM). All known two-pass streaming algorithms for MBM operate in a similar fashion: They compute a maximal matching in the first pass and find 3-augmenting paths in the second in order to augment the matching found in the first pass. Our aim is to explore the limitations of this approach and to determine whether current techniques can be used to further improve the state-of-the-art algorithms. We give the following results: We show that every two-pass streaming algorithm that solely computes a maximal matching in the first pass and outputs a $(2/3+ε)$-approximation requires $n^{1+Ω(\frac{1}{\log \log n})}$ space, for every $ε> 0$, where $n$ is the number of vertices of the input graph. This result is obtained by extending the Ruzsa-Szemerédi graph construction of [GKK, SODA'12] so as to ensure that the resulting graph has a close to perfect matching, the key property needed in our construction. This result may be of independent interest. Furthermore, we combine the two main techniques, i.e., subsampling followed by the Greedy matching algorithm [Konrad, MFCS'18] which gives a $2-\sqrt{2} \approx 0.5857$-approximation, and the computation of \emph{degree-bounded semi-matchings} [EHM, ICDMW'16][KT, APPROX'17] which gives a $\frac{1}{2} + \frac{1}{12} \approx 0.5833$-approximation, and obtain a meta-algorithm that yields Konrad's and Esfandiari et al.'s algorithms as special cases. This unifies two strands of research. By optimizing parameters, we discover that Konrad's algorithm is optimal for the implied class of algorithms and, perhaps surprisingly, that there is a second optimal algorithm. We show that the analysis of our meta-algorithm is best possible. Our results imply that further improvements, if possible, require new techniques.

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