11 Search Results for "Tardos, Jakab"

Document
DOI: 10.4230/LIPIcs.STACS.2026.38
Fully Dynamic Spectral Sparsification for Directed Hypergraphs

Authors: Sebastian Forster, Gramoz Goranci, and Ali Momeni

Published in: LIPIcs, Volume 364, 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)

Abstract
There has been a surge of interest in spectral hypergraph sparsification, a natural generalization of spectral sparsification for graphs. In this paper, we present a simple fully dynamic algorithm for maintaining spectral hypergraph sparsifiers of directed hypergraphs. Our algorithm achieves a near-optimal size of O(n² / ε ² log ⁷ m) and amortized update time of O(r² log ³ m), where n is the number of vertices, and m and r respectively upper bound the number of hyperedges and the rank of the hypergraph at any time. We also extend our approach to the parallel batch-dynamic setting, where a batch of any k hyperedge insertions or deletions can be processed with O(kr² log ³ m) amortized work and O(log ² m) depth. This constitutes the first spectral-based sparsification algorithm in this setting.
Cite as

Sebastian Forster, Gramoz Goranci, and Ali Momeni. Fully Dynamic Spectral Sparsification for Directed Hypergraphs. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 38:1-38:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{forster_et_al:LIPIcs.STACS.2026.38,
  author =	{Forster, Sebastian and Goranci, Gramoz and Momeni, Ali},
  title =	{{Fully Dynamic Spectral Sparsification for Directed Hypergraphs}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{38:1--38:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-412-3},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{364},
  editor =	{Mahajan, Meena and Manea, Florin and McIver, Annabelle and Thắng, Nguy\~{ê}n Kim},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2026.38},
  URN =		{urn:nbn:de:0030-drops-255272},
  doi =		{10.4230/LIPIcs.STACS.2026.38},
  annote =	{Keywords: Spectral sparsification, Dynamic algorithms, (Directed) hypergraphs, Data structures}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.4
Streaming Algorithms for Network Design

Authors: Chandra Chekuri, Rhea Jain, Sepideh Mahabadi, and Ali Vakilian

Published in: LIPIcs, Volume 353, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)

Abstract
We consider the Survivable Network Design problem (SNDP) in the single-pass insertion-only streaming model. The input to SNDP is an edge-weighted graph G = (V, E) and an integer connectivity requirement r(uv) for each u, v ∈ V. The objective is to find a minimum-weight subgraph H ⊆ G such that, for every pair of vertices u, v ∈ V, u and v are r(uv)-edge/vertex-connected. Recent work by [Ce Jin et al., 2024] obtained approximation algorithms for edge-connectivity augmentation, and via that, also derived algorithms for edge-connectivity SNDP (EC-SNDP). In this work we consider vertex-connectivity setting (VC-SNDP) and obtain several results for it as well as improved results for EC-SNDP. - We provide a general framework for solving connectivity problems including SNDP and others in streaming; this is based on a connection to fault-tolerant spanners. For VC-SNDP we provide an O(tk)-approximation in Õ(k^{1-1/t}n^{1 + 1/t}) space, where k is the maximum connectivity requirement, assuming an exact algorithm at the end of the stream. Using a refined LP-based analysis, we provide an O(β t)-approximation where β is the integrality gap of the natural cut-based LP relaxation. These are the first approximation algorithms in the streaming model for VC-SNDP. When applied to the EC-SNDP, our framework provides an O(t)-approximation in Õ(k^{1/2-1/(2t)}n^{1 + 1/t} + kn) space, improving the O(t log k)-approximation of [Ce Jin et al., 2024] using Õ(kn^{1+1/t}) space; this also extends to element-connectivity SNDP. - We consider vertex connectivity-augmentation in the link-arrival model. The input is a k-vertex-connected spanning subgraph G, and additional weighted links L arrive in the stream; the goal is to store the min-weight set of links such that G ∪ L is (k+1)-vertex-connected. We obtain constant-factor approximations in near-linear space for k = 1, 2. Our result for k = 2 is based on using the SPQR tree, a novel application for this well-known representation of 2-connected graphs.
Cite as

Chandra Chekuri, Rhea Jain, Sepideh Mahabadi, and Ali Vakilian. Streaming Algorithms for Network Design. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 4:1-4:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{chekuri_et_al:LIPIcs.APPROX/RANDOM.2025.4,
  author =	{Chekuri, Chandra and Jain, Rhea and Mahabadi, Sepideh and Vakilian, Ali},
  title =	{{Streaming Algorithms for Network Design}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{4:1--4:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-397-3},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{353},
  editor =	{Ene, Alina and Chattopadhyay, Eshan},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2025.4},
  URN =		{urn:nbn:de:0030-drops-243709},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.4},
  annote =	{Keywords: Streaming Algorithms, Survivable Network Design, Fault-Tolerant Spanners}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.107
A Theory of Spectral CSP Sparsification

Authors: Sanjeev Khanna, Aaron Putterman, and Madhu Sudan

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We initiate the study of spectral sparsification for instances of Constraint Satisfaction Problems (CSPs). In particular, we introduce a notion of the spectral energy of a fractional assignment for a Boolean CSP instance, and define a spectral sparsifier as a weighted subset of constraints that approximately preserves this energy for all fractional assignments. Our definition not only strengthens the combinatorial notion of a CSP sparsifier but also extends well-studied concepts such as spectral sparsifiers for graphs and hypergraphs. Recent work by Khanna, Putterman, and Sudan [SODA 2024] demonstrated near-linear sized combinatorial sparsifiers for a broad class of CSPs, which they term field-affine CSPs. Our main result is a polynomial-time algorithm that constructs a spectral CSP sparsifier of near-quadratic size for all field-affine CSPs. This class of CSPs includes graph (and hypergraph) cuts, XORs, and more generally, any predicate which can be written as P(x₁, … x_r) = 𝟏[∑ a_i x_i ≠ b mod p]. Based on our notion of the spectral energy of a fractional assignment, we also define an analog of the second eigenvalue of a CSP instance. We then show an extension of Cheeger’s inequality for all even-arity XOR CSPs, showing that this second eigenvalue loosely captures the "expansion" of the underlying CSP. This extension specializes to the case of Cheeger’s inequality when all constraints are even XORs and thus gives a new generalization of this powerful inequality which converts the combinatorial notion of expansion to an analytic property. Perhaps the most important effect of spectral sparsification is that it has led to certifiable sparsifiers for graphs and hypergraphs. This aspect remains open in our case even for XOR CSPs since the eigenvalues we describe in our Cheeger inequality are not known to be efficiently computable. Computing this efficiently, and/or finding other ways to certifiably sparsify CSPs are open questions emerging from our work. Another important open question is determining which classes of CSPs have near-linear size spectral sparsifiers.
Cite as

Sanjeev Khanna, Aaron Putterman, and Madhu Sudan. A Theory of Spectral CSP Sparsification. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 107:1-107:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{khanna_et_al:LIPIcs.ICALP.2025.107,
  author =	{Khanna, Sanjeev and Putterman, Aaron and Sudan, Madhu},
  title =	{{A Theory of Spectral CSP Sparsification}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{107:1--107:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.107},
  URN =		{urn:nbn:de:0030-drops-234840},
  doi =		{10.4230/LIPIcs.ICALP.2025.107},
  annote =	{Keywords: Sparsification, sketching, hypergraphs}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.108
Near-Optimal Hypergraph Sparsification in Insertion-Only and Bounded-Deletion Streams

Authors: Sanjeev Khanna, Aaron Putterman, and Madhu Sudan

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We study the problem of constructing hypergraph cut sparsifiers in the streaming model where a hypergraph on n vertices is revealed either via an arbitrary sequence of hyperedge insertions alone (insertion-only streaming model) or via an arbitrary sequence of hyperedge insertions and deletions (dynamic streaming model). For any ε ∈ (0,1), a (1 ± ε) hypergraph cut-sparsifier of a hypergraph H is a reweighted subgraph H' whose cut values approximate those of H to within a (1 ± ε) factor. Prior work shows that in the static setting, one can construct a (1 ± ε) hypergraph cut-sparsifier using Õ(nr/ε²) bits of space [Chen-Khanna-Nagda FOCS 2020], and in the setting of dynamic streams using Õ(nrlog m/ε²) bits of space [Khanna-Putterman-Sudan FOCS 2024]; here the Õ notation hides terms that are polylogarithmic in n, and we use m to denote the total number of hyperedges in the hypergraph. Up until now, the best known space complexity for insertion-only streams has been the same as that for the dynamic streams. This naturally poses the question of understanding the complexity of hypergraph sparsification in insertion-only streams. Perhaps surprisingly, in this work we show that in insertion-only streams, a (1 ± ε) cut-sparsifier can be computed in Õ(nr/ε²) bits of space, matching the complexity of the static setting. As a consequence, this also establishes an Ω(log m) factor separation between the space complexity of hypergraph cut sparsification in insertion-only streams and dynamic streams, as the latter is provably known to require Ω(nr log m) bits of space. To better explain this gap, we then show a more general result: namely, if the stream has at most k hyperedge deletions then Õ(n r log k/ε²) bits of space suffice for hypergraph cut sparsification. Thus the space complexity smoothly interpolates between the insertion-only regime (k = 0) and the fully dynamic regime (k = m). Our algorithmic results are driven by a key technical insight: once sufficiently many hyperedges have been inserted into the stream (relative to the number of allowed deletions), we can significantly reduce the underlying hypergraph by size by irrevocably contracting large subsets of vertices. Finally, we complement this result with an essentially matching lower bound of Ω(n r log(k/n)) bits, thus providing essentially a tight characterization of the space complexity for hypergraph cut-sparsification across a spectrum of streaming models.
Cite as

Sanjeev Khanna, Aaron Putterman, and Madhu Sudan. Near-Optimal Hypergraph Sparsification in Insertion-Only and Bounded-Deletion Streams. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 108:1-108:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{khanna_et_al:LIPIcs.ICALP.2025.108,
  author =	{Khanna, Sanjeev and Putterman, Aaron and Sudan, Madhu},
  title =	{{Near-Optimal Hypergraph Sparsification in Insertion-Only and Bounded-Deletion Streams}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{108:1--108:11},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.108},
  URN =		{urn:nbn:de:0030-drops-234851},
  doi =		{10.4230/LIPIcs.ICALP.2025.108},
  annote =	{Keywords: Sparsification, sketching, hypergraphs}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.119
Faster Semi-Streaming Matchings via Alternating Trees

Authors: Slobodan Mitrović, Anish Mukherjee, Piotr Sankowski, and Wen-Horng Sheu

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We design a deterministic algorithm for the (1+ε)-approximate maximum matching problem. Our primary result demonstrates that this problem can be solved in O(ε^{-6}) semi-streaming passes, improving upon the O(ε^{-19}) pass-complexity algorithm by [Fischer, Mitrović, and Uitto, STOC'22]. This contributes substantially toward resolving Open question 2 from [Assadi, SOSA'24]. Leveraging the framework introduced in [FMU'22], our algorithm achieves an analogous round complexity speed-up for computing a (1+ε)-approximate maximum matching in both the Massively Parallel Computation (MPC) and CONGEST models. The data structures maintained by our algorithm are formulated using blossom notation and represented through alternating trees. This approach enables a simplified correctness analysis by treating specific components as if operating on bipartite graphs, effectively circumventing certain technical intricacies present in prior work.
Cite as

Slobodan Mitrović, Anish Mukherjee, Piotr Sankowski, and Wen-Horng Sheu. Faster Semi-Streaming Matchings via Alternating Trees. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 119:1-119:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mitrovic_et_al:LIPIcs.ICALP.2025.119,
  author =	{Mitrovi\'{c}, Slobodan and Mukherjee, Anish and Sankowski, Piotr and Sheu, Wen-Horng},
  title =	{{Faster Semi-Streaming Matchings via Alternating Trees}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{119:1--119:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.119},
  URN =		{urn:nbn:de:0030-drops-234965},
  doi =		{10.4230/LIPIcs.ICALP.2025.119},
  annote =	{Keywords: streaming algorithms, approximation algorithms, maximum matching}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.116
A 0.51-Approximation of Maximum Matching in Sublinear n^{1.5} Time

Authors: Sepideh Mahabadi, Mohammad Roghani, and Jakub Tarnawski

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
We study the problem of estimating the size of a maximum matching in sublinear time. The problem has been studied extensively in the literature and various algorithms and lower bounds are known for it. Our result is a 0.5109-approximation algorithm with a running time of Õ(n√n). All previous algorithms either provide only a marginal improvement (e.g., 2^{-280}) over the 0.5-approximation that arises from estimating a maximal matching, or have a running time that is nearly n². Our approach is also arguably much simpler than other algorithms beating 0.5-approximation.
Cite as

Sepideh Mahabadi, Mohammad Roghani, and Jakub Tarnawski. A 0.51-Approximation of Maximum Matching in Sublinear n^{1.5} Time. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 116:1-116:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mahabadi_et_al:LIPIcs.ICALP.2025.116,
  author =	{Mahabadi, Sepideh and Roghani, Mohammad and Tarnawski, Jakub},
  title =	{{A 0.51-Approximation of Maximum Matching in Sublinear n^\{1.5\} Time}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{116:1--116:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.116},
  URN =		{urn:nbn:de:0030-drops-234932},
  doi =		{10.4230/LIPIcs.ICALP.2025.116},
  annote =	{Keywords: Sublinear Algorithms, Maximum Matching, Maximal Matching, Approximation Algorithm}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.89
Optimal Communication Complexity of Chained Index

Authors: Janani Sundaresan

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
We study the chain communication problem introduced by Cormode et al. [ICALP 2019]. For k ≥ 1, in the chain_{n,k} problem, there are k string and index pairs (X_i, σ_i) for i ∈ [k] such that the value at position σ_i in string X_i is the same bit for all k pairs. The input is shared between k+1 players as follows. Player 1 has the first string X₁ ∈ {0,1}ⁿ, player 2 has the first index σ₁ ∈ [n] and the second string X₂ ∈ {0,1}ⁿ, player 3 has the second index σ₂ ∈ [n] along with the third string X₃ ∈ {0,1}ⁿ, and so on. Player k+1 has the last index σ_k ∈ [n]. The communication is one way from each player to the next, starting from player 1 to player 2, then from player 2 to player 3 and so on. Player k+1, after receiving the message from player k, has to output a single bit which is the value at position σ_i in X_i for any i ∈ [k]. It is a generalization of the well-studied index problem, which is equivalent to chain_{n, 2}. Cormode et al. proved that the chain_{n,k} problem requires Ω(n/k²) communication, and they used it to prove streaming lower bounds for the approximation of maximum independent sets. Subsequently, Feldman et al. [STOC 2020] used it to prove lower bounds for streaming submodular maximization. However, it is not known whether the Ω(n/k²) lower bound used in these works is optimal for the problem, and in fact, it was conjectured by Cormode et al. that Ω(n) bits are necessary. We prove the optimal lower bound of Ω(n) for chain_{n,k} when k = o(n/log n) as our main result. This settles the open conjecture of Cormode et al., barring the range of k = Ω(n /log n). The main technique is a reduction to a non-standard index problem where the input to the players is such that the answer is biased away from uniform. This biased version of index is analyzed using tools from information theory. As a corollary, we get an improved lower bound for approximation of maximum independent set in vertex arrival streams via a reduction from chain directly.
Cite as

Janani Sundaresan. Optimal Communication Complexity of Chained Index. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 89:1-89:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{sundaresan:LIPIcs.ITCS.2025.89,
  author =	{Sundaresan, Janani},
  title =	{{Optimal Communication Complexity of Chained Index}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{89:1--89:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.89},
  URN =		{urn:nbn:de:0030-drops-227172},
  doi =		{10.4230/LIPIcs.ITCS.2025.89},
  annote =	{Keywords: communication complexity, index communciation problem}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.43
Space Complexity of Minimum Cut Problems in Single-Pass Streams

Authors: Matthew Ding, Alexandro Garces, Jason Li, Honghao Lin, Jelani Nelson, Vihan Shah, and David P. Woodruff

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
We consider the problem of finding a minimum cut of a weighted graph presented as a single-pass stream. While graph sparsification in streams has been intensively studied, the specific application of finding minimum cuts in streams is less well-studied. To this end, we show upper and lower bounds on minimum cut problems in insertion-only streams for a variety of settings, including for both randomized and deterministic algorithms, for both arbitrary and random order streams, and for both approximate and exact algorithms. One of our main results is an Õ(n/ε) space algorithm with fast update time for approximating a spectral cut query with high probability on a stream given in an arbitrary order. Our result breaks the Ω(n/ε²) space lower bound required of a sparsifier that approximates all cuts simultaneously. Using this result, we provide streaming algorithms with near optimal space of Õ(n/ε) for minimum cut and approximate all-pairs effective resistances, with matching space lower-bounds. The amortized update time of our algorithms is Õ(1), provided that the number of edges in the input graph is at least (n/ε²)^{1+o(1)}. We also give a generic way of incorporating sketching into a recursive contraction algorithm to improve the post-processing time of our algorithms. In addition to these results, we give a random-order streaming algorithm that computes the exact minimum cut on a simple, unweighted graph using Õ(n) space. Finally, we give an Ω(n/ε²) space lower bound for deterministic minimum cut algorithms which matches the best-known upper bound up to polylogarithmic factors.
Cite as

Matthew Ding, Alexandro Garces, Jason Li, Honghao Lin, Jelani Nelson, Vihan Shah, and David P. Woodruff. Space Complexity of Minimum Cut Problems in Single-Pass Streams. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 43:1-43:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{ding_et_al:LIPIcs.ITCS.2025.43,
  author =	{Ding, Matthew and Garces, Alexandro and Li, Jason and Lin, Honghao and Nelson, Jelani and Shah, Vihan and Woodruff, David P.},
  title =	{{Space Complexity of Minimum Cut Problems in Single-Pass Streams}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{43:1--43:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.43},
  URN =		{urn:nbn:de:0030-drops-226714},
  doi =		{10.4230/LIPIcs.ITCS.2025.43},
  annote =	{Keywords: minimum cut, approximate, random order, lower bound}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.44
Learning-Augmented Streaming Algorithms for Approximating MAX-CUT

Authors: Yinhao Dong, Pan Peng, and Ali Vakilian

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
We study learning-augmented streaming algorithms for estimating the value of MAX-CUT in a graph. In the classical streaming model, while a 1/2-approximation for estimating the value of MAX-CUT can be trivially achieved with O(1) words of space, Kapralov and Krachun [STOC’19] showed that this is essentially the best possible: for any ε > 0, any (randomized) single-pass streaming algorithm that achieves an approximation ratio of at least 1/2 + ε requires Ω(n / 2^poly(1/ε)) space. We show that it is possible to surpass the 1/2-approximation barrier using just O(1) words of space by leveraging a (machine learned) oracle. Specifically, we consider streaming algorithms that are equipped with an ε-accurate oracle that for each vertex in the graph, returns its correct label in {-1, +1}, corresponding to an optimal MAX-CUT solution in the graph, with some probability 1/2 + ε, and the incorrect label otherwise. Within this framework, we present a single-pass algorithm that approximates the value of MAX-CUT to within a factor of 1/2 + Ω(ε²) with probability at least 2/3 for insertion-only streams, using only poly(1/ε) words of space. We also extend our algorithm to fully dynamic streams while maintaining a space complexity of poly(1/ε,log n) words.
Cite as

Yinhao Dong, Pan Peng, and Ali Vakilian. Learning-Augmented Streaming Algorithms for Approximating MAX-CUT. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 44:1-44:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dong_et_al:LIPIcs.ITCS.2025.44,
  author =	{Dong, Yinhao and Peng, Pan and Vakilian, Ali},
  title =	{{Learning-Augmented Streaming Algorithms for Approximating MAX-CUT}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{44:1--44:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.44},
  URN =		{urn:nbn:de:0030-drops-226728},
  doi =		{10.4230/LIPIcs.ITCS.2025.44},
  annote =	{Keywords: Learning-Augmented Algorithms, Graph Streaming Algorithms, MAX-CUT}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.74
Sublinear Metric Steiner Tree via Improved Bounds for Set Cover

Authors: Sepideh Mahabadi, Mohammad Roghani, Jakub Tarnawski, and Ali Vakilian

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
We study the metric Steiner tree problem in the sublinear query model. In this problem, for a set of n points V in a metric space given to us by means of query access to an n× n matrix w, and a set of terminals T ⊆ V, the goal is to find the minimum-weight subset of the edges that connects all the terminal vertices. Recently, Chen, Khanna and Tan [SODA'23] gave an algorithm that uses Õ(n^{13/7}) queries and outputs a (2-η)-estimate of the metric Steiner tree weight, where η > 0 is a universal constant. A key component in their algorithm is a sublinear algorithm for a particular set cover problem where, given a set system (𝒰, ℱ), the goal is to provide a multiplicative-additive estimate for |𝒰|-SC(𝒰, ℱ). Here 𝒰 is the set of elements, ℱ is the collection of sets, and SC(𝒰, ℱ) denotes the optimal set cover size of (𝒰, ℱ). In particular, their algorithm returns a (1/4, ε⋅|𝒰|)-multiplicative-additive estimate for this set cover problem using Õ(|ℱ|^{7/4}) membership oracle queries (querying whether a set S ∈ 𝒮 contains an element e ∈ 𝒰), where ε is a fixed constant. In this work, we improve the query complexity of (2-η)-estimating the metric Steiner tree weight to Õ(n^{5/3}) by showing a (1/2, ε⋅|𝒰|)-estimate for the above set cover problem using Õ(|ℱ|^{5/3}) membership queries. To design our set cover algorithm, we estimate the size of a random greedy maximal matching for an auxiliary multigraph that the algorithm constructs implicitly, without access to its adjacency list or matrix. Previous analyses of random greedy maximal matching have focused on simple graphs, assuming access to their adjacency list or matrix. To address this, we extend the analysis of Behnezhad [FOCS'21] of random greedy maximal matching on simple graphs to multigraphs, and prove additional properties that may be of independent interest.
Cite as

Sepideh Mahabadi, Mohammad Roghani, Jakub Tarnawski, and Ali Vakilian. Sublinear Metric Steiner Tree via Improved Bounds for Set Cover. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 74:1-74:24, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{mahabadi_et_al:LIPIcs.ITCS.2025.74,
  author =	{Mahabadi, Sepideh and Roghani, Mohammad and Tarnawski, Jakub and Vakilian, Ali},
  title =	{{Sublinear Metric Steiner Tree via Improved Bounds for Set Cover}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{74:1--74:24},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.74},
  URN =		{urn:nbn:de:0030-drops-227029},
  doi =		{10.4230/LIPIcs.ITCS.2025.74},
  annote =	{Keywords: Sublinear Algorithms, Steiner Tree, Set Cover, Maximum Matching, Approximation Algorithm}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2022.91
Noisy Boolean Hidden Matching with Applications

Authors: Michael Kapralov, Amulya Musipatla, Jakab Tardos, David P. Woodruff, and Samson Zhou

Published in: LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Abstract
The Boolean Hidden Matching (BHM) problem, introduced in a seminal paper of Gavinsky et al. [STOC'07], has played an important role in lower bounds for graph problems in the streaming model (e.g., subgraph counting, maximum matching, MAX-CUT, Schatten p-norm approximation). The BHM problem typically leads to Ω(√n) space lower bounds for constant factor approximations, with the reductions generating graphs that consist of connected components of constant size. The related Boolean Hidden Hypermatching (BHH) problem provides Ω(n^{1-1/t}) lower bounds for 1+O(1/t) approximation, for integers t ≥ 2. The corresponding reductions produce graphs with connected components of diameter about t, and essentially show that long range exploration is hard in the streaming model with an adversarial order of updates. In this paper we introduce a natural variant of the BHM problem, called noisy BHM (and its natural noisy BHH variant), that we use to obtain stronger than Ω(√n) lower bounds for approximating a number of the aforementioned problems in graph streams when the input graphs consist only of components of diameter bounded by a fixed constant. We next introduce and study the graph classification problem, where the task is to test whether the input graph is isomorphic to a given graph. As a first step, we use the noisy BHM problem to show that the problem of classifying whether an underlying graph is isomorphic to a complete binary tree in insertion-only streams requires Ω(n) space, which seems challenging to show using either BHM or BHH.
Cite as

Michael Kapralov, Amulya Musipatla, Jakab Tardos, David P. Woodruff, and Samson Zhou. Noisy Boolean Hidden Matching with Applications. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 91:1-91:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{kapralov_et_al:LIPIcs.ITCS.2022.91,
  author =	{Kapralov, Michael and Musipatla, Amulya and Tardos, Jakab and Woodruff, David P. and Zhou, Samson},
  title =	{{Noisy Boolean Hidden Matching with Applications}},
  booktitle =	{13th Innovations in Theoretical Computer Science Conference (ITCS 2022)},
  pages =	{91:1--91:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-217-4},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{215},
  editor =	{Braverman, Mark},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.91},
  URN =		{urn:nbn:de:0030-drops-156876},
  doi =		{10.4230/LIPIcs.ITCS.2022.91},
  annote =	{Keywords: Boolean Hidden Matching, Lower Bounds, Communication Complexity, Streaming Algorithms}
}
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