13 Search Results for "Sinop, Ali Kemal"

Document
DOI: 10.4230/LIPIcs.SoCG.2026.33
Near-Optimal Bounds for Parameterized Euclidean k-Means

Authors: Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
The k-means problem is a classic objective for modeling clustering in a metric space. Given a set of points in a metric space, the goal is to find k representative points so as to minimize the sum of the squared distances from each point to its closest representative. In this work, we study the approximability of k-means in Euclidean spaces parameterized by the number of clusters, k. In seminal works, de la Vega, Karpinski, Kenyon, and Rabani [STOC'03] and Kumar, Sabharwal, and Sen [JACM'10] showed how to obtain a (1+ε)-approximation for high-dimensional Euclidean k-means in time 2^{(k/ε)^O(1)} ⋅ dn^O(1). In this work, we introduce a new fine-grained hypothesis called Exponential Time for Expanders Hypothesis (XXH) which roughly asserts that there are no non-trivial exponential time approximation algorithms for the vertex cover problem on near perfect vertex expanders. Assuming XXH, we close the above long line of work on approximating Euclidean k-means by showing that there is no 2^{(k/ε)^{1-o(1)}} ⋅ n^O(1) time algorithm achieving a (1+ε)-approximation for k-means in Euclidean space. This lower bound is tight as it matches the algorithm given by Feldman, Monemizadeh, and Sohler [SoCG'07] whose runtime is 2^O(k/ε) + O(ndk). Furthermore, assuming XXH, we show that the seminal O(n^{kd+1}) runtime exact algorithm of Inaba, Katoh, and Imai [SoCG'94] for k-means is optimal for small values of k.
Cite as

Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn. Near-Optimal Bounds for Parameterized Euclidean k-Means. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cohenaddad_et_al:LIPIcs.SoCG.2026.33,
  author =	{Cohen-Addad, Vincent and C. S., Karthik and Saulpic, David and Schwiegelshohn, Chris},
  title =	{{Near-Optimal Bounds for Parameterized Euclidean k-Means}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{33:1--33:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.33},
  URN =		{urn:nbn:de:0030-drops-258391},
  doi =		{10.4230/LIPIcs.SoCG.2026.33},
  annote =	{Keywords: k-means clustering, Euclidean space, Fine-Grained Complexity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.34
Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces

Authors: Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
The k-median and k-means clustering objectives are classic objectives for modeling clustering in a metric space. Given a set of points in a metric space, the goal of the k-median (resp. k-means) problem is to find k representative points so as to minimize the sum of the distances (resp. sum of squared distances) from each point to its closest representative. Cohen-Addad, Feldmann, and Saulpic [JACM'21] showed how to obtain a (1+ε)-factor approximation in low-dimensional Euclidean metric for both the k-median and k-means problems in near-linear time 2^{(1/ε)^O(d²)} n ⋅ polylog(n) (where d is the dimension and n is the number of input points). We improve this running time to 2^{O(1/ε)^{d-1}} ⋅ n ⋅ polylog(n), and show an almost matching lower bound: under the Gap Exponential Time Hypothesis for 3-SAT, there is no 2^o(1/ε^{d-1}) n^O(1) algorithm achieving a (1+ε)-approximation for k-means.
Cite as

Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn. Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 34:1-34:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{cohenaddad_et_al:LIPIcs.SoCG.2026.34,
  author =	{Cohen-Addad, Vincent and Karthik C. S. and Saulpic, David and Schwiegelshohn, Chris},
  title =	{{Almost-Optimal Upper and Lower Bounds for Clustering in Low Dimensional Euclidean Spaces}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{34:1--34:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.34},
  URN =		{urn:nbn:de:0030-drops-258404},
  doi =		{10.4230/LIPIcs.SoCG.2026.34},
  annote =	{Keywords: k-means clustering, k-median clustering, Euclidean space, Fine-Grained Complexity}
}
Document
DOI: 10.4230/LIPIcs.SoCG.2026.48
FPT Approximations for Capacitated Sum of Radii and Diameters

Authors: Arnold Filtser and Ameet Gadekar

Published in: LIPIcs, Volume 367, 42nd International Symposium on Computational Geometry (SoCG 2026)

Abstract
The Capacitated Sum of Radii problem involves partitioning a set of points P, where each point p ∈ P has capacity U_p, into k clusters that minimize the sum of cluster radii, such that the number of points in the cluster centered at point p is at most U_p. We begin by showing that the problem is APX-hard, and that under gap-ETH there is no parameterized approximation scheme (FPT-AS). We then construct a ≈5.83-approximation algorithm in FPT time (improving a previous ≈7.61 approximation in FPT time). Our results also hold when the objective is a general monotone symmetric norm of radii. We also improve the approximation factors for the uniform capacity case, and for the closely related problem of Capacitated Sum of Diameters.
Cite as

Arnold Filtser and Ameet Gadekar. FPT Approximations for Capacitated Sum of Radii and Diameters. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 48:1-48:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{filtser_et_al:LIPIcs.SoCG.2026.48,
  author =	{Filtser, Arnold and Gadekar, Ameet},
  title =	{{FPT Approximations for Capacitated Sum of Radii and Diameters}},
  booktitle =	{42nd International Symposium on Computational Geometry (SoCG 2026)},
  pages =	{48:1--48:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-418-5},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{367},
  editor =	{Ahn, Hee-Kap and Hoffmann, Michael and Nayyeri, Amir},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2026.48},
  URN =		{urn:nbn:de:0030-drops-258545},
  doi =		{10.4230/LIPIcs.SoCG.2026.48},
  annote =	{Keywords: clustering, sum of radii, sum of diameter, capacitated clustering, fpt}
}
Document
DOI: 10.4230/LIPIcs.STACS.2026.69
Approximating q → p Norms of Non-Negative Matrices in Nearly-Linear Time

Authors: Etienne Objois and Adrian Vladu

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

Abstract
We provide the first nearly-linear time algorithm for approximating 𝓁_{q → p}-norms of non-negative matrices, for q ≥ p ≥ 1. Our algorithm returns a (1-ε)-approximation to the matrix norm in time Õ(1/(q ε) ⋅ nnz(A)), where A is the input matrix, and improves upon the previous state of the art, which either proved convergence only in the limit [Boyd '74], or had very high polynomial running times [Bhaskara-Vijayraghavan, SODA '11]. Our algorithm is extremely simple, and is largely inspired from the coordinate-scaling approach used for positive linear program solvers. Our algorithm can readily be used in the [Englert-Räcke, FOCS '09] to improve the running time of constructing O(log n)-competitive 𝓁_p-oblivious routings.
Cite as

Etienne Objois and Adrian Vladu. Approximating q → p Norms of Non-Negative Matrices in Nearly-Linear Time. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 69:1-69:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{objois_et_al:LIPIcs.STACS.2026.69,
  author =	{Objois, Etienne and Vladu, Adrian},
  title =	{{Approximating q → p Norms of Non-Negative Matrices in Nearly-Linear Time}},
  booktitle =	{43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026)},
  pages =	{69:1--69:19},
  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.2026.69},
  URN =		{urn:nbn:de:0030-drops-255585},
  doi =		{10.4230/LIPIcs.STACS.2026.69},
  annote =	{Keywords: matrix norm, Perron-Frobenius theory, oblivious routings, input-sparsity time, lp norm}
}
Document
DOI: 10.4230/LIPIcs.DISC.2025.26
Towards Constant Time Multi-Call Rumor Spreading on Small-Set Expanders

Authors: Emilio Cruciani, Sebastian Forster, and Tijn de Vos

Published in: LIPIcs, Volume 356, 39th International Symposium on Distributed Computing (DISC 2025)

Abstract
We study a multi-call variant of the classic PUSH&PULL rumor spreading process where nodes can contact k of their neighbors instead of a single one during both PUSH and PULL operations. We show that rumor spreading can be made faster at the cost of an increased amount of communication between the nodes. As a motivating example, consider the process on a complete graph of n nodes: while the standard PUSH&PULL protocol takes Θ(log n) rounds, we prove that our k-PUSH&PULL variant completes in Θ(log_{k} n) rounds, with high probability. We generalize this result in an expansion-sensitive way, as has been done for the classic PUSH&PULL protocol for different notions of expansion, e.g., conductance and vertex expansion. We consider small-set vertex expanders, graphs in which every sufficiently small subset of nodes has a large neighborhood, ensuring strong local connectivity. In particular, when the expansion parameter satisfies ϕ > 1, these graphs have a diameter of o(log n), as opposed to other standard notions of expansion. Since the graph’s diameter is a lower bound on the number of rounds required for rumor spreading, this makes small-set expanders particularly well-suited for fast information dissemination. We prove that k-PUSH&PULL takes O(log_{ϕ} n ⋅ log_{k} n) rounds in these expanders, with high probability. We complement this with a simple lower bound of Ω(log_{ϕ} n+ log_{k} n) rounds.
Cite as

Emilio Cruciani, Sebastian Forster, and Tijn de Vos. Towards Constant Time Multi-Call Rumor Spreading on Small-Set Expanders. In 39th International Symposium on Distributed Computing (DISC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 356, pp. 26:1-26:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{cruciani_et_al:LIPIcs.DISC.2025.26,
  author =	{Cruciani, Emilio and Forster, Sebastian and de Vos, Tijn},
  title =	{{Towards Constant Time Multi-Call Rumor Spreading on Small-Set Expanders}},
  booktitle =	{39th International Symposium on Distributed Computing (DISC 2025)},
  pages =	{26:1--26:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-402-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{356},
  editor =	{Kowalski, Dariusz R.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.DISC.2025.26},
  URN =		{urn:nbn:de:0030-drops-248434},
  doi =		{10.4230/LIPIcs.DISC.2025.26},
  annote =	{Keywords: small set expansion, vertex expansion, rumor spreading, multi-call rumor spreading, push\&pull protocol}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.27
Triangles Improve 0.878 Approximation for Maxcut

Authors: Fredie George, Anand Louis, and Rameesh Paul

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

Abstract
Maxcut is a fundamental problem in graph algorithms, extensively studied for its theoretical and practical significance. The goal is to partition the vertex set of a graph G = (V, E) into disjoint subsets S and V⧵S so as to maximize the number of edges crossing the cut (S,V⧵S). The seminal work of Goemans and Williamson [Goemans and Williamson, 1995] introduced a semidefinite programming (SDP) based algorithm achieving a α_{GW} ≈ 0.87856-approximation for general graphs, guaranteed to be optimal under the Unique Games Conjecture [Khot, 2002; Khot et al., 2007]. We revisit the Goemans–Williamson SDP and prove that the standard Maxcut SDP achieves a (α_{GW} + Ω(1))-approximation whenever the input graph contains Ω(|E|) edge-disjoint triangles. Our analysis builds on classical rounding techniques studied in [Goemans and Williamson, 1995; Zwick, 1999] and introduces a refined understanding of the SDP solution structure in regimes where the previous guarantees are tight. Our result identifies a simple combinatorial property that may be satisfied by many natural graph classes. As applications, we show that unit ball graphs and graphs satisfying a spectral transitivity condition (as studied in [Gupta et al., 2016; Basu et al., 2024]) meet our structural criterion, and therefore we get better than α_{GW} approximation guarantees for them. Our algorithm runs in nearly linear time 𝒪̃(|E|), offering a more practical alternative to the PTAS of [Jansen et al., 2005] for unit ball graphs, which has exponential dependence on the approximation parameter.
Cite as

Fredie George, Anand Louis, and Rameesh Paul. Triangles Improve 0.878 Approximation for Maxcut. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 27:1-27:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{george_et_al:LIPIcs.APPROX/RANDOM.2025.27,
  author =	{George, Fredie and Louis, Anand and Paul, Rameesh},
  title =	{{Triangles Improve 0.878 Approximation for Maxcut}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{27:1--27:25},
  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.2025.27},
  URN =		{urn:nbn:de:0030-drops-243931},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.27},
  annote =	{Keywords: Approximation Algorithms, Maxcut, Semidefinite Programming, Edge-disjoint Triangles, Unit Ball Graphs, Spectral Triadic Graphs}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.5
Min-CSPs on Complete Instances II: Polylogarithmic Approximation for Min-NAE-3-SAT

Authors: Aditya Anand, Euiwoong Lee, Davide Mazzali, and Amatya Sharma

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

Abstract
This paper studies complete k-Constraint Satisfaction Problems (CSPs), where an n-variable instance has exactly one nontrivial constraint for each subset of k variables, i.e., it has binom(n,k) constraints. A recent work started a systematic study of complete k-CSPs [Anand, Lee, Sharma, SODA'25], and showed a quasi-polynomial time algorithm that decides if there is an assignment satisfying all the constraints of any complete Boolean-alphabet k-CSP, algorithmically separating complete instances from dense instances. The tractability of this decision problem is necessary for any nontrivial (multiplicative) approximation for the minimization version, whose goal is to minimize the number of violated constraints. The same paper raised the question of whether it is possible to obtain nontrivial approximation algorithms for complete Min-k-CSPs with k ≥ 3. In this work, we make progress in this direction and show a quasi-polynomial time polylog(n)-approximation to Min-NAE-3-SAT on complete instances, which asks to minimize the number of 3-clauses where all the three literals equal the same bit. To the best of our knowledge, this is the first known example of a CSP whose decision version is NP-Hard in general (and dense) instances while admitting a polylog(n)-approximation in complete instances. Our algorithm presents a new iterative framework for rounding a solution from the Sherali-Adams hierarchy, where each iteration interleaves the two well-known rounding tools: the conditioning procedure, in order to "almost fix" many variables, and the thresholding procedure, in order to "completely fix" them. Finally, we improve the running time of the decision algorithms of Anand, Lee, and Sharma and show a simple algorithm that decides any complete Boolean-alphabet k-CSP in polynomial time.
Cite as

Aditya Anand, Euiwoong Lee, Davide Mazzali, and Amatya Sharma. Min-CSPs on Complete Instances II: Polylogarithmic Approximation for Min-NAE-3-SAT. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 5:1-5:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{anand_et_al:LIPIcs.APPROX/RANDOM.2025.5,
  author =	{Anand, Aditya and Lee, Euiwoong and Mazzali, Davide and Sharma, Amatya},
  title =	{{Min-CSPs on Complete Instances II: Polylogarithmic Approximation for Min-NAE-3-SAT}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{5:1--5:23},
  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.2025.5},
  URN =		{urn:nbn:de:0030-drops-243712},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.5},
  annote =	{Keywords: Constraint Satisfiability Problems, Approximation Algorithms, Sherali Adams}
}
Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.16
Sparsest Cut and Eigenvalue Multiplicities on Low Degree Abelian Cayley Graphs

Authors: Tommaso d'Orsi, Chris Jones, Jake Ruotolo, Salil Vadhan, and Jiyu Zhang

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

Abstract
Whether or not the Sparsest Cut problem admits an efficient O(1)-approximation algorithm is a fundamental algorithmic question with connections to geometry and the Unique Games Conjecture. Revisiting spectral algorithms for Sparsest Cut, we present a novel, simple algorithm that combines eigenspace enumeration with a new algorithm for the Cut Improvement problem. The runtime of our algorithm is parametrized by a quantity that we call the solution dimension SD_ε(G): the smallest k such that the subspace spanned by the first k Laplacian eigenvectors contains all but ε fraction of a sparsest cut. Our algorithm matches the guarantees of prior methods based on the threshold-rank paradigm, while also extending beyond them. To illustrate this, we study its performance on low degree Cayley graphs over Abelian groups - canonical examples of graphs with poor expansion properties. We prove that low degree Abelian Cayley graphs have small solution dimension, yielding an algorithm that computes a (1+ε)-approximation to the uniform Sparsest Cut of a degree-d Cayley graph over an Abelian group of size n in time n^O(1) ⋅ exp{(d/ε)^O(d)}. Along the way to bounding the solution dimension of Abelian Cayley graphs, we analyze their sparse cuts and spectra, proving that the collection of O(1)-approximate sparsest cuts has an ε-net of size exp{(d/ε)^O(d)} and that the multiplicity of λ₂ is bounded by 2^O(d). The latter bound is tight and improves on a previous bound of 2^O(d²) by Lee and Makarychev.
Cite as

Tommaso d'Orsi, Chris Jones, Jake Ruotolo, Salil Vadhan, and Jiyu Zhang. Sparsest Cut and Eigenvalue Multiplicities on Low Degree Abelian Cayley Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 16:1-16:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{dorsi_et_al:LIPIcs.APPROX/RANDOM.2025.16,
  author =	{d'Orsi, Tommaso and Jones, Chris and Ruotolo, Jake and Vadhan, Salil and Zhang, Jiyu},
  title =	{{Sparsest Cut and Eigenvalue Multiplicities on Low Degree Abelian Cayley Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{16:1--16:20},
  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.2025.16},
  URN =		{urn:nbn:de:0030-drops-243827},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.16},
  annote =	{Keywords: Sparsest Cut, Spectral Graph Theory, Cayley Graphs, Approximation Algorithms}
}
Document
DOI: 10.4230/LIPIcs.CCC.2025.11
Switching Graph Matrix Norm Bounds: From i.i.d. to Random Regular Graphs

Authors: Jeff Xu

Published in: LIPIcs, Volume 339, 40th Computational Complexity Conference (CCC 2025)

Abstract
In this work, we give novel spectral norm bounds for graph matrix on inputs being random regular graphs. Graph matrix is a family of random matrices with entries given by polynomial functions of the underlying input. These matrices have been known to be the backbone for the analysis of various average-case algorithms and hardness. Previous investigations of such matrices are largely restricted to the Erdős-Rényi model, and tight matrix norm bounds on regular graphs are only known for specific examples. We unite these two lines of investigations, and give the first result departing from the Erdős-Rényi setting in the full generality of graph matrices. We believe our norm bound result would enable a simple transfer of spectral analysis for average-case algorithms and hardness between these two distributions of random graphs. As an application of our spectral norm bounds, we show that higher-degree Sum-of-Squares lower bounds for the independent set problem on Erdős-Rényi random graphs can be switched into lower bounds on random d-regular graphs. Our main conceptual insight is that existing Sum-of-Squares lower bounds analysis based on moment methods are surprisingly robust, and amenable for a light-weight translation. Our result is the first to address the general open question of analyzing higher-degree Sum-of-Squares on random regular graphs.
Cite as

Jeff Xu. Switching Graph Matrix Norm Bounds: From i.i.d. to Random Regular Graphs. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{xu:LIPIcs.CCC.2025.11,
  author =	{Xu, Jeff},
  title =	{{Switching Graph Matrix Norm Bounds: From i.i.d. to Random Regular Graphs}},
  booktitle =	{40th Computational Complexity Conference (CCC 2025)},
  pages =	{11:1--11:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-379-9},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{339},
  editor =	{Srinivasan, Srikanth},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2025.11},
  URN =		{urn:nbn:de:0030-drops-237054},
  doi =		{10.4230/LIPIcs.CCC.2025.11},
  annote =	{Keywords: Semidefinite programming, random matrices, average-case complexity}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.29
Guessing Efficiently for Constrained Subspace Approximation

Authors: Aditya Bhaskara, Sepideh Mahabadi, Madhusudhan Reddy Pittu, Ali Vakilian, and David P. Woodruff

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

Abstract
In this paper we study constrained subspace approximation problem. Given a set of n points {a₁,…,a_n} in ℝ^d, the goal of the subspace approximation problem is to find a k dimensional subspace that best approximates the input points. More precisely, for a given p ≥ 1, we aim to minimize the pth power of the 𝓁_p norm of the error vector (‖a₁-Pa₁‖,…,‖a_n-Pa_n‖), where P denotes the projection matrix onto the subspace and the norms are Euclidean. In constrained subspace approximation (CSA), we additionally have constraints on the projection matrix P. In its most general form, we require P to belong to a given subset 𝒮 that is described explicitly or implicitly. We introduce a general framework for constrained subspace approximation. Our approach, that we term coreset-guess-solve, yields either (1+ε)-multiplicative or ε-additive approximations for a variety of constraints. We show that it provides new algorithms for partition-constrained subspace approximation with applications to fair subspace approximation, k-means clustering, and projected non-negative matrix factorization, among others. Specifically, while we reconstruct the best known bounds for k-means clustering in Euclidean spaces, we improve the known results for the remainder of the problems.
Cite as

Aditya Bhaskara, Sepideh Mahabadi, Madhusudhan Reddy Pittu, Ali Vakilian, and David P. Woodruff. Guessing Efficiently for Constrained Subspace Approximation. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 29:1-29:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{bhaskara_et_al:LIPIcs.ICALP.2025.29,
  author =	{Bhaskara, Aditya and Mahabadi, Sepideh and Pittu, Madhusudhan Reddy and Vakilian, Ali and Woodruff, David P.},
  title =	{{Guessing Efficiently for Constrained Subspace Approximation}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{29:1--29:20},
  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.2025.29},
  URN =		{urn:nbn:de:0030-drops-234068},
  doi =		{10.4230/LIPIcs.ICALP.2025.29},
  annote =	{Keywords: parameterized complexity, low rank approximation, fairness, non-negative matrix factorization, clustering}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.96
Asymptotically Optimal Inapproximability of Maxmin k-Cut Reconfiguration

Authors: Shuichi Hirahara and Naoto Ohsaka

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

Abstract
k-Coloring Reconfiguration is one of the most well-studied reconfiguration problems, which asks to transform a given proper k-coloring of a graph to another by repeatedly recoloring a single vertex. Its approximate version, Maxmin k-Cut Reconfiguration, is defined as an optimization problem of maximizing the minimum fraction of bichromatic edges during the transformation between (not necessarily proper) k-colorings. In this paper, we demonstrate that the optimal approximation factor of this problem is 1 - Θ(1/k) for every k ≥ 2. Specifically, we prove the PSPACE-hardness of approximating the objective value within a factor of 1 - ε/k for some universal constant ε > 0, whereas we develop a deterministic polynomial-time algorithm that achieves the approximation factor of 1 - 2/k. To prove the hardness result, we propose a new probabilistic verifier that tests a "striped" pattern. Our approximation algorithm is based on a random transformation that passes through a random k-coloring.
Cite as

Shuichi Hirahara and Naoto Ohsaka. Asymptotically Optimal Inapproximability of Maxmin k-Cut Reconfiguration. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 96:1-96:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{hirahara_et_al:LIPIcs.ICALP.2025.96,
  author =	{Hirahara, Shuichi and Ohsaka, Naoto},
  title =	{{Asymptotically Optimal Inapproximability of Maxmin k-Cut Reconfiguration}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{96:1--96:18},
  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.2025.96},
  URN =		{urn:nbn:de:0030-drops-234733},
  doi =		{10.4230/LIPIcs.ICALP.2025.96},
  annote =	{Keywords: reconfiguration problems, graph coloring, hardness of approximation}
}
Document
DOI: 10.4230/LIPIcs.ICALP.2018.84
Spectrally Robust Graph Isomorphism

Authors: Alexandra Kolla, Ioannis Koutis, Vivek Madan, and Ali Kemal Sinop

Published in: LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Abstract
We initiate the study of spectral generalizations of the graph isomorphism problem. b) The Spectral Graph Dominance (SGD) problem: On input of two graphs G and H does there exist a permutation pi such that G preceq pi(H)? c) The Spectrally Robust Graph Isomorphism (kappa-SRGI) problem: On input of two graphs G and H, find the smallest number kappa over all permutations pi such that pi(H) preceq G preceq kappa c pi(H) for some c. SRGI is a natural formulation of the network alignment problem that has various applications, most notably in computational biology. G preceq c H means that for all vectors x we have x^T L_G x <= c x^T L_H x, where L_G is the Laplacian G. We prove NP-hardness for SGD. We also present a kappa^3-approximation algorithm for SRGI for the case when both G and H are bounded-degree trees. The algorithm runs in polynomial time when kappa is a constant.
Cite as

Alexandra Kolla, Ioannis Koutis, Vivek Madan, and Ali Kemal Sinop. Spectrally Robust Graph Isomorphism. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 84:1-84:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kolla_et_al:LIPIcs.ICALP.2018.84,
  author =	{Kolla, Alexandra and Koutis, Ioannis and Madan, Vivek and Sinop, Ali Kemal},
  title =	{{Spectrally Robust Graph Isomorphism}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{84:1--84:13},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.84},
  URN =		{urn:nbn:de:0030-drops-90887},
  doi =		{10.4230/LIPIcs.ICALP.2018.84},
  annote =	{Keywords: Network Alignment, Graph Isomorphism, Graph Similarity}
}
Document
DOI: 10.4230/LIPIcs.SOCG.2015.754
The Hardness of Approximation of Euclidean k-Means

Authors: Pranjal Awasthi, Moses Charikar, Ravishankar Krishnaswamy, and Ali Kemal Sinop

Published in: LIPIcs, Volume 34, 31st International Symposium on Computational Geometry (SoCG 2015)

Abstract
The Euclidean k-means problem is a classical problem that has been extensively studied in the theoretical computer science, machine learning and the computational geometry communities. In this problem, we are given a set of n points in Euclidean space R^d, and the goal is to choose k center points in R^d so that the sum of squared distances of each point to its nearest center is minimized. The best approximation algorithms for this problem include a polynomial time constant factor approximation for general k and a (1+c)-approximation which runs in time poly(n) exp(k/c). At the other extreme, the only known computational complexity result for this problem is NP-hardness [Aloise et al.'09]. The main difficulty in obtaining hardness results stems from the Euclidean nature of the problem, and the fact that any point in R^d can be a potential center. This gap in understanding left open the intriguing possibility that the problem might admit a PTAS for all k, d. In this paper we provide the first hardness of approximation for the Euclidean k-means problem. Concretely, we show that there exists a constant c > 0 such that it is NP-hard to approximate the k-means objective to within a factor of (1+c). We show this via an efficient reduction from the vertex cover problem on triangle-free graphs: given a triangle-free graph, the goal is to choose the fewest number of vertices which are incident on all the edges. Additionally, we give a proof that the current best hardness results for vertex cover can be carried over to triangle-free graphs. To show this we transform G, a known hard vertex cover instance, by taking a graph product with a suitably chosen graph H, and showing that the size of the (normalized) maximum independent set is almost exactly preserved in the product graph using a spectral analysis, which might be of independent interest.
Cite as

Pranjal Awasthi, Moses Charikar, Ravishankar Krishnaswamy, and Ali Kemal Sinop. The Hardness of Approximation of Euclidean k-Means. In 31st International Symposium on Computational Geometry (SoCG 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 34, pp. 754-767, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{awasthi_et_al:LIPIcs.SOCG.2015.754,
  author =	{Awasthi, Pranjal and Charikar, Moses and Krishnaswamy, Ravishankar and Sinop, Ali Kemal},
  title =	{{The Hardness of Approximation of Euclidean k-Means}},
  booktitle =	{31st International Symposium on Computational Geometry (SoCG 2015)},
  pages =	{754--767},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-83-5},
  ISSN =	{1868-8969},
  year =	{2015},
  volume =	{34},
  editor =	{Arge, Lars and Pach, J\'{a}nos},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SOCG.2015.754},
  URN =		{urn:nbn:de:0030-drops-51178},
  doi =		{10.4230/LIPIcs.SOCG.2015.754},
  annote =	{Keywords: Euclidean k-means, Hardness of Approximation, Vertex Cover}
}
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