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Documents authored by d'Orsi, Tommaso

Document
APPROX
DOI: 10.4230/LIPIcs.APPROX/RANDOM.2025.11
On Finding Randomly Planted Cliques in Arbitrary Graphs

Authors: Francesco Agrimonti, Marco Bressan, and Tommaso d'Orsi

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

Abstract
We study a planted clique model introduced by Feige [Uriel Feige, 2021] where a complete graph of size c⋅ n is planted uniformly at random in an arbitrary n-vertex graph. We give a simple deterministic algorithm that, in almost linear time, recovers a clique of size (c/3)^O(1/c) ⋅ n as long as the original graph has maximum degree at most (1-p)n for some fixed p > 0. The proof hinges on showing that the degrees of the final graph are correlated with the planted clique, in a way similar to (but more intricate than) the classical G(n,1/2)+K_√n planted clique model. Our algorithm suggests a separation from the worst-case model, where, assuming the Unique Games Conjecture, no polynomial algorithm can find cliques of size Ω(n) for every fixed c > 0, even if the input graph has maximum degree (1-p)n. Our techniques extend beyond the planted clique model. For example, when the planted graph is a balanced biclique, we recover a balanced biclique of size larger than the best guarantees known for the worst case.
Cite as

Francesco Agrimonti, Marco Bressan, and Tommaso d'Orsi. On Finding Randomly Planted Cliques in Arbitrary Graphs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 353, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{agrimonti_et_al:LIPIcs.APPROX/RANDOM.2025.11,
  author =	{Agrimonti, Francesco and Bressan, Marco and d'Orsi, Tommaso},
  title =	{{On Finding Randomly Planted Cliques in Arbitrary Graphs}},
  booktitle =	{Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2025)},
  pages =	{11:1--11:18},
  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.2025.11},
  URN =		{urn:nbn:de:0030-drops-243774},
  doi =		{10.4230/LIPIcs.APPROX/RANDOM.2025.11},
  annote =	{Keywords: Computational Complexity, Planted Clique, Semi-random, Unique Games Conjecture, Approximation Algorithms}
}
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.2023.27
A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs

Authors: Tommaso d'Orsi and Luca Trevisan

Published in: LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

Abstract
We define a novel notion of "non-backtracking" matrix associated to any symmetric matrix, and we prove a "Ihara-Bass" type formula for it. We use this theory to prove new results on polynomial-time strong refutations of random constraint satisfaction problems with k variables per constraints (k-CSPs). For a random k-CSP instance constructed out of a constraint that is satisfied by a p fraction of assignments, if the instance contains n variables and n^{k/2} / ε² constraints, we can efficiently compute a certificate that the optimum satisfies at most a p+O_k(ε) fraction of constraints. Previously, this was known for even k, but for odd k one needed n^{k/2} (log n)^{O(1)} / ε² random constraints to achieve the same conclusion. Although the improvement is only polylogarithmic, it overcomes a significant barrier to these types of results. Strong refutation results based on current approaches construct a certificate that a certain matrix associated to the k-CSP instance is quasirandom. Such certificate can come from a Feige-Ofek type argument, from an application of Grothendieck’s inequality, or from a spectral bound obtained with a trace argument. The first two approaches require a union bound that cannot work when the number of constraints is o(n^⌈k/2⌉) and the third one cannot work when the number of constraints is o(n^{k/2} √{log n}). We further apply our techniques to obtain a new PTAS finding assignments for k-CSP instances with n^{k/2} / ε² constraints in the semi-random settings where the constraints are random, but the sign patterns are adversarial.
Cite as

Tommaso d'Orsi and Luca Trevisan. A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 27:1-27:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{dorsi_et_al:LIPIcs.CCC.2023.27,
  author =	{d'Orsi, Tommaso and Trevisan, Luca},
  title =	{{A Ihara-Bass Formula for Non-Boolean Matrices and Strong Refutations of Random CSPs}},
  booktitle =	{38th Computational Complexity Conference (CCC 2023)},
  pages =	{27:1--27:16},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-282-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{264},
  editor =	{Ta-Shma, Amnon},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.27},
  URN =		{urn:nbn:de:0030-drops-182979},
  doi =		{10.4230/LIPIcs.CCC.2023.27},
  annote =	{Keywords: CSP, k-XOR, strong refutation, sum-of-squares, tensor, graph, hypergraph, non-backtracking walk}
}
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