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Documents authored by Xu, Jeff

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.2023.78
Ellipsoid Fitting up to a Constant

Authors: Jun-Ting Hsieh, Pravesh K. Kothari, Aaron Potechin, and Jeff Xu

Published in: LIPIcs, Volume 261, 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)

Abstract
In [Saunderson, 2011; Saunderson et al., 2013], Saunderson, Parrilo, and Willsky asked the following elegant geometric question: what is the largest m = m(d) such that there is an ellipsoid in ℝ^d that passes through v_1, v_2, …, v_m with high probability when the v_is are chosen independently from the standard Gaussian distribution N(0,I_d)? The existence of such an ellipsoid is equivalent to the existence of a positive semidefinite matrix X such that v_i^⊤ X v_i = 1 for every 1 ⩽ i ⩽ m - a natural example of a random semidefinite program. SPW conjectured that m = (1-o(1)) d²/4 with high probability. Very recently, Potechin, Turner, Venkat and Wein [Potechin et al., 2022] and Kane and Diakonikolas [Kane and Diakonikolas, 2022] proved that m ≳ d²/log^O(1) d via a certain natural, explicit construction. In this work, we give a substantially tighter analysis of their construction to prove that m ≳ d²/C for an absolute constant C > 0. This resolves one direction of the SPW conjecture up to a constant. Our analysis proceeds via the method of Graphical Matrix Decomposition that has recently been used to analyze correlated random matrices arising in various areas [Barak et al., 2019; Bafna et al., 2022]. Our key new technical tool is a refined method to prove singular value upper bounds on certain correlated random matrices that are tight up to absolute dimension-independent constants. In contrast, all previous methods that analyze such matrices lose logarithmic factors in the dimension.
Cite as

Jun-Ting Hsieh, Pravesh K. Kothari, Aaron Potechin, and Jeff Xu. Ellipsoid Fitting up to a Constant. In 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 261, pp. 78:1-78:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{hsieh_et_al:LIPIcs.ICALP.2023.78,
  author =	{Hsieh, Jun-Ting and Kothari, Pravesh K. and Potechin, Aaron and Xu, Jeff},
  title =	{{Ellipsoid Fitting up to a Constant}},
  booktitle =	{50th International Colloquium on Automata, Languages, and Programming (ICALP 2023)},
  pages =	{78:1--78:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-278-5},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{261},
  editor =	{Etessami, Kousha and Feige, Uriel 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.2023.78},
  URN =		{urn:nbn:de:0030-drops-181304},
  doi =		{10.4230/LIPIcs.ICALP.2023.78},
  annote =	{Keywords: Semidefinite programming, random matrices, average-case complexity}
}
Document
DOI: 10.4230/LIPIcs.CCC.2022.11
Certifying Solution Geometry in Random CSPs: Counts, Clusters and Balance

Authors: Jun-Ting Hsieh, Sidhanth Mohanty, and Jeff Xu

Published in: LIPIcs, Volume 234, 37th Computational Complexity Conference (CCC 2022)

Abstract
An active topic in the study of random constraint satisfaction problems (CSPs) is the geometry of the space of satisfying or almost satisfying assignments as the function of the density, for which a precise landscape of predictions has been made via statistical physics-based heuristics. In parallel, there has been a recent flurry of work on refuting random constraint satisfaction problems, via nailing refutation thresholds for spectral and semidefinite programming-based algorithms, and also on counting solutions to CSPs. Inspired by this, the starting point for our work is the following question: What does the solution space for a random CSP look like to an efficient algorithm? In pursuit of this inquiry, we focus on the following problems about random Boolean CSPs at the densities where they are unsatisfiable but no refutation algorithm is known. 1) Counts. For every Boolean CSP we give algorithms that with high probability certify a subexponential upper bound on the number of solutions. We also give algorithms to certify a bound on the number of large cuts in a Gaussian-weighted graph, and the number of large independent sets in a random d-regular graph. 2) Clusters. For Boolean 3CSPs we give algorithms that with high probability certify an upper bound on the number of clusters of solutions. 3) Balance. We also give algorithms that with high probability certify that there are no "unbalanced" solutions, i.e., solutions where the fraction of +1s deviates significantly from 50%. Finally, we also provide hardness evidence suggesting that our algorithms for counting are optimal.
Cite as

Jun-Ting Hsieh, Sidhanth Mohanty, and Jeff Xu. Certifying Solution Geometry in Random CSPs: Counts, Clusters and Balance. In 37th Computational Complexity Conference (CCC 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 234, pp. 11:1-11:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hsieh_et_al:LIPIcs.CCC.2022.11,
  author =	{Hsieh, Jun-Ting and Mohanty, Sidhanth and Xu, Jeff},
  title =	{{Certifying Solution Geometry in Random CSPs: Counts, Clusters and Balance}},
  booktitle =	{37th Computational Complexity Conference (CCC 2022)},
  pages =	{11:1--11:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-241-9},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{234},
  editor =	{Lovett, Shachar},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2022.11},
  URN =		{urn:nbn:de:0030-drops-165735},
  doi =		{10.4230/LIPIcs.CCC.2022.11},
  annote =	{Keywords: constraint satisfaction problems, certified counting, random graphs}
}
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