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DOI: 10.4230/LIPIcs.ITCS.2026.12

Semi-Random Graphs, Robust Asymmetry, and Reconstruction

Authors: Julian Asilis, Xi Chen, Dutch Hansen, and Shang-Hua Teng

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

The Graph Reconstruction Conjecture famously posits that any undirected graph on at least three vertices is determined up to isomorphism by its family of (unlabeled) induced subgraphs. At present, the conjecture admits partial resolutions of two types: 1) casework-based demonstrations of reconstructibility for families of graphs satisfying certain structural properties, and 2) probabilistic arguments establishing reconstructibility of random graphs by leveraging average-case phenomena. While results in the first category capture the worst-case nature of the conjecture, they play a limited role in understanding the general case. Results in the second category address much larger graph families, but it remains unclear how heavily the necessary arguments rely on optimistic distributional properties. Drawing on the algorithmic notions of smoothed and semi-random analysis, we study the robustness of what are arguably the two most fundamental properties in this latter line of work: asymmetry and uniqueness of subgraphs. Notably, we find that various natural semi-random graph distributions exhibit these properties asymptotically, much like their Erdős-Rényi counterparts. In particular, Bollobás [Bollob{á}s, 1990] demonstrated that almost all Erdős-Rényi random graphs G = (V, E) ∼ G(n, p) enjoy the property that their induced subgraphs on n - Θ(1) vertices are asymmetric and mutually non-isomorphic, for 1 - p, p = Ω(log(n) / n). As our primary result, we demonstrate that this property is robust against perturbation - even when an adversary is permitted to add/remove each vertex pair in V^{(2)} with (independent) arbitrarily large constant probability. Exploiting this result, we derive asymptotic characterizations of asymmetry in random graphs with large planted structure and bounded adversarial corruptions, along with improved bounds on the probability mass of nonreconstructible graphs in G(n, p).

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Julian Asilis, Xi Chen, Dutch Hansen, and Shang-Hua Teng. Semi-Random Graphs, Robust Asymmetry, and Reconstruction. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 12:1-12:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{asilis_et_al:LIPIcs.ITCS.2026.12,
  author =	{Asilis, Julian and Chen, Xi and Hansen, Dutch and Teng, Shang-Hua},
  title =	{{Semi-Random Graphs, Robust Asymmetry, and Reconstruction}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{12:1--12:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.12},
  URN =		{urn:nbn:de:0030-drops-252993},
  doi =		{10.4230/LIPIcs.ITCS.2026.12},
  annote =	{Keywords: Graph reconstruction, random graphs}
}

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DOI: 10.4230/LIPIcs.ITCS.2026.48

Auditability and the Landscape of Distance to Multicalibration

Authors: Nathan Derhake, Siddartha Devic, Dutch Hansen, Kuan Liu, and Vatsal Sharan

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract

Calibration is a critical property for establishing the trustworthiness of predictors that provide uncertainty estimates. Multicalibration is a strengthening of calibration which requires that predictors be calibrated on a potentially overlapping collection of subsets of the domain. As multicalibration grows in popularity with practitioners, an essential question is: how do we measure how multicalibrated a predictor is? Błasiok et al. [Błasiok et al., 2023] considered this question for standard calibration by introducing the distance to calibration framework (dCE) to understand how calibration metrics relate to each other and the ground truth. Building on the dCE framework, we consider the auditability of the distance to multicalibration of a predictor f. We begin by considering what are perhaps the two most natural generalizations of dCE to multiple subgroups: worst group dCE (wdMC), and distance to multicalibration (dMC). Using wdMC and dMC as a guiding path, we argue that there are two essential properties of any multicalibration error metric: 1) the metric should capture how much f would need to be modified in order to be perfectly multicalibrated; and 2) the metric should be auditable in an information theoretic sense (i.e., with some finite sample complexity). We show that wdMC and dMC each fail to satisfy one of these two properties, and that similar barriers arise when considering the auditability of general distance to multigroup fairness notions (e.g. multiaccuracy or low-degree multicalibration). We then propose two (equivalent) multicalibration metrics which do satisfy these requirements: 1) a continuized variant of dMC; and 2) a distance to intersection multicalibration, which leans on intersectional fairness desiderata. Along the way, we shed light on the loss-landscape of distance to multicalibration and the geometry of the set of perfectly multicalibrated predictors. We also demonstrate that the loss surface of any metric which captures how much f would need to be modified to be perfectly multicalibrated often satisfies a local minima are global minima property. Our findings may have implications for the development of stronger multicalibration algorithms, as well as multicalibration auditing more generally.

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Nathan Derhake, Siddartha Devic, Dutch Hansen, Kuan Liu, and Vatsal Sharan. Auditability and the Landscape of Distance to Multicalibration. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 48:1-48:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{derhake_et_al:LIPIcs.ITCS.2026.48,
  author =	{Derhake, Nathan and Devic, Siddartha and Hansen, Dutch and Liu, Kuan and Sharan, Vatsal},
  title =	{{Auditability and the Landscape of Distance to Multicalibration}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{48:1--48:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.48},
  URN =		{urn:nbn:de:0030-drops-253351},
  doi =		{10.4230/LIPIcs.ITCS.2026.48},
  annote =	{Keywords: Multicalibration, Auditability, Fairness, Classification, Calibration}
}

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Documents authored by Hansen, Dutch

Semi-Random Graphs, Robust Asymmetry, and Reconstruction

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Auditability and the Landscape of Distance to Multicalibration

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