Theory of Computing Report

Yesterday on Good Morning America, NFL Hall of Famer Chris Johnson announced that he had ALS. Johnson, nicknamed CJ2K for his 2006 rushing yards in 2009, was one of the most electrifying players I’ve ever seen. He was impossibly fast. Every one of his carries was must-watch football, as you’d never know when he’d break a run for a touchdown. Johnson still holds the record for the most yards from scrimmage in a season, with his total of 2509 a comfortable 80 yards in front of second place.

Johnson retired in 2018, and, like most football players, disappeared from the national spotlight. So it was beyond heartbreaking to see him yesterday, at 40 years old, speaking to Michael Strahan with an eye-tracking device. ALS attacks the body’s nerve cells, progressively inhibiting a person’s ability to move their limbs, speak, and breathe. Though highly variable, the progression can be shockingly rapid. Johnson told GMA that he had been diagnosed less than a year ago. He wanted to share his story to bring more attention to ALS and the search for radical new treatments.

In 2023, Gideon Lewis-Kraus wrote a moving investigation into this search for The New Yorker. People with ALS and their families are, for quite obvious reasons, passionately dedicated to finding any reasonable treatments for the disease. The hope is to find therapies that at least slow disease progression and improve the patient’s quality of life. The challenge is this tends to require drugs, and ALS lies outside the sweet spot of modern drug discovery.

ALS is a rare but devastating condition, with around 5,000 new cases diagnosed in the United States every year. There is no clear established cause for ALS. Though it typically manifests in people’s late fifties or early sixties, Johnson’s case exemplifies how unpredictable its onset can be. Sometimes ALS progression takes years; sometimes it rapidly advances in months. There are no clear genetic signals for ALS, and family history isn’t predictive. There is a biomarker present in the nerve cells of almost all ALS patients, but current technology only lets us detect this biomarker after death.

Without access to a marker, measuring progression or reversal is a subjective challenge. Current ALS surrogates numerically assess bodily functions like speech, swallowing, breathing, and motor skills, and combine these different tests into a single number. For example, the ALSFRS-R score is the sum of 12 different assessments, each scored on a scale from 0 to 4, where 0 indicates total loss of function, and 4 indicates no loss of function.

All of this adds up to make drug testing a major challenge. It’s hard to define reasonable endpoints, and most trials aim to find a statistically significant difference between treatment and control in ordinal assessments made by clinicians. Though patients and their families urgently seek new possible therapies, there are not enough ALS patients to meet the harsh statistical standards of clinical trials. Moreover, drugs require trials, and the ethics of potentially denying the control group something beneficial are complex and fraught.1

Lewis-Kraus draws connections to the disease that set the standard for patient advocacy against institutionalized medicine: AIDS. AIDS activists in the 1980s were not only vocal in spreading awareness of the disease and its various treatments, but they actively engaged in scientific research and drug policy. It’s a remarkable example of a citizen group not only challenging, but shaping scientific expertise. To this day, AIDS activists remain go-to consultants for helping people and families affected by rare diseases to organize and advocate for the afflicted.

In hindsight, HIV treatment is similar to the treatment of other progressive diseases like ALS. None of the treatments cured an HIV infection, but they allowed patients to manage their infection. Today, HIV can be treated pharmaceutically with an excellent long-term prognosis. Though initially a terrifying death sentence, people with HIV infections can live a normal life with access to the proper medication.

The differences between AIDS and ALS highlight what makes a progressive, terminal condition treatable. AIDS was caused by an infection with the virus HIV. The first effective HIV treatment, AZT, targeted the virus by fooling it to bind to a byproduct of the drug rather than the person’s DNA manufacturing system. This eliminated HIV’s ability to replicate. Moreover, AIDS had an easily measurable surrogate marker to monitor progress. HIV killed a particular type of white blood cell, and counting these cells was a reasonable way to test if the drug was having an effect.

Sadly, while researchers are hopeful that we’re making rapid progress, we’re still missing many crucial causal pieces to the ALS story that might illuminate an appropriate target for the pharmaceutical industry. Johnson hopes that raising awareness of ALS might help us find the missing link in understanding and treating the disease. He wants to keep fighting. My heart goes out to CJ2K, his family, and everyone else suffering from this terrible condition. And I hope his inspiration will help us find the missing gap for a breakthrough treatment.

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By Ben Recht

June 14-18, 2027 Institute for Advanced Study, Princeton, NJ, USA https://www.ias.edu/math/events/celebrating-100-years-avi-70-csdm-30 The Institute for Advanced Study’s School of Mathematics is pleased to announce a conference celebrating Avi Wigderson’s 70th birthday and retirement, together with 30 years of the Computer Science and Discrete Mathematics program (CSDM) at IAS. The conference will honor both Avi’s extraordinary scientific … Continue reading Celebrating 100 Years: Avi 70 + CSDM 30

By shacharlovett

[Guest post by Amir Shpilka and Irit Dinur]

https://www.ias.edu/math/events/celebrating-100-years-avi-70-csdm-30

The Institute for Advanced Study’s School of Mathematics is pleased to announce a conference celebrating Avi Wigderson’s 70th birthday and retirement, together with 30 years of the Computer Science and Discrete Mathematics program (CSDM) at IAS. The conference will honor both Avi’s extraordinary scientific legacy and his transformative role in shaping the intellectual community around theoretical computer science and discrete mathematics at IAS.

Avi Wigderson has made foundational contributions across an exceptional range of areas in theoretical computer science and mathematics. His work has had a profound impact on complexity theory broadly, and especially on cryptography, circuit and proof complexity, communication complexity, pseudorandomness, and expander graphs; in recent years, he has also opened deep and unexpected connections to invariant theory. These contributions have shaped major research directions and influenced generations of mathematicians and theoretical computer scientists.

Confirmed Speakers (all alumni of CSDM):

Boaz Barak, Harvard University

Maria Chudnovsky, Princeton University

Julia Chuzhoy, Toyota Technological Institute at Chicago

Gil Cohen, Tel Aviv University

Yotam Dikstein, Tel Aviv University

Pavel Hrubes, Charles University

Valentine Kabanets, Simon Fraser University

Tali Kaufman, Bar Ilan University

Subhash Khot, NYU

Swastik Kopparty, University of Toronto

Parvesh Kothari, Princeton University

Dor Minzer, MIT

Ankur Moitra, MIT

Shay Moran, Technion Israel Institute of Technology

Ryan O’Donnell, Carnegie Mellon University

Jinyoung Park, NYU

Noga Ren-Zewi, University of Haifa

Shubhangi Saraf, University of Toronto

Benjamin Sudakov, ETH Zurich

Avishay Tal, University of California, Berkeley

Salil Vadhan, Harvard University

Emanuele Viola, Northeastern University

R. Ryan Williams, MIT

Amir Yehudayoff, University of Copenhagen

Or Zamir, Tel Aviv University

Jeroen Zuiddam, University of Amsterdam

By Boaz Barak

A robust combiner for a cryptographic primitive $P$ takes multiple candidate constructions of $P$ and produces a secure construction of $P$ provided that sufficiently many of the candidates are secure. A closely related notion is that of a security amplifier, where given a weakly secure construction of $P$, we aim to obtain a (strongly) secure one. Intuitively, one may expect that any robust combiner should act as an amplifier by thinking of "good randomness" as inducing secure instances, and of "bad randomness" as inducing insecure instances. Formalizing this intuition, however, has turned out to be challenging. Despite significant progress, general results remain limited and confined either to specific primitives or only to the statistical setting. We establish a new framework of robust indistinguishability combiners, which greatly extends the class of combiners covered by prior work, and prove that they inherently act as security amplifiers. Our results extend to the computational setting, provided that the combiner makes a single query to each candidate. The new framework allows us to rederive previously known amplification results in a simplified manner, as well as prove new amplification results that have so far been out of reach. As our main application, we present the first security amplifier for functional encryption, resolving an open question that first arose in constructions of indistinguishability obfuscation, and for which a gap was discovered in previous proofs. Our amplifier transforms a weak scheme with any constant indistinguishability error into one with full negligible security.

Authors: Ben Foxman, Alex Lombardi, Fermi Ma, Barak Nehoran, John Wright

A central challenge in quantum algorithms and cryptography is reasoning about algorithms with oracle access to a random group element (e.g. a random function, permutation, or unitary). Can we efficiently simulate such algorithms? Can we determine what they know after t queries? A classical tool for this is lazy sampling: the oracle does not commit to the full group element upfront, but rather samples partial information about it on the fly. We study a quantum analog of lazy sampling: compressed oracles (or recording oracles). These are quantum data structures that allow on-the-fly simulation for quantum queries, originally introduced by Zhandry (CRYPTO '19) for random functions, and generalized to unitaries by Ma-Huang (STOC '25) and permutations by Carolan (STOC '26), and used to great effect in security proofs and lower bounds due to their interpretability. We define and analyze a general-purpose and interpretable path-recording oracle, derived from first principles, that perfectly simulates random elements of any closed subgroup of $U(N)$. Our oracle stores, in superposition, t input-output pairs, with updates described in terms of the commutant of the group's tensor power representation. This transparently records the information the algorithm has learned. Our oracle builds on recent work of Grinko-Yoshida (QIP '26), who gave a different general-purpose compressed oracle without clear interpretability. One interesting application of our path-recording is allowing direct comparisons between compressed oracles of different groups, giving a new technique for proving pseudorandomness results. For example, comparing $S_N$ and $U(N)$ yields what is arguably the simplest construction to date of pseudorandom unitaries: the product PC of a pseudorandom permutation and a random Clifford, improving on the prior PFC construction (Metger-Poremba-Sinha-Yuen, FOCS '24; Ma-Huang, STOC '25).

Authors: Alexander Drobyshev, Grigoriy Bokov

Boolean networks are finite discrete nonlinear systems whose long-term behaviour is organised by fixed-point and cyclic attractors. Detecting such recurrent states is important in applications ranging from gene regulation and neural computation to complex-network models, but the computational boundary between tractable and intractable attractor analysis is still not fully understood. We study that boundary from the perspective of local logical rules. We consider Boolean networks under parallel update whose coordinate functions are given by circuits over a fixed finite basis of a closed Boolean-function class, and ask whether the network has a cyclic attractor of prescribed exact period $k$. For every fixed $k\ge 2$, we obtain a complete complexity dichotomy over Post's lattice. The problem is $\mathrm{NP}$-complete whenever the local rule class contains majority-like self-dual rules or one of the two mixed conjunctive-disjunctive monotone families. In all remaining Post classes it is polynomial-time solvable, with affine rules and pure conjunctive or pure disjunctive rules with constants providing the boundary tractable cases. The results show that exact attractor detection is governed not only by the network architecture but also by the logical mechanism of local update: affine and one-sided rules preserve algebraic or order structure, whereas majority-like and mixed monotone rules can encode global Boolean consistency constraints.

Authors: Michal Ajdarów, A. R. Balasubramanian, Łukasz Orlikowski

Vector Addition System with States (VASS) are a ubiquitous model of infinite-state systems consisting of a set of non-negative counters which can be incremented and decremented. It is known that the reachability problem for VASS is Ackermann-complete. Because of this huge complexity, various over-approximations of VASS have been studied in the literature. One such over-approximation is continuous VASS (CVASS), in which the counters are (non-negative) rational numbers and whenever a vector is added to the current counter values, it is first scaled with an arbitrarily chosen rational factor between zero and one. It is known that the reachability problem for CVASS is $\mathsf{NP}$-complete. In this paper, we initiate the study of fixed-dimensional CVASS, i.e., CVASS with a fixed number of counters. We study both the reachability and coverability problems, under both unary and binary encodings as well as over both the non-negative and the rational semantics. This gives rise to a collection of eight different problems. As our main result, we prove a complexity dichotomy for all of these eight problems when the transition vectors are over the rationals: For dimension 1, all of the eight problems are in $\mathsf{AC}^1$, whereas for any dimension at least 2, all of the eight problems are $\mathsf{NP}$-complete. Furthermore, the hardness holds even when the underlying automaton is acyclic. To achieve this result, we present a new technique called the Egyptian prime fractions technique. Finally, we also study these problems when the transition vectors are over the integers. Except for dimension 2, we classify the complexity of these problems over the non-negative semantics: For dimension 1, all of the problems are in $\mathsf{AC}^1$, whereas for dimensions 3 and above, all of the problems are $\mathsf{NP}$-complete.

Authors: Sanyou Mei, Chunlin Sun, Yinyu Ye

We study Turn-Based Deterministic Forward Games (TBDFGs), the subclass of turn-based deterministic zero-sum games in which no directed cycle contains actions controlled by both players. This forward condition is strictly weaker than acyclicity: recurrent behavior may be arbitrarily rich within one player's states, while mixed-player feedback cycles are excluded. Our main contribution separates two algorithmic consequences of this structure. First, we analyze the simple strategy-iteration method of [11,14], a generic method for TBSGs whose execution neither tests for nor uses the TBDFG property. We prove that this structure-oblivious algorithm nevertheless has a strongly polynomial guarantee on every TBDFG. In particular, it terminates after at most $O(n^6m^4\log^4 n)$ simplex pivot steps. Thus, the forward property acts as a structural certificate for convergence even when the algorithm is not informed that the input has this property. Second, when the TBDFG structure is known in advance, a backward SCC propagation algorithm is proposed that solves a sequence of deterministic-MDP subproblems and improves the bound to $O(n^3m^2\log^2 n)$ simplex pivot steps. Together, these results show that forward structure both regularizes the convergence of a general strategy-iteration method and supports a sharper structure-aware algorithm.

Authors: Max Hopkins

The KKL Theorem, a seminal result in boolean function analysis, characterizes the structure of low-influence (non-expanding) functions on the hypercube. While recent years have seen breakthrough results across a variety of areas relying on analogs of the KKL Theorem beyond the cube (e.g., on product spaces, Grassmann graphs), further progress has been inhibited by our poor understanding of the phenomenon across more general domains. Motivated in this context, Bafna, Hopkins, Kaufman, and Lovett (STOC 2022) and Gur, Lifshitz, and Liu (STOC 2022) proved a generalized KKL-type Theorem for spectral high dimensional expanders (HDX). Their results, however, remain highly restricted due to strong quantitative expansion requirements on the underlying complex. In this work, we introduce a simple local-to-global method for analyzing low influence functions on simplicial complexes. Using this method we prove a local-to-global KKL-type Theorem: any simplicial complex whose links satisfy a KKL-Theorem also satisfies such a result globally. Building on Gotlib and Kaufman (RANDOM 2023), we also prove a weaker dimension-dependent KKL-type Theorem for simplicial complexes with any non-trivial (two-sided) expansion. As concrete applications of our framework, we give the first characterization of non-expanding functions on `combinatorial' HDX such as dense clique complexes and a corresponding Kruskal-Katona Theorem, as well as a small-set expansion theorem for the Ramanujan Complexes of Lubotzky, Samuels, and Vishne (EJC '05).

Authors: Marcelo Arenas, María Alejandra Schild, Bernardo Subercaseaux

We study the computational complexity of several counting problems on graphs. Each of these problems consists of counting orderings of the vertices or edges with adjacency constraints. We show $\#P$-completeness for all of them via a common new technique. Given a counting function $C$ of interest, we define a parameterized family of instances $G_q$, where the parameter $q$ controls the amplification of a simple gadget. After multiplying by an explicit factor $f(q)$, we show that the values of $f(q) \cdot C(G_q)$, for positive integers $q$, agree with a rational function in $q$ whose numerator and denominator can be interpolated in polynomial time. We then recover a $\#P$-hard function by evaluating this rational function symbolically at a limiting value $L \in \mathbb{Q} \cup \{\infty, -\infty\}$. With this methodology, we show $\#P$-completeness for the following counting problems: (a) successive vertex orderings of bipartite graphs, (b) st-numberings of graphs, (c) shellings of bipartite graphs, (d) linear extensions of N-free posets of height $3$, and (e) linear extensions of posets of height $2$. Result (d) settles a conjecture of Felsner and Manneville (2015). Although result (e) was first proved by Dittmer and Pak (2018), we include an alternative proof, using our technique, that does not rely on the result of Brightwell and Winkler (1991) about the hardness of counting linear extensions for general posets.

Authors: Henry Siegel

This paper explores the hardness of two popular recreational chess puzzles: The Knight's Tour and the Knight Exchange (Swap). The problem of finding a Knight's Tour is known to be NP-hard for any chessboard with holes and constant-time decidable for rectangular chessboards, so a natural direction is to explore the hardness of the problem for intermediate chessboard restrictions. In this paper, we show that Knight's Tour is NP-hard for connected boards. We also give a short polynomial-time reduction between the two problems, showing that the optimality version of Knight Exchange is NP-hard.

Authors: Francesco Carboni, Daniele Muscillo

Quoridor is a popular award-winning board game whose computational complexity, listed among the open problems of the Demaine-Hearn survey, remained open for nearly two decades. It was settled only recently, via a reduction from the formula game $G_{pos}$ tailored to Quoridor. We give a shorter and more general proof: a single reduction from Reisch's planar graph-Hex, in which wall placement encodes the path-connection structure of Hex. The same construction settles three closely related games -- Maze Attack and Pinko Pallino with no change, and Blockade with only minor adaptations -- showing that all four are PSPACE-complete, the latter three for the first time. More generally, our reduction shows that any race-and-wall game is PSPACE-complete.

Authors: Jose Pascual Gumbau Mezquita

This article establishes the foundational mathematical limits of Artificial General Intelligence (AGI) safety, proving that the core barrier is not the impossibility of an aligned state, but its structural unverifiability. We formalize this boundary through two central impossibility results: the Unverifiability Theorem of Alignment and the Theorem of Finite Structural Unverifiability of AGI Alignment. We ground this boundary at Trakhtenbrot's Wall, demonstrating that contemporary engineering defenses relying on finite hardware or halting architectures fail to escape logical obstructions. This failure manifests as an inescapable triad of containment failures: open domains yield fundamental undecidability (Rice and Gödel); universal finite verification collapses into algorithmic incomputability (Trakhtenbrot); and particular bounded environments trap the supervisor within intractable bounds in the worst case. As a direct structural corollary of these results, we derive the Soundness--Completeness--Tractability Trilemma, establishing that the mutual incompatibility of these three properties is a necessary consequence of descriptive complexity rather than an empirical anomaly. Finally, we map these theoretical bounds onto practical AI engineering, demonstrating that modern containment strategies are not temporary patches, but mandatory sacrifices of logical expressivity required to secure decidable fragments of safety.

Authors: Michael T. M. Emmerich

The continuous integral R2 indicator is a Pareto-compliant refinement of the classical finite-weight-vector R2 indicator, used in performance assessment, bounded archiving for a-posteriori multi-objective optimization, and skyline selection in databases. This work introduces a bidirectional perspective mapping between continuous integral R2 computation and integration over unions of anchored axis-aligned boxes. After translating the ideal point of a minimization problem to the origin, approximation points become strictly positive loss vectors, and the subgraph of the lower weighted Tchebycheff envelope over the weight simplex maps to the complement of an anchored-box union in reciprocal objective space. The Jacobian gives an absolute R2 formula as a weighted complement volume with density $(x_1+\cdots+x_N)^{-(N+1)}$, while differences of R2 values become finite weighted hypervolume differences. Hence, hypervolume algorithms that emit box decompositions can be reused by replacing ordinary box volumes with closed-form weighted box integrals. For $N$ objectives, this gives an output-sensitive overhead $O(2^N M)$ for an $M$-box decomposition, or $O(M)$ for fixed $N$. Using existing box-decomposition approaches, the integral R2 can be computed in $O(n \log n)$ for $N=2,3$, in $O(n^2)$ for $N=4$, and in $O\left(n^{\lfloor (N-1)/2\rfloor+1}\right)$ for $N\geq4$, with $n$ denoting the size of the approximation set. On the lower-bound side, exact value computation has an $Ω(n\log n)$ lower bound in the algebraic decision-tree model already in two objectives, this bound lifts to every fixed $N\geq2$, and exact computation is $\#P$-hard when $N$ is part of the input. Together, the proposed perspective mapping provides a powerful tool for transferring algorithmic and structural results between anchored-box union and hypervolume theory and integral R2 computation.

Authors: Srinivasa Rao P., Vangmayi P Reddy

Why overparameterised deep networks generalise so remarkably well remains one of the most stubborn open questions in machine learning theory. Classical frameworks like VC dimension and Rademacher complexity predict catastrophic overfitting in modern models, leaving a massive theoretical gap between theory and reality. In this paper, we bridge this divide by introducing a unified framework that links information theory, topology, and statistical mechanics to map the hard limits of deep learning. Central to our approach is the Entropic Learnability Horizon (ELH): a fundamental law stating that a network can only truly learn a target function if the Shannon entropy of the data manifold outpaces the topological entropy of the function's decision boundary, balanced by the von Neumann entropy of the network's weight space. We establish the Shannon-Topological Bottleneck Theorem, proving that when a target boundary's geometric complexity exceeds this informational horizon, the system undergoes a sudden entropic phase transition. It falls into a state of Informational Frustration - a glassy, rigid memorization phase where generalization becomes thermodynamically impossible. Using this lens, we show that the enigmatic phenomenon of "grokking" is actually an Entropic Release, where weights abruptly reorganise to unlock the bottleneck. Finally, we translate this theory into practice with Entropic Gradient Descent (EGD), an optimization algorithm that dynamically manages weight entropy to keep learning on track. Ultimately, this work repositions entropy not just as a tool for tracking uncertainty but as the fundamental physical currency that dictates whether a machine can learn.

Authors: Zhikai Shen, Jiasheng Qu, Chenyu Xu, Zhuo Huang, Chengkai Dai, Yongzhe Li, Guoxin Fang

Redundant robotic multi-axis additive manufacturing (MAAM) enables support-free and conformal fabrication, but trajectory optimization for long-horizon paths remains challenging under strict deposition-position constraints and time-varying collision constraints. This work proposes a computational framework for collision-aware trajectory optimization in redundant robotic MAAM. We first formulate nozzle-workpiece relative kinematics using a relative Jacobian, and develop a differentiable SDF-based collision model that captures fabrication-induced geometry evolution and provides optimization gradients. The deposition position is then enforced as a hard waypoint-wise equality constraint through iterative projection onto the self-motion manifold, with the loss gradient restricted to the corresponding tangent space. Experiments on an 8-DOF robotic MAAM platform with diverse long-horizon support-free and conformal toolpaths show that our method maintains a mean nozzle-position error below 10μm, reduces maximum joint jerk by up to $77.6\%$, and eliminates all sampled collision and orientation violations. Compared with the SQP-based baseline, it achieves up to a 10.2x speedup and improved convergence. Physical fabrication experiments further verify that the resulting smooth, collision-free trajectories enable successful printing of complex geometries with fewer visible deposition artifacts.

Authors: Kristijan Bartol, Frieda Hentschel, Nataliya Sadretdinova, Benjamin Russig, Melinos Averkiou, Yordan Kyosev, Stefan Gumhold

Garment fit and comfort depend critically on ease, the local allowance of excess material relative to the body. In existing design pipelines, ease is typically a byproduct of geometry or simulation rather than an independent design variable, making it difficult to specify, edit, transfer, or redistribute without re-running simulation or optimization. We propose a garment representation that embeds meshes directly on the surface of a parametric human body model and represents ease explicitly as spatially varying, anisotropic per-triangle scales. These scales act as primary design variables, decoupling the specification of material allowance from its physical deformation. Given a design specified by parametric and user-defined surface cuts together with local scale fields, we optimize sewing patterns that enforce the prescribed ease distribution while satisfying geometric and seam constraints. The representation enables three capabilities that are unavailable without explicit ease control: (1) direct specification and editing of local material allowance on the body surface; (2) intent-preserving transfer to new body shapes that reproduces the specified ease distribution without re-running simulation; and (3) intent-modifying pose adaptation that redistributes ease to relieve strain in high-stretch regions. We verify each of these experimentally: ease is closely retained after optimization, excessive strain is significantly mitigated for target poses, and the ease distribution is accurately transferred to target shapes. The approach is implemented as a virtual try-on framework, with physics-based cloth simulation used for final garment visualization. We will publicly release our framework and detailed documentation.

Authors: Panteleimon Rodis

This paper presents reproducible experimental evidence on unit-distance graph density that surpasses recent theoretical lower bounds. Our approach is based on a novel algorithmic exploration of the rational plane for the generation of unit-distance graphs. An efficient algorithm for this utility must perform a local-breadth search on a bounded and finite set of elements and generate a graph that potentially encompasses the general properties of a unit-distance graph, not affected by restrictions on its generation. To this end, we show that our approach accomplishes this purpose by overcoming the limitations of grid-based structures used in the literature for generating unit-distance graphs. Furthermore, the scaling exponent of the generated graph surpasses recent results.

Authors: Arun Vignesh Malarkkan, Manan Roy Choudhury, Utkarsh Byahut, Yash Ravindra Charde, Vivek Gupta, Yanjie Fu

Latent reasoning models perform multi-step inference directly in hidden-state space, yet the structure of these latent reasoning trajectories remains poorly understood. We show that contrastive refinement signals between stronger and weaker reasoning trajectories exhibit a highly concentrated low-rank structure, while unconstrained latent updates remain sensitive to paraphrases, checkpoint choice, and trajectory perturbations. These observations suggest that latent reasoning trajectories contain stable invariant directions mixed with unstable instance-specific variation. We introduce \textbf{Trajectory-Invariant Latent Refinement (TILR)}, a training-free intervention framework for identifying and manipulating stable reasoning directions in latent space. TILR first learns a low-rank invariant subspace from contrastive trajectory differences across inputs, then constrains latent interventions to this subspace while suppressing poorly aligned updates through an adaptive alignment gate. Across six reasoning benchmarks, we find that a small number of latent directions explain most variation between strong and weak reasoning trajectories. Interventions on these directions causally improve reasoning consistency and reduce trajectory instability under paraphrases and perturbations. TILR improves answer consistency under paraphrase by ~10% and reduces latent trajectory variance by up to $50\%$ while preserving reasoning accuracy. These results support a geometric view of latent reasoning in which transferable reasoning behavior emerges from stable low-dimensional structure within hidden-state trajectories.

Authors: Scott Duke Kominers

We construct a reduced planar convex body $R$ with thickness $Δ(R)=1$ and \[\operatorname{area}(R)=0.786215\ldots>0.785398\ldots=\fracπ{4}.\] Thus $R$ is a counterexample to Lassak's conjectured upper bound $\operatorname{area}\le(π/4)Δ^2$ for planar reduced bodies. The construction is given by an explicit support function, and the proofs use only elementary support-function, width, area, and contact-point computations.

Authors: Piotr Mitosek, Miriam Backens

Measurement-based quantum computing is a universal model of quantum computation in which successive product measurements of an entangled resource state drive the computation. The non-deterministic nature of measurements necessitates adaptivity to ensure an overall deterministic computation. Flow structures characterise cases in which such an adaptive correction procedure is possible. Recently, flow has been defined in a setting where the resource states are prime-dimensional qudit graph states rather than the usual qubit graph states. Yet, this qudit flow definition is more burdensome to work with than analogous definitions for qubits. Here, we give a simpler definition of qudit flow and consider various useful properties of this flow, drawing on results for the qubit case. In particular, we show how to focus qudit flow and argue that focused flow is canonical. We improve the previous algebraic formulation to capture focused flow and use it to obtain an $O(n^3)$ flow-finding algorithm (where $n$ is the number of qudits), matching the best known complexity for qubit flows and improving on the previous $O(n^4)$ result for qudits. Furthermore, we explore multiple flow-preserving transformations, thus opening a pathway to using flow for optimisation. These transformations include pivoting, removal and insertion of certain types of vertices, and reversibility of flow. Lastly, we propose an algorithmic approach to generating large qudit computations with flow, for testing or machine learning.

Authors: Themistoklis Haris, Diptaksho Palit

We initiate the study of property testing for $k$-submodular functions, a higher-dimensional analogue of submodular functions defined on partial partitions of a ground set. While $k$-submodularity retains the diminishing-returns flavor of ordinary submodularity, it also introduces a pairwise monotonicity constraint comparing competing assignments of the same element. This additional local structure makes the testing problem qualitatively different from the classical case. Our results show a sharp contrast between distance regimes. In the $\ell_p$ regime for $p \geq 1$, we prove that every bounded $k$-submodular function is close to a junta on the hypergrid. Combined with an implicit-learning tester for hypergrid domains, this yields a constant-query tester for $k$-submodularity. In the Hamming distance regime, $k$-submodularity admits two qualitatively different local witnesses -- violated squares for diminishing marginal gains, and violated triangles for pairwise-monotonicity failures -- and the latter has no counterpart at $k=1$. We prove density theorems for both witness types via repair on filters and ideals of partial partitions, yielding non-adaptive, one-sided sub-exponential-query testers for the two component properties of $k$-submodularity. We then exhibit a configuration in which the two repair directions are forced into opposition on a shared vertex, identifying a structural barrier to combining these into a tester for the full property. Finally, for bounded-range functions, we give an adaptive tester for monotone $k$-submodularity via a pseudo-DNF representation and learning on the hypergrid. Several of the structural and learning tools developed here may be useful for testing other properties over product domains.

Authors: Laura Lewis, Ewin Tang, John Wright

We design an algorithm for learning the coefficients of an $n$-qubit constant-local Lindbladian to $\varepsilon$ error with $O(g d^2 \log(n) / \varepsilon^2)$ total evolution time, where $g$ is the single-site energy and $d$ is the (approximate) degree of the interaction graph. Though Lindbladians present new challenges not present in the special case of Hamiltonians, our algorithm achieves the suite of desiderata attained by state-of-the-art Hamiltonian learning algorithms: (1) it uses non-adaptive, ancilla-free randomized Pauli measurement circuits with a time resolution of only $Θ(1/g)$; (2) it works without knowledge of the structure of the unknown Lindbladian; (3) it depends on a smooth form of degree, thereby supporting the learning of quasi-local and power-law Lindbladians. Our algorithm is a simple iterative method, where the objective function consists of Fourier coefficients of the Lindbladian restricted to few-site regions. Its analysis identifies the difficulty unique to open systems, which we call "confusing" terms. For settings where the "confusion" is limited, the performance of the algorithm improves. We demonstrate this for the case of structure learning of Hamiltonians from access to real-time evolution, where we obtain a new algorithm that is significantly simpler than previous work. In addition, using the same iterative method, we design the first efficient algorithm for structure learning Hamiltonians from high-temperature Gibbs states.

Authors: Amir Carmel, Robert Krauthgamer, Nir Petruschka

The geometric median problem asks to find a point in $\mathbb{R}^d$ that minimizes the sum of Euclidean distances to an input set. It is a classical problem in computational geometry and appears as a subroutine in numerous optimization tasks, many of which require the solution to satisfy additional structural constraints. A common approach to reduce the input size is to construct a coreset, which is a small weighted subset that faithfully represents the input for a specific optimization problem. Strong coresets preserve the cost of every candidate solution but require linear time to construct; weak coresets admit sublinear construction, in fact by uniform sampling, but only preserve near-optimal solutions, which is insufficient when the solution is constrained. To address this, we focus instead on the recently introduced intermediate notion of a \emph{stable coreset}, which simultaneously handles all constrained variants. Currently, there is a large gap between the known sample sizes for stable and weak coresets. Our main result is that a uniform sample of size $O(ε^{-2} \log \tfrac{1}ε)$ is a stable $(ε, O(ε))$-coreset for the geometric median, with high constant probability, and this bound is tight up to the logarithmic factor. Our analysis adapts recent machinery of Carmel and Krauthgamer (ICLR 2026) for constructing stable coresets, which incurs an $O(\log d)$ factor. We show an iterative argument that progressively reduces the sample size, and eliminates this dependence on the dimension $d$. At a high level, this approach resembles the technique of iterative size reduction, which is applicable for strong coresets but not for weak coresets.

Authors: Jung-hun Kim, Vianney Perchet

We study prophet inequalities with discounted rewards, where i.i.d. base rewards are multiplicatively discounted over time. Our main message is that even this structured and arbitrarily weak form of nonstationarity can erase the classical advantage of the stationary i.i.d. setting. Focusing on single-quantile threshold policies, we show that the competitive ratio transitions from the classical $1-1/e$ guarantee to a fundamental $1/2$ barrier as discounting accumulates over many phases in a canonical regime with a common-decay factor and equal-length phases. We further show that, in the same regime, the $1/2$ barrier persists even for arbitrary stopping rules. Consequently, i.i.d. base rewards under discounting can be as hard as the fully non-i.i.d. case. On the algorithmic side, we design single-quantile threshold rules that attain the tight bounds by calibrating acceptance decisions to an effective horizon induced by discounting, and we extend this calibration to heterogeneous decay factors and unequal phase lengths. We further show that a similar discontinuous breakdown persists in an infinite-horizon continuous-decay benchmark, where arbitrarily weak decay collapses the stationary benchmark from $1$ to $1/2$.

Authors: Buxin Su, Weijie Su, Chendi Wang

In 2020, the U.S. Census Bureau adopted differential privacy for the Decennial Census by injecting integer-valued Gaussian noise into published census tabulations. Exactly evaluating the privacy guarantees of these data releases would enable the Bureau to determine the absolute minimum noise required to satisfy a given privacy budget, preventing the injection of unnecessary excess noise and thereby substantially enhancing the statistical utility of the data for downstream applications such as federal funding allocation and political redistricting. In this paper, we introduce a computationally efficient and mathematically rigorous quadrature method to evaluate the exact privacy profile of practical, large-scale census releases under the composition of heterogeneous discrete Gaussian mechanisms. Mathematically, this problem reduces to evaluating the tail probabilities of high-dimensional convolutions of integer-valued random variables sampled from heterogeneous discrete Gaussian distributions under exceptionally stringent numerical error tolerances (e.g., $10^{-35}$). By recasting the exact privacy accounting as a numerical integration problem via the discrete Fourier transform, we explicitly exploit the exponential convergence of the trapezoidal rule for complex analytic, periodic characteristic functions. Furthermore, to overcome the computational bottleneck of evaluating highly oscillatory integrands in high dimensions, we develop a sieve algorithm that identifies and prunes negligible quadrature nodes, accelerating the computation by three orders of magnitude. Taken together, these numerical innovations enable the first exact, assumption-free privacy accounting for the 2020 Census Demographic and Housing Characteristics File, achieving a 1,824-fold speedup over prior methods while maintaining census-mandated error tolerances.

Authors: Seokbeom Kim, O-joung Kwon, Myounghwan Lee

Cycle rank is a depth parameter for digraphs introduced by Eggan in 1963. Gruber (DMTCS 2012) and Giannopoulou, Hunter, and Thilikos (DAM 2012) asked whether the problem of determining if a given digraph has cycle rank at most $w$ is fixed-parameter tractable parameterized by $w$. We provide such algorithms for semi-complete digraphs, and for digraphs of bounded directed clique-width. Specifically, we show that given an $n$-vertex semi-complete digraph $G$ and an integer $w$, one can in time $\mathcal{O}(9^{(w+1)4^{w+2}} \cdot n^2)$ determine whether $G$ has cycle rank at most $w$. The proof is reduced to the case of bounded directed clique-width, and we then show that given an $n$-vertex digraph $G$ with a directed clique-width $k$-expression and an integer $w$, one can in time $\mathcal{O}(9^{(w+1) 4^k} \cdot n)$ determine whether $G$ has cycle rank at most $w$. Additionally, we consider the \textsc{Minimum Feedback Arc Set} problem on semi-complete digraphs, and show that it can be solved in time $n^{\mathcal{O}(w)}$, where $w$ is the cycle rank of the given semi-complete digraph.

Authors: Sander Gribling, Aaron Sidford, Chenyi Zhang

We provide algorithms that compute $ε$-estimates of the $\ell_p$-Lewis weights of a matrix $A \in \mathbb{R}^{m \times n}$ for $p \geq 4$ using $O(p^2 \log(m/ε))$ rounds of leverage score computation, where $\ell_p$-Lewis weights and leverage scores are both standard measures of row importance. This improves upon the state-of-the-art round complexity of $O(p^3 \log(m/ε))$ due to Fazel, Lee, Padmanabha, and Sidford (2022). We obtain our results by carefully applying a local variant of relatively smooth gradient descent to primal and dual forms of the $\ell_p$-Lewis weight optimization problem and providing tools to convert between different notions of approximate $\ell_p$-Lewis weights.

Authors: Chao Wu, Marek Chrobak

We study the problem of gossiping (all-to-all information exchange) in ad-hoc radio networks. Such a network is represented by a strongly-connected directed graph with \(n\) vertices, whose topology is initially unknown to the protocol. In 2004, Gasieniec, Radzik, and Xin gave a \(\tilde O(n^{4/3})\)-time deterministic protocol for this problem, and closing the gap between their upper bound and the \(\tildeΩ(n)\) lower bound on the time complexity of gossiping remains a central open problem. We develop a deterministic protocol for gossiping in ad-hoc radio networks that achieves running time \(\tilde O((mn)^{3/5})\) for directed graphs with at most \(m\) edges. Our protocol improves on the \(\tilde O(n^{4/3})\) bound when \(m = O(n^c)\), for \(c < 11/9\). We also present a \(\tilde O(Δ^{1/2} n)\)-time gossiping protocol for \(Δ\)-regular graphs.

Authors: Vitaly Aksenov, Rene van Bevern, Artem Shilkin

Most work on efficient concurrent ordered indices, such as concurrent binary search trees, B-trees, skip lists, etc., has focused on data structures that provide good \emph{worst-case} guarantees. In real workloads, objects are often accessed at different rates, since access distributions may be non-uniform. Many efficient distribution-adaptive data structures exist in the sequential case; however, they are often complicated to make efficient in the concurrent case. The most prominent distribution-adaptive data structure is Splay Tree. Its most important advantage is that it does not store any balancing information and provides a reasonable performance improvement on extremely skewed workloads, such as Zipfian workloads. This paper proposes a splay-like rotation design for concurrent binary search trees. Instead of moving an accessed node to the root, rotations use two depth thresholds that are based on the static-optimality complexity computed from the number of accesses to the node: a node is rotated only when it is substantially deeper than the upper threshold, and rotations of the node stop before reaching the lower threshold. This design aims to preserve the main practical benefit of splaying on skewed workloads while reducing contention near the root. We present two variants of the rotation design: one using an exact 64-bit access counter per node and one using a 6-bit approximate counter. We prove static optimality for the corresponding sequential read-only tree and evaluate both rotation designs by implementing them on top of the concurrent AVL tree of Bronson et al. Our experiments show that the approach can improve throughput on several skewed workloads.

Authors: Marcin Bienkowski, Joakim Blikstad, Jarosław Byrka, Martín Costa, Yann Disser, Annette Lutz

We consider submodular maximization under increasing cardinality constraint and ask for a good incremental solution, i.e., an ordering of the ground set such that each prefix of the ordering yields a good solution for its respective cardinality. A classical result in this setting is that the greedy algorithm achieves a competitive ratio, i.e., an approximation guarantee across all cardinalities, of $\mathrm{e}/(\mathrm{e}-1) \approx 1.582$. No better general guarantee was previously known. We present an adaptive scaling algorithm achieving a competitive ratio of $1.373$. We complement our result by a deterministic lower bound of $1.25$ on the best possible competitive ratio for incremental submodular maximization.

Our research group at Aalto University is hiring 3 postdocs + 3 doctoral students to work on distributed quantum algorithms in projects funded by an ERC Advanced Grant and a QuantERA grant. Application deadline: July 31, 2026. Start date: October 2026–October 2027 (negotiable).

Website: https://www.aalto.fi/en/open-positions/doctoral-students-theoretical-foundations-of-distributed-quantum-computing
Email: jukka.suomela@aalto.fi

By shacharlovett

Our research group at Aalto University is hiring 3 postdocs + 3 doctoral students to work on distributed quantum algorithms in projects funded by an ERC Advanced Grant and a QuantERA grant. Application deadline: July 31, 2026. Start date: October 2026–October 2027 (negotiable).

Website: https://www.aalto.fi/en/open-positions/postdoctoral-researchers-theoretical-foundations-of-distributed-quantum-computing
Email: jukka.suomela@aalto.fi

By shacharlovett

Spreading the Gospel of Theoretical Computer Science to an Ω(1) Fraction of Humanity (Or, How We Can Do Like the Physicists)
Scott Aaronson’s Trevisan Award Acceptance Speech
Salt Lake City, Utah, June 23, 2026

Thank you so much! It’s one of the highlights of my life, frankly, to accept the first-ever Luca Trevisan Award for Expository Work in Theoretical Computer Science—because of, firstly, what this entire STOC community means to me, but also what Luca Trevisan in particular meant to me. Luca was one of the main people who taught me complexity theory—first at an IAS summer school in 2000, then at UC Berkeley, where I took two of his courses and TA’ed for him. As a member of my dissertation committee, Luca once stood on a street corner in San Francisco to meet my friend to sign the signature page of my thesis, as I struggled to get the thing in by the deadline. Later, Luca’s theoretical computer science blog, In Theory, bounced off of my blog.

I wish Luca were here now. But knowing him as well as I did for a quarter century, I feel like I know what he’d say if he learned that I had received the inaugural prize that bears his name. I imagine he’d slap his forehead and say “Seriously, there was no other option??” But I’d like to think that he’d eventually reconcile himself to the choice!

By the way, I noticed that in the committee’s prize announcement, which I found so moving, they added a special paragraph at the end that basically said, “please don’t imagine that to win this prize in the future, you need to behave the way Aaronson behaves. You can just write beautiful textbooks or survey articles or whatnot, and be normal and sane.”

The foundation of my career is that I realized 25 years ago that there were better theoretical computer scientists than me—like many in this room, or like Ryan Williams or Andris Ambainis, both of whom I knew at the time. Certainly there were better quantum physicists than me. There were better writers, better expositors, better performers. On the other hand, if you looked specifically at the intersection of computational complexity and quantum physics and standup comedy, that was just this totally uncontested territory!

I’ll let you in on a secret: pretty much everything I’ve done for decades has just been drawing out one joke. That joke is, basically, “computer scientists they be like this, but physicists they be like that.” The physicists they be like [exaggerated doofus voice] “duhhhh, NP, what’s that stand for? Not Polynomial?” See, but then there’s also a Rodney Dangerfield aspect to it, because it’s like, how come we never get as much respect as the physicists get? (Though when we do get that respect, I confess that I complain all the more, because then I lose my shtick…)

It’s true that the physicists have certain built-in advantages. They had Einstein, Stephen Hawking, the atom bomb—and just the fact that they’re ultimately talking about, or trying to talk about, the world that we can see and touch. A black hole is an actual place that you could visit, even though I wouldn’t recommend it. But physicists also have much better names for things than we do. I mean, black hole? Big Bang? Quark? Gluon? Supersymmetry? Dark matter?

Meanwhile, what names have we got? TFNP. NC¹. And worst of all, PP. These are names that you want to flush down the toilet. But also, the concept of a zero-knowledge protocol, or a two-source extractor, just inherently take longer to explain to people than the concept of a particle, or even a field—even though the latter also turn out to be extremely abstract and mathematical when you push on them. Ask a physicist what a particle is, they’ll tell you that it’s an irreducible representation of the Poincaré group. See, but people think they know what a particle is, it’s just a tiny little hard sphere that moves around, and that’s good enough for them.

So then, how can we win the grand popularity contest against the physicists? How can we, as I put it in my title, spread the gospel of CS theory to a constant fraction of the human race? In my view, the first step is to reframe who we are and what we’re about. We’re not this obscure little community off to the side, proving its little theorems about derandomization and catalytic space. No! What we are is the conceptual and mathematical core of computer science, the field that’s changing the face of civilization in obvious and undeniable ways.

This was even true a long time ago. The physicists had Galileo and Einstein? Well, we had Turing, a figure so heroic and so tragic that no one would’ve believed him as a fictional character. And while we’re at it, we’ll claim Gödel and Shannon and von Neumann, Leibniz and Babbage and Ada Lovelace—they’re all ours too.

That’s our proud history. But then when we turn to today, it’s like, holy crap! Even the densest ignoramus can now see how deep intellectual ideas originating in CS are changing the world.

Blockchains—some people might wish they’d never been invented, but they were invented, so we all need to think about how they change the world’s economy for better and worse. And of course, they’re fundamentally based on hardness assumptions; they couldn’t exist in a world where NP was easy.

Part of my outreach job these days is to explain to finance people, over and over, why a quantum computer could break the elliptic curve signature schemes used by Bitcoin and many other coins, but would have only a more modest effect on the proof-of-work part, the hash function. And it’s like, if you actually want to know, then we need to talk about BQP versus NP, Grover’s algorithm and its optimality, black-box problems with and without abelian group structure—and now we’re deep into TCS!

Speaking of quantum computing—even if we set aside the question of whether quantum computing is going to revolutionize materials science or chemistry or pharmaceuticals design—or whether it will revolutionize AI and machine learning and optimization [I shake my head, make a thumb-down, and blow a raspberry]—even if we set aside those practical questions, quantum computing plausibly represents the most dramatic test of quantum mechanics itself that we’re ever going to see. And it now looks clear that we will see that test within the next decade or sooner. One way or the other, we’re going to learn the truth.

People sometimes ask me, why did it take until the 1980s for anyone to propose the idea of quantum computing? You know, Heisenberg and Schrödinger were in the 1920s, Turing was in the 1930s, so it seems like all the ingredients were in place a half-century earlier! In my Quantum Computing Since Democritus book, I reflected on this, and I think the deepest answer is that not quite all of the intellectual ingredients were in place. Quantum computing is something that it doesn’t make a great deal of sense even to ask about until you’ve established polynomial versus exponential, and even P and NP and NP-hard, as central concepts. And that’s what didn’t happen until the 1970s.

But of course, the biggest thing that our CS concepts have unleashed on humanity—the thing that the entire world now realizes holds even greater promise and greater peril than nuclear energy did in the last century—is [pause for effect] the Razborov-Smolensky lower bound method.

No, I’m kidding of course. It’s generative AI.

Twenty years ago, I remember people in our community—was it Fortnow? Impagliazzo? I’m not sure—saying, “you know the real reason why P vs. NP is such an important problem? Suppose P=NP, via an algorithm that was fast in practice. Then it’s not just that you could break all the encryption systems, or have your computer find a proof of the Riemann Hypothesis, or whatever. No, it’s that you could program your computer to find the shortest efficient compression of, for example, the full text of Wikipedia. For in order to create that compression, it seems plausible that your computer would need to create an AGI as a byproduct.”

I remember thinking to myself: “that’s an amusing thought experiment, I’ll need to steal it sometime, but still, what an utterly simplistic vision of the nature of intelligence! There has to be more to intelligence than sheer data compression!”

Fast forward to spring 2022, when I accepted an invitation to go on leave for a couple of years, to join what was then a relatively obscure little nonprofit foundation by the name of … err … OpenAI. When I flew to San Francisco to start my assignment, I had lunch with Ilya Sutskever, the cofounder of OpenAI and then its chief scientist. And Ilya said to me, “Scott, let me explain to you how we think about things here at OpenAI. For us, intelligence is fundamentally about prediction, and prediction is fundamentally about compressing your training data. As you know, Kolmogorov complexity is uncomputable, but one can get better and better computable upper bounds on it. We conjectured that, in order to get sufficiently good at predicting and compressing all the text on the Internet, you’d need to build a model of the entire world that had led to that text being written. And we made a gamble that large neural nets would do that well enough, despite the problem’s worst-case intractability.”

That conversation was when it hit me that, if only we in CS theory had taken our own concepts and thought experiments more seriously, one of us could’ve started OpenAI 15 or 20 years ago. So OK, we didn’t, and that’s why I flew coach to get here. But this is the kind of story that it seems to me we could be singing from the rooftops.

(Incidentally, the reason why OpenAI wanted me back in 2022, was to use theoretical computer science to figure out how to make AI safe for humanity. Alas, that problem is still open! But I’m thrilled that there are so many sessions about exactly this question at STOC this year, and I hope many of you will choose to get involved.)

In the rest of this talk, I’d like to offer some advice—such as I have—for any of you who’d like to try your hand at speaking or writing or blogging or podcasting about theoretical computer science for a broad audience. You see how my hair is starting to gray? Yeah, that’s what authorizes me to go into advice mode.

Let’s start with the obvious: meeting the audience where they are. This is something that I learned years ago from Steven Rudich, who along with Luca, was another irreplaceable figure who our community recently lost, and lost too soon. I remember 26 years ago, at that same IAS summer school where I learned from Luca, Rudich gave the students a talk about how to give talks. In it, he showed a cartoon of someone lecturing. And there were little thought bubbles that said:

What the speaker thinks the audience is thinking: MORE! HARDER! FASTER! Ah yes, QED, truth is beauty and beauty is truth!

What the audience is actually thinking: What the hell are they talking about? When is this over? Can I get a date with the person sitting next to me?

You know, this misconception that because something has become obvious to you, after thinking about it for years, therefore it should be equally obvious to your readers or listeners encountering it for the first time? This is what Steven Pinker dubbed “the Curse of Knowledge,” and calls the most fundamental problem of all exposition. (I could mention the related misconception that because something has become interesting to you, therefore it’s interesting to your audience. But you can make just about anything that’s interesting to you interesting to your audience, by telling a suitable story about it.)

What can you do about the Curse of Knowledge? Practice giving a buttload of talks to undergrads, high school clubs, even physicists, and listen to the feedback you get. If the same weird confusion shows up at least twice, it’s a safe bet that it’s going to keep showing up—which means, now you can anticipate and preempt it the next time you explain the same concept.

But it’s not just misconceptions that you should listen for. Listen for which of your metaphors and anecdotes actually land. Certainly listen for which of your jokes get a laugh. Use those more the next time. And if saying, for example, “hur hur, I’m in a quantum superposition of two different topics that I could talk about next”—if that fails to get a laugh, then DROP IT.

Eventually, you’ll build up what Carl Sagan once called “consumer-tested stepping stones”: that is, a library of jokes, anecdotes, and metaphors that can get you from wherever you see the audience is to wherever you need them to be. Here’s an intentionally tiny example of one of my stepping stones: “Why is P contained in NP? Because the verifier just says to the prover, dude, take a hike, you’re not needed here.” Or another stepping stone, for when we reach the question of the likelihood of P=NP: “look, if we were physicists, we would’ve declared P≠NP to be a law of nature. We would’ve given ourselves Nobel Prizes for the discovery of that law. And if it later turned out that P=NP? We’d just give ourselves more Nobel Prizes for the law’s overthrow!”

This brings me to a broader point. CS theory is unusually rich with facts that are true for silly or absurd or ironic reasons. Lean into that! Don’t hide it!

I mean, “if NP has small circuits then this theorem is true, but if NP doesn’t have small circuits, the theorem is again true, but now for a totally different reason”? That’s sidesplittingly hilarious! OK, maybe only to some of us.

Or why does IP=PSPACE? An alien lands and is like, “I COME TO EARTH TO TELL YOU THAT WHITE HAS THE WIN IN YOUR GAME CALLED CHESS.” And we’re like, “why should we believe that?” So the alien is like, “LET US PLAY A GAME. I’LL PLAY WHITE AND WILL WIN.” And we’re like, “oh, we assume you’re smarter than us! You came all the way here in a spaceship and all! But that still doesn’t prove it.” So the alien is like, “THEN LET US PLAY A DIFFERENT GAME, MATHEMATICALLY EQUIVALENT TO CHESS, INVOLVING SUMS OF POLYNOMIALS OVER A FINITE FIELD. IN THIS TRANSFORMED GAME, THE BEST YOU CAN DO IS TO MOVE RANDOMLY. SO IF I STILL WIN, YOU’RE STATISTICALLY CERTAIN I WOULD’VE WON REGARDLESS OF HOW YOU PLAYED.” It’s like, dude. Dude!

Of course, our founding irony, our founding absurdity, was self-reference and diagonalization. Like, “you can’t predict what any human brain will do 5 seconds from now, because if you could, you could predict what you yourself were going to do 5 seconds from now, and then do the opposite of that!” BOOM! Therefore the halting problem is undecidable and the Time Hierarchy Theorem is true, QED. But beyond that: “black-box program obfuscation is impossible, because one thing you can always do, if given the actual code of a program, is to run the program on its own code and see what happens.” Dude! Or: the reason why it’s so hard to prove P≠NP, is that it’s presumably true that P≠NP. That’s a wisecrack that, in the context of the Natural Proofs barrier, becomes so much more than a wisecrack.

One special case of leaning into absurdity concerns the central role in our field played by asymptotics. I’m always slightly at a loss when someone asks me, “so, how many times faster would a quantum computer be than a classical computer? A million times faster? A billion?”

Part of me wants to reply: “I must educate you about polynomial versus exponential scaling until you see the profound error of your question, and retract it.” But another part of me simply wants to say: “depending on the problem, a quantum computer could be anywhere from not faster at all to, let’s say, 10¹⁰⁰⁰⁰ times faster.”

The truth is, I think we need to do both. Anytime you’re talking about asymptotics to laypeople, if you can plug in some representative numbers, it will help them understand what you’re talking about. And then, if the asymptotics are what really control the real-world numbers, so much the better! If, on the other hand, the asymptotics are comically disconnected from the real-world numbers—if, for example, you’re trying to improve something from log*(n) to Ackermann^-1(n) or whatever—well then, you can lean into that as an additional source of humor.

Alright, one last piece of advice. Tell true stories about how you came to understand or discover whatever it is that you’re talking about. Don’t be like the mathematicians who love to cover their tracks.

When people ask me how I proved the lower bound on the number of steps needed for a quantum computer to find collisions in a list—a centerpiece of my PhD thesis, and one of the two or three hardest technical things I’ve done in my career—I say, look, I was 20 years old and I had no social life. So I just pulled many all-nighters trying every possible approach. Eventually, I came across some complicated expression that had no right to be a polynomial. But somehow, every term in the denominator cancelled against a corresponding term in the numerator, and it was a polynomial! And that let me use the polynomial method to prove a lower bound. Why was it a polynomial? I still don’t really understand, a quarter-century later! My point is, people want the truth.

The secret of blogging is that, even if people despise what you’re saying, even if they think it’s wrong, offensive, problematic, cringe, you name it, they need to trust that you’re telling them the truth of what you know or believe or remember about the subject at hand, the same as you’d tell your closest friend.

In summary: we, the CS theory community, are sitting on top of one of the greatest conceptual and intellectual goldmines of our whole civilization. I exhort everyone here: please help tell the world about it! As you do so, think about how to honor Luca’s memory and make him proud. But also think about how to make me, and my silly little blog, superfluous and obsolete.

Thank you for this honor, thank you for the incredible privilege of being part of the CS theory community, and thank you for listening.

By Scott

Authors: Noah Fleming, Stefan Grosser, Toniann Pitassi, Robert Robere

We introduce a new family of propositional proof systems, denoted , for an arbitrary TFNP search problem $R$. Informally, a refutation of a CNF formula $F$ in is given by a polynomial-time reduction from the false-clause search problem $Search_F$ to $R$, combined with an Extended Frege proof that the reduction is correct. These are motivated in two ways: 1. They are the propositional translations of witnessing theorems in bounded arithmetic, by which proofs of $\forall Σ^b_1$ formulas $φ$ in a theory $T$ imply algorithms solving the search problem for $φ$ in a TFNP class corresponding to $T$. 2. They are a white-box analogue of the characterizations of proof systems using decision tree reductions to black-box TFNP problems. We consider the proof system , where Iter is a complete problem for PLS. We prove that is polynomially equivalent to the sequent calculus $G_1$, and also to the implicit Resolution proof system [EF, Resolution]. Hence $G_1$ and [EF, Resolution] are equivalent, which is the first characterization of an implicit proof system by a classical proof system beyond the work of Wang. We also consider for general TFNP relations $R$. We observe that if EF can prove that a search problem $R$ is in FP, then is polynomially equivalent to EF. This contrasts to our above result, which shows that Extended-Frege provable reductions to $Iter$, a problem widely believed not to be in FP, yields a proof system ($G_1$) that is believed to be stronger than Extended Frege. Finally, we show that for any proof system $P$ which is sufficiently strong, there is a polynomial-time computable search problem $R_P \in $ FP such that is polynomially equivalent to $P$. Letting $P =$ [EF, Resolution] and combining our two results shows that is polynomially equivalent to .

Authors: Therese Biedl

In a recent paper, Antić et al.~studied collections of planar orthogonal drawings of a graph where every edge is unbent in at least one drawing. This paper generalizes this concept to non-planar drawings, and shows that then two drawings always suffice (for planar drawings three drawings are sometimes needed). The results can also be generalized to $s$-grid drawings for $s\geq 3$.

Authors: Neha Pant, Ryan Williams

For several well-studied triangle detection problems in the literature, the trivial enumeration algorithms are known to be optimal (up to the exponent) assuming popular fine-grained conjectures. For example, All-Edges Sparse Triangle and Sparse Monochromatic Triangle where each node has degree $n^δ$ for some $δ< 1$, and the Exact Triangle where edges have arbitrary weights, all have this property under the 3SUM Conjecture. However, as there are slightly nontrivial algorithms for 3SUM, it is natural to wonder if the trivial algorithm for these tricky triangle tasks might also be improved. Applying a variety of techniques from randomized algorithms, circuit complexity, and communication complexity, we present the first improvements over the trivial algorithms for each of these problems in the Word RAM model. Moreover, our algorithms can be implemented with only polysize $AC0$ operations on words. Extending our techniques, we also show how to solve the notorious 4-cycle detection problem on $n$-node graphs in $o(n^2)$ time, in a Word-RAM model with word size $w > ω(\log^2 n)$. Along the way, we show how to sort $n$ items over a universe of size $2^u$ using only $AC0$ word operations in $O(n u \log n)/w$ time.

Authors: Kijung Jeon, Thuy-Duong Vuong, Molei Tao

Inference-time scaling is a promising paradigm to improve generative models, especially when outputs must satisfy structural constraints or optimize downstream rewards. We consider Masked Diffusion Model (MDM) and introduce MDM-VGB, a discrete diffusion sampler that augments unmasking generation with theoretically principled reward-guided remasking. Inspired by the recent success of the classical Jerrum-Sinclair backtracking Markov chain in reward-tilted generation, MDM-VGB extends the backtracking random walk from a fixed prefix tree to a masked-state graph, allowing tokens to be unmasked and remasked at arbitrary positions. The resulting sampler favors unmasking and remasking moves that lead to higher-value partial configurations, enabling both effective high-reward generation and efficient repair of low-reward samples. We prove that MDM-VGB is robust to process-verifier noise and achieves quadratic complexity, while popular test-time heuristics such as best-of-$N$ can incur exponential complexity due to error accumulation. Our theoretical findings are corroborated by strong empirical performance, particularly on popular constraint-satisfaction and scientific benchmarks such as Sudoku and QM9.

Authors: Pan Peng, Yuyang Wang, Joy Qiping Yang, Yichun Yang

We study lower bounds for estimating the spectral density of the normalized adjacency matrix of a graph. Previously, Cohen-Steiner et al. [KDD 2018] proposed an algorithm for $\varepsilon$-approximate spectral density estimation in the Wasserstein-1 distance, using $2^{O(1/\varepsilon)}$ random walks initiated from uniformly random nodes in the graph. Later, Jin et al. [COLT 2023] established a nearly matching exponential lower bound for \emph{weighted} graphs, assuming the algorithm has access to samples from random walks started at random nodes. It was left open whether this lower bound could be extended to \emph{unweighted} graphs. In this paper, we answer this question in the affirmative by proving an exponential lower bound for unweighted graphs. Specifically, we show that no algorithm can compute an $\varepsilon$-approximation to the spectrum of a normalized graph adjacency matrix with constant success probability, even when given the full transcripts of $2^{Ω(1/\varepsilon^{1/6})}$ random walks, each of length $2^{Ω(1/\varepsilon^{1/6})}$, started from uniformly random nodes.

Authors: Mohsen Ghaffari, Magnús M. Halldórsson, Yannic Maus, Alexandre Nolin

Graph shattering is a central technique underlying sublogarithmic-time distributed algorithms in the LOCAL model. Its analysis typically relies on bounding the probability that large sets of distant nodes remain unresolved, often via independence assumptions justified by locality. We show that these assumptions fail for pre-shattering procedures that run for super-constant rounds, where dependencies accumulate over time. As a result, several standard shattering arguments in the literature are incomplete, including those for maximal independent set, $(Δ+1)$-coloring, and the distributed Lovász Local Lemma (LLL). We provide a systematic repair of these analyses. Our main contribution is a corrected shattering analysis of the Fischer--Ghaffari LLL algorithm. In addition, we develop general tools that capture common patterns in modern algorithms and yield the required decay bounds without relying on independence. We also present explicit counterexamples to commonly used shattering lemmas. Overall, we establish a robust and reusable foundation for shattering arguments in the presence of long-range dependencies.

Authors: Andreas Göbel, Matthew Jenssen, Marcus Michelen, Marcus Pappik, Will Perkins, Leon Schiller

A recent breakthrough of Chen, Chen, Chen, Yin, and Zhang shows rapid mixing for Glauber dynamics for the hard-core model on random regular graphs beyond the tree uniqueness threshold. Their approach builds upon the literature of various local-to-global techniques and applies to a more general setting of discrete distributions supported on downward-closed set families. We give a short and self-contained proof via a Bochner--Bakry--Émery approach and directly show a Poincaré inequality by expanding the Dirichlet form in terms of the $L^2$-norm of the generator applied to a test function and eliminating a sum of squares term. Our proof is a streamlined version of an argument of Kondratiev, Kuna, and Ohlerich used to study spatial birth-and-death dynamics for Gibbs point processes in the continuum, which we adapt to the discrete setting.

Authors: Ilan Doron Arad, Jonathan Gal, Seffi Naor

Dominating Set is a fundamental problem in graph theory: given a graph, find a minimum-weight subset of vertices such that every vertex is either selected or adjacent to a selected vertex. In online settings where vertices arrive sequentially, comparing algorithms against an offline optimum with full knowledge of the input leads to extremely strong lower bounds, where even a simple star graph shows that any online algorithm must have competitive ratio $Ω(Δ)$, with $Δ$ the largest degree of any vertex in the graph, matching the trivial strategy of selecting all vertices. We study the incremental dominating set problem, where the optimal algorithm is constrained to the same choices available to online algorithms. This introduces a benchmark that enables a meaningful comparison between algorithms. We present the first results for vertex-weighted graphs and randomized algorithms in this model. For incremental dominating set, we give an $O(Δ)$-competitive deterministic algorithm and an $O(\log^2Δ)$-competitive randomized algorithm. We extend these results to the Connected Dominating Set problem using a linear-programming formulation that captures connectivity through local constraints. When the neighborhood of each arriving vertex is known \textit{in advance}, deterministic algorithms achieve similar polylogarithmic competitive ratios as their randomized counterparts. Finally, we establish matching lower bounds, showing that our results are optimal up to constant factors.

Authors: Mark Glass, Meir Feder

We demonstrate that the problem of finding the maximum cut of a planar graph with arbitrary weights can be easily mapped to a minimum T-join problem in the absolute dual graph - the dual graph with absolute weights, as opposed to the known mapping to a maximum T-join problem with an empty set in the dual graph. By enabling the use of the shortest paths, this approach allows for the straightforward adaptation of the first efficient Max-Cut algorithm, designed by Hadlock in 1975 for planar graphs with non-negative weights, to handle the general case of planar graphs with arbitrary weights. Furthermore, we prove that applying a planar Max-Cut algorithm to a higher genus graph, such as a toroidal graph, while disregarding its topology, provides an upper bound for its maximum cut. Employing this methodology, we derive upper bounds for the maximum cut across all toroidal graphs within the GSet benchmark. We report that the known maximum cut values for part of those GSet toroidal problems including the three largest instances, which were previously documented in the literature, are the maximum possible because they match their upper bound values. Additionally, we introduce a novel heuristic algorithm for finding Max-Cut of toroidal graphs, which is based on the planar graph algorithm. Applying this algorithm to all seventeen toroidal Max-Cut problems in the GSet benchmark successfully reproduces all the best-known results, and for problem #62, it yields a new, previously unknown best Max-Cut value.

Peter Brass is a prior NSF theory director. He has written an intelligent guest post on the new NSF guidelines that we present here.

You have received many mails regarding the proposed OMB Uniform Guidance for federal grant making. It is a very long document. So far the OMB has received more than 32,000 comments, which nominally all will be read and taken into account, but actually most are general comments along the lines of “this will destroy scientific research,” and unspecific comments will achieve nothing. You can sample the comments, they are public, and you will be disappointed: much heat but no illumination.

It is my aim to bring a bit more content to the debate, and encourage you to file comments, but on specific issues, in my opinion primarily on the passage against “promoting anti-American Values,” which might indeed be a catch-all for political pressure on research. Otherwise there is little new: reviews were always only advisory.

The relevant section is https://www.federalregister.gov/d/2026-10817/p-amd-155, the revision of section 200.205, especially section (b) “pre-issuance review,” which states (underline and boldface mine):

(b) Pre-issuance review. As part of the merit review process, Federal agencies must perform pre-issuance reviews to ensure that Federal award proposals selected for funding are consistent with applicable law, Federal agency priorities, and the national interest. In doing so, Federal agencies heads must designate one or more senior appointees to conduct a pre-issuance review of all discretionary awards. As part of this pre-issuance review for discretionary awards, senior appointees (or their designee) must, as relevant and to the extent consistent with applicable law, apply the following principles when reviewing Federal award proposals:

This is followed by a list of criteria, but to summarize this: The agency head (a presidential appointment) designates a senior appointee (possibly a presidential appointment) who might designate another person inside the agency, who checks all intended awards that certain criteria are satisfied. This is not really different from the state before 2025: any proposal I proposed for award needed to be checked by the division director.

So if there is a problem, it must be in the criteria. It turns out there might be, depending on the interpretation.

1. Discretionary awards must, where applicable, demonstrably advance the President’s policy priorities.
   Whether this is applicable is the agency’s decision.
2. Discretionary awards must not be used to fund, promote, encourage, subsidize, or facilitate:
   • (i) Racial preferences or other forms of racial discrimination by the recipient, including activities where race or intentional proxies for race will be used as a selection criterion for employment or program participation;
   • (ii) Denial by the recipient of the sex binary in humans or the notion that sex is a chosen or mutable characteristic;
   • (iii) Illegal immigration; or
   • (iv) Any other initiatives that compromise public safety or promote anti-American values.
   That last one is a catch-all, which can indeed be used by politics to suppress research. This is the passage against which we should protest.
3. All else being equal, preference for discretionary awards should be given to institutions with lower indirect cost rates.
   All else is never equal, but it is in the program director’s discretion to consider how much funding actually reaches science.
4. Discretionary awards should be given to a broad range of recipients. Research grants should be awarded to a mix of recipients likely to produce immediately demonstrable results and recipients with the potential for potentially longer-term, breakthrough results, in a manner consistent with the notice of funding opportunity.
   I am quite happy about this explicit recognition of the importance of basic research.
5. In performing activities under Federal awards, applicants should commit to complying with administration policies, procedures, and guidance respecting Gold Standard Science.
   It is fairly unclear what Gold Standard Science is, but that seems to affect only lab or simulation sciences with results that are advisory to politics.
6. Discretionary awards should include benchmarks for measuring success and progress towards relevant goals and, as relevant for awards pertaining to scientific research, a commitment to achieving Gold Standard Science.
   Same. For us the benchmark measuring success is publication, so no news.
7. To the extent institutional affiliation is considered in making discretionary awards, agencies should prioritize an institution’s commitment to rigorous, reproducible scholarship over its historical reputation or perceived prestige. For science grants, agencies should prioritize institutions that have demonstrated success in implementing Gold Standard Science.
   No news either. We should not give extra credit to a proposal coming from a famous institution. Of course that is in the discretion of the program director.

(c) Procedure for pre-issuance review. When conducting a pre-issuance review, senior appointees (or their designee) must not ministerially ratify or routinely defer to the recommendations of others, but must instead use their independent judgment when evaluating Federal award proposals.

(so the designee should actually look at the proposal before approving the award)

(d) Use of peer review. Nothing in this part must be construed to discourage or prevent the use of peer review methods to evaluate proposals for discretionary awards or otherwise inform agency decision making, provided that peer review recommendations remain advisory and are not ministerially ratified, routinely deferred to, or otherwise treated as de facto binding by senior appointees or their designees. Further, nothing in this part must be construed to create any rights to any particular level of review or consideration for any funding applicant except as consistent with applicable law.

(good news: nothing changes)

(e) Agency discretion to reissue funding opportunities. A Federal agency is not required to issue a discretionary award as a result of a NOFO if doing so would fund low-quality proposals or be inconsistent with the principles of this part. The agency may, at its discretion, repost a funding opportunity.

(so the agency can decide not to make any award)

This is the entire section which causes so much outcries. I believe that the proposed budget cuts are a much larger danger to research than these recommended OMB rules.

By gasarch

One of the most popular posts in this blog’s history was Common Knowledge and Aumann’s Agreement Theorem, based on a lecture that I gave to high-school students 11 years ago. One of the impacts of that post, I’m proud to say, is that (according to Steven Pinker) it helped to inspire Steve’s excellent recent popular book, which you should read, entitled What Everyone Knows That Everyone Knows…: Common Knowledge and the Mysteries of Money, Power, and Everyday Life.

Two weeks ago, I was privileged to attend a workshop in Paris on “50 Years of Agreeing to Disagree,” where (among other things) I got to meet the 96-year-old Economics Nobel Laureate Robert Aumann for the first time.

I got to catch up there with Steven Pinker as well, who gave a phenomenal talk on the psychology of common knowledge. My own talk was entitled The Complexity of Agreement, with New Directions and Applications (link goes to my PowerPoint slides).

Aran Nayebi has graciously posted on YouTube some partial video from the meeting, including his talk, brief snippets from my talk, and Aumann’s own remarks:

Meanwhile, here were the Aumannian insights that I remembered to write down:

AUDIENCE QUESTION: What questions did people ask you after you published your famous agreement theorem in 1976?

AUMANN: I don’t remember what happened yesterday, let alone 1976.

Also:

ME: I thought you might enjoy knowing that I just came here from a meeting of rationalists…

AUMANN: A meeting of who?

ME: Rationalists, they call themselves, at a beautiful venue called Lighthaven in Berkeley, and that they named the main building there “Aumann Hall” in your honor.

AUMANN: OK, so I’ve made it then.

One recent result announced at the workshop, for those who care, is that the proof of Aumann’s Theorem has now been formalized in Lean, by Scott Kominers at Harvard and a group from the startup company Axiom Math.

Thanks so much to Christina Katt-Pawlowitsch, Ziv Hellman, and others for organizing the workshop and for including me in it.

Happy to field questions in the comments, although if someone wants to call me an idiot like usual, we’ll just need to agree to disagree!

By Scott

We introduce a new family of propositional proof systems, denoted $\langle EF, R \rangle$, for an arbitrary TFNP search problem $R$. Informally, a refutation of a CNF formula $F$ in $\langle EF, R \rangle$ is given by a polynomial-time mapping reduction from the false-clause search problem ${\mathrm{Search}}_F$ to $R$, combined with an Extended Frege proof that the reduction is correct. These new systems are naturally motivated in two ways: 1. They are the propositional translations of witnessing theorems in bounded arithmetic, by which proofs of $\forall \Sigma^b_1$ formulas $\phi$ in a theory $T$ imply algorithms solving the search problem for $\phi$ in a TFNP subclass corresponding to $T$ [Bus86, KP90, BK94, KST07, BB09]. 2. They form a white-box analogue of the recent characterizations of proof systems using decision tree reductions to black-box TFNP problems [GHJ+22b, BIK+94, DR23, LPR24, FGJ+26, FIM25]. We consider the proof system $\langle EF, Iter \rangle$, where $Iter$ is the complete problem for the classical TFNP subclass PLS. We prove that $\langle EF, Iter \rangle$ is polynomially equivalent to the quantified boolean sequent calculus $G_1$, and also to the implicit Resolution proof system $[EF, Resolution]$ introduced by Krajícek [Kra04]. Hence $G_1$ and $[EF, Resolution]$ are polynomially equivalent, which is the first natural characterization of an implicit proof system by a classical propositional proof system beyond the work of Wang [Wan13]. We further observe our characterization can be extended to capture $G_i$ via the game induction principles of [ST07]. We also calibrate the strength of $\langle EF, R \rangle$ for general TFNP relations $R$. We observe that if Extended Frege can prove that a search problem $R$ is in FP, then $\langle EF, R \rangle$ is polynomially equivalent to $EF$. This contrasts to our above result, which shows that Extended-Frege provable reductions to $Iter$, a problem widely believed not to be in FP, yields a proof system ($G_1$) that is believed to be stronger than Extended Frege. Finally, and somewhat paradoxically, we show that for any proof system $P$ which is sufficiently strong, there is a polynomial-time computable search problem $R_P \in $ FP such that $\langle EF, R_P \rangle$ is polynomially equivalent to $P$. Letting $P = [EF, Resolution]$ and combining our two results shows that $\langle EF, Iter \rangle$ is polynomially equivalent to $\langle EF, R_{[EF, Resolution]} \rangle$. Hence, despite the widely-believed conjecture that $Iter \not \in $ FP, the problems which $EF$-provably reduce to $Iter$ are exactly the problems which $EF$-provably reduce to a fixed polynomial-time computable set.

This post is a map of the lower-bound posts on Decentralized Thoughts. The Start Here page already lists many of these results by topic, and the lowerbound tag lists the posts themselves. Here we turn that list into a chapter: first the main fault-threshold barriers, then communication, round complexity, latency, validity, privacy, and liveness. In distributed computing, an upper bound is often a condition and a protocol for the honest...

By Ittai Abraham

We design a deterministic algorithm that, given blackbox access to the product $f=\prod_{i=1}^{\ell}{h_i}$ of $\ell$ irreducible $s$-sparse $n$-variate polynomials of bounded individual degree $d$, over fields of characteristic zero, and more generally over fields of sufficiently large positive characteristic, recovers the $h_i$'s and their multiplicities in time $\mathrm{poly}(n,(s\ell d)^d)$. For any constant $d>2$, this is the first \emph{polynomial-time} algorithm for this problem, resolving for the bounded-individual-degree regime an open question of Dutta, Sinhababu, and Thierauf (Random 2026). The previous best deterministic algorithm, due to the authors, runs in $\mathrm{poly}(n,d^d,s^{d\log \ell},\ell^d)$ time, which is quasi-polynomial in $\ell$. The improvement is enabled by a new sparse rational interpolation theorem in the bounded-individual-degree setting, based on reconstructing the denominator from its logarithmic derivatives. Given blackbox access to rational functions $a_1/b,\ldots,a_N/b$ where the $a_j$'s and $b$ are $s$-sparse of individual degree at most $d$ with $\gcd(a_1,\ldots,a_N,b)=1$, we recover the $a_j$'s and $b$ in time $\mathrm{poly}(n,d!,s^d,N)$. In contrast to the rational interpolation theorem of Chuyoon-Shpilka (2026), our algorithm reconstructs the denominator directly and does not require a precomputed list of its irreducible factors.

The Hamiltonian Cycle polynomial, denoted as $HC_n$, is defined to be the sum of the weighted Hamiltonian Cycles in an $n$-vertex complete digraph, with vertices labeled $1$ to $n$ and edges weighted by formal variables $x_{i,j}$. The Permanent and $HC$, defined as the family $\{HC_n | \ n \geq 1\}$, were studied by Valiant (STOC 1979), with the former shown to be VNP-complete over all fields of characteristic other than $2$, and the latter to be VNP-complete over every field. Since its introduction, $HC$ has been studied from the perspective of circuit lower bounds by Jerrum-Snir (JACM 1982), determinantal complexity by Huttenhain-Ikenmeyer (LAA 2016), and its connection with the Permanent and the Determinant polynomials by Goulden-Jackson (EJC 1981) and Grochow (ToC 2017). It has been the most prominent choice for generalising results to all fields, such as in Malod (CCC 2007) and Grochow-Mulmuley-Qiao (ICALP 2016), owing to its VNP-completeness over every field. Hrubes (ToCT, 2016) showed the VNP-completeness of many graph-based polynomial families over every field by using $HC$. In Kayal (STOC 2012), a randomised polynomial time algorithm was given for the following problem: Given an $n^2$-variate degree-$n$ polynomial $f(\mathbf{x}) \in \mathbb{F}[\mathbf{x}]$ as a black box, decide if there exists $A \in \mathrm{GL}_{n^2}(\mathbb{F})$ such that $f(\mathbf{x}) = Perm_n(A\mathbf{x})$. Here, the Permanent polynomial $Perm_n$ computes the permanent of the $n \times n$ symbolic matrix $(x_{i,j})$. This problem is known as testing equivalence to the Permanent, or alternatively, ET for Permanent. In this work, we study ET for $HC$. While both families are VNP-complete, the efficient ET algorithm for Permanent does not imply the same for $HC$. Besides, there are crucial differences between the two polynomials that make studying the complexity of ET for $HC$ interesting: The underlying decision problem corresponding to the Permanent is in P (detecting perfect matchings in a bipartite graph), but that for $HC$ (detecting Hamiltonian cycles in a digraph) is NP-complete. The Permanent polynomial is known to be characterised by its symmetries as shown by Mulmuley-Sohoni (SIAM J. Computing, 2001). This property yields an efficient algorithm for the circuit-testing problem for the Permanent, a special case of ET for the Permanent, in which we check whether a given circuit computes the Permanent. In contrast, we show $HC_n$ is not characterised by its symmetries. In this work, we give a randomised polynomial time ET algorithm for $HC$ with mild constraints on the underlying field. The algorithm is obtained by studying and completely characterising the Lie algebra and the symmetries of $HC_n$. We show that, like the Permanent polynomial, the symmetries of $HC_n$ are generated by permutation and scaling matrices over large enough fields. However, we also show that, unlike the Permanent polynomial, $HC_n$ is not characterised by its symmetries. Nevertheless, like the Permanent polynomial, $HC_n$ is downward self-reducible, as shown in Zhang-Bai (TCS 2011), which implies $HC_n$ is characterised by circuit identities and that we can efficiently test whether a given circuit $\mathrm{C}$ computes $HC_n$. We also get a Flip theorem for $HC_n$ as a result of its circuit identities.

We introduce an approach to distinguishing isomorphism types of graphs based on vector spaces of polynomials that are set-wise invariant under permutations (“separating modules,” which are representations of the symmetric group), inspired by the Geometric Complexity Theory approach to separating complexity classes (Mulmuley & Sohoni, SIAM J. Comput., 2001). We characterize the power of this method for distinguishing non-isomorphic graphs under several different complexity measures: - We show that separating modules of "support-degree" $k$ (each monomial touches at most $k$ vertices) are equivalent to the counts of $O(k)$-vertex subgraphs. This is strictly weaker than $O(k)$-dimensional Weisfeiler--Leman (Fürer, ICALP '01). - We show that separating modules of symmetric circuit size $n^{\Theta(k)}$ are equivalent to $\Theta(k)$-WL. This generalizes and strengthens a result of Dawar & Wilsenach (CSL '18; ICALP '20; ACM Trans. Comput. Log., 2022; Theory Comput., 2025): they proved one direction of this equivalence for invariant polynomials; we generalize to separating modules and prove both directions. - When considering only the multiplicities of separating modules (as was proposed in GCT by Mulmuley & Sohoni, ibid., rather than the polynomials themselves), we show that two graphs are separated by multiplicities if and only if their automorphism groups have different cycle indices. The latter result is notable in the analogy with GCT, as it is the only result we are aware of in which the multiplicity approach to separating isomorphism types of objects has been given an "intrinsic" characterization in terms of the objects themselves. We use this to show that for graphs, multiplicity obstructions are stronger than occurrence obstructions. We also connect invariant polynomials to the Graph Reconstruction Conjectures and Forman's “invariants of finite type” (Adv. Math., 2004).