Theory of Computing Report

I more or less said what I was going to do in my tongue-in-cheek, cryptic post on Tuesday, but let me dive into the full details of how I’m planning on structuring this grad seminar. I’m going to log the course content here, which now lists a rough schedule for the class sessions. Taking inspiration from Matt Jones and Chris Wiggins, I’m going to do a weekly split between culture and engineering. Focusing on a particular application domain each week, we’ll spend one session discussing the purpose of forecasts in that domain and the other on the methods.

I arranged things into a thematic arc, starting with the weather—forecasting’s biggest success story—then moving to shakier ground in seismology and epidemiology, and ending with, well, millenarianism. As we move across timescales, our ability to predict nature dissipates: we can make accurate ten-day forecasts, but predicting large-scale climate disruptions is far more qualitative. We can predict immediate earthquake impacts at a distance once an earthquake has happened, but we can’t nail down precisely when a big one will happen. What we do with precise, short-term forecasts is completely different from what we do with imprecise long-term forecasts. Short-term forecasts dictate actions; long-term forecasts of discrete shocks inform risk management, preparedness, and rapid-response policies. I’m hoping that by the end of the semester I can better articulate what long-term forecasts of the end of the world do.

We’ll look at forecasts in governance and how they influence and shape policy. I’m particularly interested in discussing the rise of cost-benefit analysis in US governance. Cost-benefit analysis puts a specific number on something completely unknowable, but is now mandatory for any bill or law to pass. I want to trace how we became so reliant on a particular set of methods for guessing costs and benefits. I’m also interested in how economists became convinced you could forecast “the economy.” This required inventing something called “the economy” that could be forecast in the first place. How our system of government got so tied to a particular style of economic prediction will occupy several weeks of the class.

We’ll also get into why we want to predict “the public.” I want to examine how opinion polls went from a question of legibility to one of prediction. How did we get obsessed with using surveys to predict outcomes like elections? In parallel, we’ll look at the history of attempts to simulate the public. Simulation is nice because you don’t have to talk to people, right? We’ll look at the many misses over the history of human simulation in policy scenarios, and dig into the current obsessions with using LLMs to predict what people might do.

Finally, we’ll get into the weird culture of competitive forecasting. We’ll engage with ideas from superforecasting and prediction markets and ask why people think these are useful information-processing systems. We’ll talk about punditry and how it’s not always interested in minimizing a Brier Score. We have to talk about the relationship between forecasts and gambling. And we’ll try to piece together why putting odds on outcomes makes people feel better about the future.

For the methods, I did my topic matching so you could extract a logically ordered half-semester course on forecasting from those lectures alone. This is not a class on how to be a rational forecaster. I want to problematize those methods more than tell you how to implement them. You can ask your friend Claude if you need an honest, load-bearing implementation.

Our methods survey starts with a refresher on my idiosyncratic views of machine learning as optimization-driven algorithmic pattern recognition. This will lead to a lot of discussion of the optimization problems themselves and why people like them. We’ll cover scoring rules, calibration, maximum likelihood, and utility maximization. We will discuss the role of models and look at probabilistic recurrence models, dynamical system models, differential equations, and other simulation-based tools. We’ll spend time on uncertainty quantification and how people come up with error bars (part of being a good forecaster is plausible deniability). Then we’ll look at offline and online optimization methods that let you fill in predictions based on your cost functions and modeling assumptions. I’m interested in highlighting the metrical determinism. The cost functions and models more or less tie your hands algorithmically, and most of the cleverness goes into how you evaluate.

Hopefully this arc will feel coherent as we go. I’m not into predictions, so don’t get mad if the story changes as I go. I’ll blog through it, and then we can reflect on where we land at the end of the semester.

Enrolled students (and those dedicated to following along at home) have an important first assignment: pick something to forecast. I don’t care what it is. Throughout the course, the goal is to learn the practical techniques by making predictions. Every week I’ll ask you to try to apply the tools to your problem. Or at least find how other people have applied those same tools to your problem. At the end of the semester, we’ll present our full findings and see how accurate we can be.

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By Ben Recht

Authors: Martin Koutecký, Nikolaos Melissinos, Tung Anh Vu, Lluís Sabater

Computational social choice seeks algorithmic answers to questions about preference aggregation, safety of elections, robustness of outcomes, stability, etc. It overwhelmingly models societies as composed of discrete agents. We propose to study computational social choice problems in a society continuum} setting, where a society is modeled as a distribution of infinitely many infinitesimal agents of different types. An analogous approach has been very useful in physics (it is the basis of statistical mechanics), economics (mean field games), and other fields. As an initial case study, we focus on election attacks (bribery and control), which have been extensively studied in the discrete setting. We show that a broad class of standard election attacks becomes polynomial-time solvable in the society continuum. The class contains problems that are NP-hard discretely, among them Borda- and Bucklin-CCDV and unit-cost Borda-SWAP BRIBERY. Furthermore, we give polynomial-time algorithms for $k$-Approval-SWAP BRIBERY when $k$ is constant for general costs, and when $k$ varies and the cost function is additively separable. The latter result contrasts with the discrete problem, which we prove NP-complete for additively separable costs and every fixed $k\ge 2$. In contrast, we prove that Borda-SWAP BRIBERY and $k$-Approval-SWAP BRIBERY, both with general costs, remain computationally hard in the society continuum. To obtain these results, we use both continuous and discrete optimization techniques, such as the Configuration LP framework and dynamic programming. Of particular note is the technique underlying our hardness proofs, which shows how to ''reverse the flow of hardness'' between LP formulations and pricing problems.

Authors: Youlong Ding, Aayush Jain, Ilan Komargodski

We present a new generic transformation from weak PRFs computable in depth $d(n) = Ω(\log n)$ to strong PRFs computable in depth $O(d(n))$. This construction refines the classical tree-based paradigm of GGM by {tapering} the internal state so the per-level depth decreases geometrically. We complement the above with new depth-efficient weak PRF constructions based on various standard assumptions. As a corollary, we obtain new $\mathsf{NC}^1$-computable PRFs from various classical assumptions, resolving several long-standing open problems. Concretely, for the first time, we obtain $\mathsf{NC}^1$-computable PRFs: (1) from the \textbf{Learning With Errors (LWE)} assumption with a polynomial modulus-to-noise ratio, improving upon prior low-depth constructions that required Ring-LWE with super-polynomial ratios [Banerjee-Peikert-Rosen, EUROCRYPT 2012]; (2)from the standard \textbf{Learning Parity with Noise (LPN)} assumption, removing the need for structured LPN variants [Boyle et al., FOCS 2020], [Ding-Jain-Komargodski, STOC 2025]; (3) from the \textbf{Computational Diffie-Hellman (CDH)} assumption; prior works relied on the stronger Decisional Diffie-Hellman (DDH) or generalized Diffie-Hellman (GDH) assumptions [Naor-Reingold, FOCS '97, J. ACM '04].

Authors: Pedro M. M. de Castro

Let $p_0,p_1,\ldots,p_N$ be points of the unit ball of $\mathbb R^d$, processed in a prescribed order. We study the insertion cost $\sum_{i=1}^N\lVert p_i-x_{i-1}\rVert^α$, where each $x_{i-1}$ is computed from the previously observed points. The input-order path is sensitive to the input distribution but can repeatedly pay the diameter under adversarial input. The center star has controlled worst-case scale but ignores the observed sequence. We compress the past into one point through $x_0=p_0$ and $x_i=γx_{i-1}+(1-γ)p_i$, where $0\leqγ\leq1$. Thus $x_i$ is an exponentially weighted memory of the input, maintained with one $d$-dimensional point of working state. For independent uniform points, the stationary insertion length is nonincreasing in the usual stochastic order as $γ$ increases. If $d\geq2$ and $α>0$, every optimal constant parameter for $N$ insertions satisfies $1-γ_N^*=Θ(N^{-1/2})$. We determine its asymptotic constant and the resulting $\sqrt N$ correction, with explicit bounds in $d$ and $α$. For $α=1$, the leading expected tree length equals that of the center star and is strictly smaller than those of the endpoint constructions. For $α=2$, the minimizer is unique for $N\geq2$, with $1-γ_N^*=N^{-1/2}-\tfrac12N^{-1}+O(N^{-3/2})$. For arbitrary input sequences and fixed $0\leqγ<1$, the largest asymptotic mean cost is $(2/(1+γ))^α$ for $0<α\leq3$, strictly below the path value when $γ>0$. Among fixed nonnegative weighting rules whose contributing points have the same average distance in the input order from the most recent point, exponential weighting is within a factor smaller than $1.161^α$ of the best adversarial value in dimension at least two; this ratio tends to one as that average distance grows.

Authors: Édouard Bonnet, Julien Duron, Marcin Pilipczuk, Marek Sokołowski, Szymon Toruńczyk

The notion of linear neighborhood complexity is a very general structural assumption on a graph class, covering most classes of sparse graphs such as planar graphs, graphs excluding a fixed (topological) minor, or bounded expansion graphs, as well as many structured classes of dense graphs, such as graphs of bounded clique-width, twin-width, merge-width, or flip-width. In this work, we present $O(n^2)$-time optimal algorithms for $n$-vertex graphs coming from a class of linear neighborhood complexity for the following problems: $\bullet$ All-Pairs Shortest Paths, $\bullet$ the multiplication of the adjacency matrix $M$ of the input graph with any $n \times n$ matrix. More specifically, after a quadratic preprocessing, we can multiply $M$ with any $n$-vector in $O(n)$ time. This solves several questions raised in [Bonnet, Kim, Geniet, Moon; ICALP '26], and improves and generalizes results in several other recent papers [Bonnet, Giocanti, Ossona de Mendez, Thomassé; STACS '23], [Bannach, Marwitz, Tantau; STACS '24], [Anand, van den Brand, McCarty; NeurIPS '26], [Kozma, Opler '26], and [Cardinal, McCarty, Yuditsky '26]. We also extend our results to classes of bounded VC density. In classes of linear neighborhood complexity, we also give a triangle-detection algorithm in randomized linear time $O(n+m)$ in $n$-vertex $m$-edge graphs, a $K_4$-detection algorithm in randomized $O(n \log^5 n + m \log n)$ or deterministic $O(n^2)$ time, and a $K_5$-detection algorithm in randomized $O(n \log^9 n + m \log^5 n)$ time.

Authors: Honghao Lin, Vahab Mirrokni, David P. Woodruff

We study strong coresets for $\ell_p$ subspace approximation. Given $\A\in\R^{n\times d}$, the goal is to sample and rescale a small number of its rows to form $\Smat\A$ such that \[ \norm{\Smat\A(\I-\Pmat_F)}_{p,2}^p =(1\pm\eps)\norm{\A(\I-\Pmat_F)}_{p,2}^p \] simultaneously for every subspace $F\subseteq\R^d$ of dimension at most $k$, where $\Pmat_F$ is the orthogonal projector onto $F$. Woodruff and Yasuda (FOCS 2025)~\cite{WY25} obtained coreset sizes $\wtO_p(k\eps^{-4/p})$ for $1\leq p<2$ and $\wtO_p(k^{p/2}\eps^{-p})$ for $p>2$. We improve these bounds to $\wtO_p(k\eps^{-2})$ and $\wtO_p(k^{p/2}\eps^{-2})$, respectively. For $1\leq p<2$, our algorithm runs in $\wtO_p(\nnz(\A)+d^ω+k\eps^{-2})$ time. The resulting coreset size matches the known sampling lower bound~\cite{LWW21} up to logarithmic factors when $k+1\geq C\log(1/\eps)$ for an absolute constant $C$. For $p>2$, our algorithm runs in $\wtO_p(\nnz(\A)+d^ω)$ time, matching the running time of the Woodruff--Yasuda framework. We use different techniques in the two regimes. For $1\leq p<2$, we combine a bicriteria low-rank split with Lewis-weight sampling and empirical-process bounds independent of the output dimension. For $p>2$, we give a sharper analysis of the Woodruff--Yasuda construction. By retaining the truncation in its sampling probabilities throughout the row-count recurrence, we show that it achieves the improved $\eps^{-2}$ dependence.

Authors: Gramoz Goranci, Rasmus Kyng, Maximilian Probst Gutenberg, Yibin Zhao, Gernot Zöcklein

We give a randomized data structure for undirected weighted graphs that are partially dynamic, i.e., that undergo either only edge insertions or only edge deletions. The data structure maintains $(1\pmε)$-approximations to the maxflow value and effective resistance between any queried pair of vertices, with total update time $\widetilde{O}_ε(n^2)$ and worst-case query time $\widetilde{O}_ε(1)$. Thus, for dense graphs where $m = Ω(n^2)$, our guarantees are near-optimal. Our algorithms succeed with high probability against an adaptive adversary. Our result follows from a simple stability principle for partially dynamic graphs. We show how to partition an online sequence of $m$ updates into $\widetilde{O}(n/ε)$ epochs such that every graph within an epoch is a $(1\pm O(ε))$-spectral approximation of the graph at the beginning of the epoch. The epochs are determined by the cumulative leverage score of the updated edges: small leverage-score mass implies small spectral change, while the total leverage-score mass over a monotone update sequence is $\widetilde{O}(n)$. Consequently, a spectral sparsifier needs to be recomputed only once per epoch. Applying known static all-pairs maxflow and effective-resistance oracles to these sparsifiers then yields the result.

Authors: Roberto Bruno, Ugo Vaccaro

Given a probability distribution $p = (p_1, \dots, p_n)$ and an integer $1\leq m \leq n$, a contiguous aggregation of $p$ is a probability distribution $q = (q_1, \dots, q_m)$ such that each $q_i$ is a sum of consecutive elements of $p$. Given $p$ and a positive number $R$, we consider the problem of computing a maximum entropy contiguous aggregation $q$ of $p$, under the constraint that its Shannon entropy $H(q)$ is at most $R$. We devise a dynamic programming algorithm that solves the problem exactly, and two time-efficient greedy algorithms that provide close-to-optimal solutions. We discuss a few scenarios where our problem arises.

Authors: Narek Bojikian, Alexander Firbas, Robert Ganian, Hung P. Hoang, Krisztina Szilagyi

We study the computation of minimum spanning trees subject to local degree constraints. Recent work (ICALP 2026) established that three natural formalizations of this problem share the exact same parameterized complexity under standard structural graph parameters, including treewidth, pathwidth and clique-width. This applies to the cases where every vertex has a single target degree (Specified Degree MST), or a degree upper bound (Bounded Degree MST), or is equipped with a set of admissible degrees (Set of Degrees MST). In this paper, we investigate these problems under more restrictive parameterizations and reveal that their complexity landscapes fundamentally diverge on bounded-treedepth graphs. Specifically, we prove that the former two problems are fixed-parameter tractable when parameterized by the treedepth of the input graph. In sharp contrast, we show that Set of Degrees MST remains W[1]-hard parameterized by treedepth, even when combined with the feedback vertex number (i.e., deletion distance to treewidth $1$). Finally, we show that this divergence seems to be specific to treedepth: we exclude an analogous W[1]-hardness result for Set of Degrees MST w.r.t. the vertex cover number and also rule out fixed-parameter algorithms for the former two problems w.r.t. deletion distance to constant pathwidth.

Authors: Xiaoyu Chen, Kuikui Liu

It is proved that, for every $δ\in (0,1)$, the Glauber dynamics for the uniform distribution on proper $q$-colorings is rapidly mixing when $q \geq (1+δ)Δ$ and the underlying graph has girth at least $5$ and maximum degree $Δ= Ω_δ(1)$. This result also extends to general multi-spin systems satisfying a $\textit{local spectral contraction}$ condition, including the anti-ferromagnetic Potts model with $q\geq (1+δ)(1-β)Δ$. These results are achieved by a new spectral local-to-global principle on graphs with girth at least five for general multi-spin systems, and a novel Fourier analysis for Glauber dynamics on a star. The main ideas behind all the proofs were developed through several rounds of interaction with GPT-5.6 Sol Ultra.

Authors: Tianhang Lu, Runtian Ren, Shengcai Liu

Classical paging couples every miss to an immediate replacement. We ask what remains of its algorithmic structure when a miss may wait. In our per-replacement maximum-delay model, loading a pending page costs one unit of movement plus the age of its oldest outstanding request and clears the whole page-specific episode. Equivalently, the instantaneous holding rate is the number of pending pages, rather than the number of pending requests. The classical competitive hierarchy survives this change. For cache size $k$, we give a deterministic $(5k+3)$-competitive threshold-LRU algorithm and a randomized $5H_k$-competitive algorithm against an oblivious adversary; classical lower-bound instances give matching $Ω(k)$ and $Ω(H_k)$ orders. The randomized algorithm uses cache-independent temporal windows to create an ordinary-paging sequence fixed before any random choices; a shadow paging algorithm is then projected onto nonproactive physical replacements. The offline picture is less classical. We give an exact $O(nk)$ dynamic program with one hole, an exact configuration dynamic program for a fixed number of holes, and a deterministic nonproactive polynomial-time $5$-approximation without fixing that number. Yet farthest-next-use victim selection can be suboptimal in the physical delayed problem already with three pages.

Authors: Zhao Song, Lichen Zhang

We analyze exact-metric, Metropolis-adjusted Dikin walks by keeping the proposal determinant and reverse quadratic form together. Their leading uncentered terms cancel in the complete logarithmic acceptance ratio, leaving centered fluctuations that can be controlled with second-order tools. For a polytope given by $n$ inequalities and a convex $L$-Lipschitz potential, this yields warm-start mixing in $\widetilde O((d^{2}+dL^{2}R^{2})\log(w/δ))$ steps for the regularized Lee--Sidford walk. For a spectrahedron with $n\times n$ blocks, the log-det walk mixes in $\widetilde O((ψ^\star nd+dL^{2}R^{2})\log(w/δ))$ steps, where $ψ^\star$ measures matrix leverage. The two analyses share an acceptance-to-mixing reduction. A proposal-comparison argument transfers the polytope bound to an appropriately padded $O(1/d)$-accurate metric computed from high-precision Lewis weights. For spectrahedra, given $\widehatψ\geψ^\star$, a direct-or-two-seed TensorSRHT construction gives an exact-arithmetic implementation with $ψ^\star$ replaced by $\widehatψ$ in the mixing bound.

Authors: László Kozma, Junqi Tan

The optimal discretization problem asks, given two disjoint sets of points $R$ and $B$ in the plane, for a minimal family of horizontal and vertical lines that separate the two sets, so that no cell delimited by the lines contains points from both sets. The problem arises as a pre-processing in supervised machine learning, and has received significant attention in parameterized algorithmics. Answering the question raised by Bonnet, Giannopoulos, and Lampis [IPEC 2017] and Froese [PhD thesis, 2018], it was shown by Kratsch, Masařík, Muzi, Pilipczuk, and Sorge [SODA 2021] that optimal discretization admits a fixed-parameter algorithm with running time $2^{\mathcal{O}(k^2 \log k)} \cdot n^{\mathcal{O}(1)}$, where $k$ is the solution size and $n = |R| + |B|$. In this paper we give an algorithm for optimal discretization that runs in time $\mathcal{O}(1.9602^n)$. We also study the related point separation problem that asks to separate all input points by axis-parallel lines. For this problem we obtain an algorithm with runtime $\mathcal{O}(1.8906^n)$. Our guarantees follow from structural observations about bichromatic and monochromatic point sets, and hold even if points are allowed to share coordinates. To our knowledge, these are the first improvements over the trivial $2^n$ bound for both problems.

Authors: Michael T. Goodrich, Gonzalo Navarro, Claire A. To

Suppose two parties, Alice and Bob, hold long character strings, $X$ and $Y$, respectively, and they are interested in determining how similar $X$ and $Y$ are. {Moreover, they want to exchange the strings with cost proportional to their degree of dissimilarity.} Such problems arise, for example, in database and file system synchronization operations, as well as in DNA sequence comparisons. Since the strings are long, we are interested in methods that are communication-efficient and have low overhead in terms of the computations that Alice and Bob must perform, when the strings are similar enough. In this paper, we provide a simple low-overhead communication-efficient algorithms for such string reconciliation and edit distance problems, determining the edit distance $k$ between $X$ and $Y$ using only $O(k\log^3 n)$ bits of communication and $O(n\log k)$ time overhead, with high probability.

Authors: Armen Kostanyan, Arevik Harmandayan

The problem of pattern matching, that is, finding all occurrences of a given pattern in a string, is one of the fundamental problems in computer science that has applications in many areas. In this paper, we consider fuzzy patterns, defined as sequences of fuzzy properties over the basic alphabet. We first consider fuzzy pattern matching for sequences of elements of the basic alphabet and then extend the problem to partially ordered sets of nodes labeled by elements of the basic alphabet. For sequences, we seek segments that match the pattern, whereas for partially ordered structures, we seek saturated chains of nodes that match the pattern. The key concept underlying the solutions to these problems is the notion of a trajectory, which generalizes the concept of the prefix function used in the Knuth--Morris--Pratt (KMP) algorithm. A trajectory is processed together with the corresponding data structure, allowing the proposed algorithms to be represented as transition systems whose states are trajectories for sequences and trajectories associated with nodes for partially ordered structures. The trajectory-based approach provides a unified framework for fuzzy pattern matching in various data structures.

I have just posted this report, which contains two previous reports, the counterexample to the dream xor lemma and the simple proof using majority, together with a new proof which appears to improve the parameters of all previous proofs of the xor lemma. Specifically, if a function has correlation epsilon with circuits of size S, the xor of two copies has correlation about epsilon square with circuits of size about S times epsilon square. By contrast, it seems to me that all previous proofs lost at least epsilon to the four in circuit size, and some also had a dependence on N. This loss arose from the need to estimate the final correlation, as is evident, for example, in Levin’s proof. The proof with the hard core set incurs this loss for similar reasons.

The new proof in the report does not do this estimate. Instead, it uses an object which I call BMA for bounded mean amplifier. It is a function that, given iid variables with a small mean returns a variable whose mean is amplified. Majority is a decent BMA but doesn’t quite get to the square of the correlation. A randomized variant of majority does get you that. The function is very similar to what’s used, for example, in Levin’s proof and, I’m sure, in many other places, but as far as I can tell, the analysis is different. I also find it simpler.

By Manu

There's a scene in Apollo 13 where Jim Lovell, played by Tom Hanks, asks Houston control to check his calculations, which they do using a slide rule. 

By Lance Fortnow

Authors: Aritra Das, Vincent Froese, Moritz Grillo, Debayan Gupta, Christoph Hertrich, Tharrshann Jayan Logarajah, Georg Loho, Mihir More, Moritz Stargalla

Lipschitz constants are a standard way to quantify the sensitivity of neural networks to small input perturbations, but computing them is difficult even for shallow ReLU networks. We study this problem for two-layer input-convex neural networks (ICNNs), a restricted architecture where nonnegative output weights enforce convexity. Computing the $L_p$-Lipschitz constant for these networks is equivalent to maximizing the dual norm over a zonotope. While $L_1$- and $L_\infty$-norm maximization on zonotopes admit fixed-parameter and polynomial-time algorithms, respectively, the parameterized complexity of the remaining $L_p$-norms was open. We prove that, for every fixed $p\in (1,\infty)\cap \mathbb{Q}$, maximizing the $L_p$-norm over a zonotope in $\mathbb{R}^d$ is W[1]-hard with respect to the dimension $d$. Moreover, our hardness results imply that brute-force enumeration algorithms are essentially optimal for this problem under the Exponential Time Hypothesis. By duality, the same hardness results hold for computing the $L_p$-Lipschitz constant of two-layer ReLU ICNNs. Our proof first establishes the result for the $L_2$-norm and then transfers the construction to arbitrary fixed $p\in (1,\infty)\cap\mathbb{Q}$ using a suitable Taylor approximation. These results resolve the corresponding questions regarding the parameterized complexity status for zonotope norm maximization and two-layer ICNN Lipschitz constants. Our paper resolves an open problem posted at COLT'25. There are several independent concurrent papers resolving the same problem. Our paper prioritizes a clear exposition of the underlying mathematics and conceptual intuitions behind the proof. Additionally, we explicitly describe our research process including the use of LLMs.

Authors: A. R. Balasubramanian, Dmitry Chistikov, Rupak Majumdar

Many problems in the verification of recursive programs can be reduced to pushdown model checking. In this problem, we are given as input a pushdown automaton (PDA) over a constant-sized stack alphabet, and a description of undesirable behaviors given by an intersection of NFAs, and the problem is to decide if there is a behavior of the PDA that belongs to the set of undesirable behaviors. It is well-known that there is an algorithm for this problem that runs in time $O(n^{2k} |Σ| + n^{3k})$, where $n$ is the maximum number of states of the PDA and the NFAs, $Σ$ is the common input alphabet, and $k-1$ is the number of NFAs. Despite the importance of this problem, no better algorithm is known for it. In this paper, we provide an explanation for this lack of progress using the lens of fine-grained complexity theory. We prove that if the $3k$-Clique hypothesis (resp. combinatorial $3k$-Clique hypothesis) is true, then for any $ε> 0$, there is no algorithm (resp. combinatorial algorithm) that solves this problem in time $O((n^{(ω-1)k} |Σ| + n^{ωk})^{1-ε})$ (resp. $O((n^{2k} |Σ| + n^{3k})^{1-ε})$) where $ω$ is the matrix multiplication exponent. Furthermore, using the combinatorial hypothesis, we also show that pushdown model checking over constant-sized input alphabets cannot be solved in time faster than $O(n^{3(k-1)-ε})$ for any $ε> 0$. Finally, we investigate the possibility of an $O(N^{3k-ε})$ time algorithm for this problem where $N$ is the total bit size of the input. We formulate a new hypothesis, the 2NPDA$(k)$ hypothesis, that helps explain the lack of $O(N^{3k-ε})$ time algorithms for this problem. To corroborate this hypothesis, we show a web of linear-time reductions between the 2NPDA$(k)$ hypothesis, pushdown model checking, and other problems in formal language and automata theory.

Authors: Benedek Nagy

The Post Correspondence Problem is as follows: having a set of dominoes, is there any (maybe repeating) sequence of them such that the words formed by the upper parts and the lower parts by the sequence of dominoes are identical. It is one of the most known problems that is algorithmically undecidable. In this paper, the reverse Post Correspondence Problem is defined, that is, where the two assembled words of the dominos are reversals of each other. Undecidability about this new, modified problem is proven. Further, based on this result, it is also proven that the emptiness problem for 5'->3' String Assembly Systems is undecidable too. 5'->3' String Assembly Systems belong to String Assembly Systems type formal language generating models. The 5'->3' denotes that, in these variants, the derivations of the generated words start from the two extremes. The notation and the new model are bio-motivated as any double stranded DNA has two opposite oriented 5'->3' strands.

Authors: Yota Maeda, Hiroshi Yano

We study the approximation of the number of solutions of Laurent polynomials over finite fields. For a Laurent polynomial \[f(x)=\sum_{j=1}^{s}a_jx^{u_j}\in \mathbb{F}_q[x_1^{\pm1},\ldots,x_n^{\pm1}], \] let $U$ be its augmented support matrix whose columns are $(1,u_j)$ with rank $ρ$ and $N(f) := \# \{x\in (\mathbb{F}_q^\times)^n \mid f(x)=0\}$ be its torus point count. Our first main result is a quantum algorithm that outputs $\widehat{N}(f)$ satisfying \[ |\widehat{N}(f) - N(f)| \le \varepsilon q^{n+s/2-ρ} \] with success probability $1-δ$. Provided that $ρ$ and $\|U\|_\infty$ are bounded, the algorithm runs in both classical bit and quantum gate complexity $\mathrm{poly}(n, s, \log q, 1/\varepsilon, \log(1/δ))$. It provides finer resolution than relative-error approximations in general settings. To the best of our knowledge, in the explicit finite-field input model considered here, no previous algorithm achieves this additive accuracy with running time polynomial in $\log q$. Van Dam (arXiv:quant-ph/0405081) conjectured the existence of such an algorithm under the assumption of an oracle reflecting the algebraic properties of the polynomial. In contrast, by exploiting a point-counting formula derived from character sums over finite fields, we develop an alternative approach that efficiently approximates the number of points without assuming the existence of such an oracle. As a second main result, we prove that the same approximation problem becomes $\#$P-hard under randomized polynomial-time Turing reductions when the support matrix $U$ varies freely as part of the input. Thus, taken together, our results clarify how the effectiveness of the quantum approach depends on the tradeoff between the accuracy scale and the support parameters of the input polynomial.

Authors: Hari Mohan Pandey

This paper proposes the Coronavirus Optimization Algorithm (COA), a SARS-CoV-2-inspired success-history adaptive evolutionary optimizer for box-constrained continuous global optimization. COA does not model disease transmission; instead, it maps selected coronavirus mechanisms to explicit search operators, including elite-guided attraction, trial-vector generation, adaptive parameter variation, stagnation recovery, and population-size scheduling. The algorithm combines opposition-based initialization, current-to-pbest mutation, binomial crossover, an external archive, success-history adaptation, population reduction, and partial restart. COA is evaluated on 29 CEC 2017 benchmark functions at 10, 30, and 50 dimensions against 15 competitive optimizers. Results show that COA achieves the best overall Friedman rank across all dimensions, with particularly strong performance on composition functions. The findings demonstrate that COA is a compact, transparent, and competitive adaptive evolutionary optimizer, while also highlighting limitations on some hybrid functions and the need for further high-dimensional validation.

Authors: Jiaqi Liu, Yansong Feng, Yanbin Pan

We prove that exact Euclidean decision-CVP is $\mathsf{NP}$-complete on coefficient lattices of nonzero principal ideals in the power-of-two cyclotomic rings $R_d=\mathbb{Z}[y]/(y^d+1)$. A deterministic reduction from X3C produces an integral target and squared threshold $Δ$ such that the closest squared distance is exactly $Δ$ in YES instances and at least $Δ+4$ in NO instances. The ideal elements within squared distance $Δ$ are in bijection with exact covers, which also gives $\mathsf{NP}$-hardness of exact search-CVP under polynomial-time Turing reductions. We further lift these instances to full-rank principal ideals of $\mathbb{Z}[X]/(X^D-1)$, where $D=2d$. The lift preserves principality, doubles the dimension, and scales corresponding squared distances by eight. Hence exact decision-CVP is $\mathsf{NP}$-complete and exact search-CVP is $\mathsf{NP}$-hard on principal cyclic ideal lattices. Both results admit uniformly computable fixed-family forms: for each X3C universe size, the principal cyclotomic and cyclic ideals can be fixed before the triple collection is known, with only the targets and thresholds depending on the collection. If exact decision-CVPP were polynomial-time solvable on either family, then $\mathsf{NP}\subseteq\mathsf{P}/\mathrm{poly}$; by Karp--Lipton, the polynomial hierarchy would collapse to $Σ_2^{\mathsf P}$. To our knowledge, the cyclic results resolve the exact decision versions of Micciancio's questions for cyclic lattices and fixed cyclic-lattice families.

Authors: Vincent Delecroix, Vincent Despré, Camille Lanuel, Hugo Parlier, Monique Teillaud

Hyperbolic surfaces are a fundamental object in mathematics and play an increasingly important role in computational geometry and topology. A key ingredient in the design of efficient algorithms on such surfaces is the availability of a geometric discretization of controlled complexity. In this paper, we present the first algorithm for constructing e-nets on hyperbolic surfaces starting from a fundamental polygon representation. Our approach is based on Delaunay refinement and relies on maintaining Delaunay triangulations through edge flips. The size of an e-net cannot be bounded solely as a function of the genus because of the presence of arbitrarily long collars around short geodesics. To overcome this difficulty, we introduce the notion of a pseudo e-net, which decomposes the surface into e-thin cylinders together with a Delaunay triangulation over an e-net of the remaining thick part. As applications, we obtain algorithms for computing the length spectrum of an e-thick hyperbolic surface and for computing the systole from a pseudo log(sqrt(2))-net. These results demonstrate that Delaunay-based discretizations provide a practical and versatile framework for algorithmic computations on hyperbolic surfaces.

Authors: Emerson G. Escolar, Yuta Shimada

In topological data analysis, in particular persistent homology analysis, extracting "optimal" representatives for homology classes is crucial for identifying geometric regions of interest. In prior work, optimality is defined in terms of minimizing length or volume. In this work, we restrict our attention to a single homology class in degree $1$ and introduce the total absolute curvature of cycles as the cost function. We show that this cost function, based on angles between edges of cycles, penalizes departures from planarity, convexity, and simple-ness of the cycle representative. We formulate the "angle-optimal homologous cycle problem", recast it as a binary quadratic optimization problem, and show the results of experiments on artificial toy data.

Authors: Jiaheng Wang

It is shown in this manuscript that a random graph $G$ drawn from the Erdős--Rényi model $\mathcal{G}(n,p)$ with \[ p=p(n)\leq 1/2, \qquad \lim_{n\to+\infty}(np-\log n-\log\log n)=+\infty, \] is a homomorphic core, i.e., every homomorphism from $G$ to itself is an automorphism. This implies tight ETH-based lower bounds of the subgraph isomorphism problem for almost all $k$-vertex patterns with polynomial average degree.

Authors: Rolando D. Somma, Ronald de Wolf

The guided Hamiltonian problem is the following: given access to the unitary $U=e^{i H}$ for some Hamiltonian $H$, and given access to a unitary that prepares a guiding state promised to have overlap at least $γ>0$ with the ground space of $H$, estimate the ground-state energy of $H$ within additive error $δ> 0$ and success probability at least $1-\varepsilon $, $\varepsilon>0$. How many applications of $U$ and its inverse $U^{-1}$ are necessary and sufficient? An upper bound $O(\log(1/\varepsilon)\log(1/γ)/γδ)$ was known, and was improved to $O(\log(1/\varepsilon)/γδ)$ very recently [JW26]. A matching lower bound was known whenever one of the three parameters $δ,γ,\varepsilon$ was held constant [MdW26]. In this paper we prove the joint lower bound $Ω(\log(1/\varepsilon)/γδ)$ with the tight $\varepsilon$-dependence provided the dimension of $H$ is at least $\log(1/\varepsilon)/γ^2$. Furthermore, we show that this same lower bound (with slightly larger dimension) holds for both the special case in which the ground state is guaranteed to be unique and $H$ has a gap of $δ$ between its first and second eigenvalue; and for ground-state preparation, where $δ$ denotes the spectral gap and $\varepsilon$ now is the approximation error. The lower bounds also apply when the Hamiltonian can be accessed via its block-encoding, and when fractional powers of $U$ are allowed, as in continuous-time Hamiltonian simulation. Lastly, improved upper bounds are known when $H$ is nonnegative and presented as a sum of squares; and our results imply the lower bound $Ω(\log(1/\varepsilon)/γ\sqrtδ)$ for this case.

Authors: Stacey Jeffery, Freek Witteveen

In the problem of ground-state energy estimation, one aims to estimate the smallest eigenvalue of a Hamiltonian, often given a guiding state, with some promised overlap $γ$ with the ground space. The main approach to this problem is to simulate its evolution, and estimate the smallest (or equivalently, largest) eigenphase of the resulting unitary $U$. We give a quantum algorithm that estimates the largest eigenphase of a unitary $U$ in this guided setting using a factor of $\log\frac{1}γ$ fewer queries to $U$ than the previous best approach. The result matches an existing lower bound, and answers an open question from Mande and de Wolf. The algorithm is based on transducers, which often allow composition of quantum algorithms without overhead from error reduction.

Authors: Ainesh Bakshi, Alex Conway, Hanna Komlós, William Kuszmaul, Alek Westover

Affine modular linear hashing is one of the simplest classical hash families. For a prime $p > u$, the hash function is obtained by choosing $s,t$ uniformly from $\mathbb{Z}_p$ and mapping each key $x \in \{0,\ldots,u-1\}$ to one of $n$ bins by $h(x) = [(sx+t) \bmod p] \bmod n$. Despite its simplicity, the maximum load of linear hashing remains poorly understood. For $n$ keys hashed into $n$ bins, the best known upper bound is $O((n \log n)^{1/3})$, whereas the best known lower bound is only $Ω(\log n / \log\log n)$. We prove a lower bound of $\exp(Ω(\log n / \log\log n))$ for universes of size $n^{1+o(1)}$. Surprisingly, there is a key set for which this load holds not just in expectation, but for every random seed. The proof is driven by two simple reductions: one transfers lower bounds from a real version of linear hashing to modular linear hashing, and the other transfers arithmetic Kakeya constructions to real hashing. We further show that, for sufficiently large $p$, the expected maximum loads in the modular and real settings are essentially the same, giving an alternative route to an $n^{1/3+o(1)}$ upper bound. Finally, we show that any uniform subpolynomial upper bound for either setting would imply a polynomial-length arithmetic Kakeya conjecture and hence the Kakeya conjecture for upper Minkowski dimension.

Authors: Gabriel Marques Domingues, Minh Hang Nguyen, Shay Solomon

We study the \emph{fully dynamic edge orientation problem}, focusing on \emph{worst-case} time bounds. An undirected graph undergoes edge insertions and deletions, and the goal is to maintain an orientation with small {\em maximum outdegree} (hereafter, outdegree) and small worst-case update time. The outdegree of any orientation is at least $α-1$, where $α$ is the graph's \emph{arboricity}, i.e., the minimum number of forests into which its edge set can be partitioned. When $α= O(1)$, it is long known that both the outdegree and the worst-case update time can be bounded by $O(\log n)$. Despite numerous follow-ups, no $o(\log^3 n)$ worst-case update time is known for maintaining constant outdegree, even for very basic graph families---with a notable exception, \emph{forests}. For forests, a \emph{simple folklore} algorithm maintains outdegree 2 via \emph{random walks}: When an insertion creates a vertex of outdegree 3, the algorithm repeatedly chooses a uniformly random outgoing edge until reaching a vertex of outdegree at most 1, and then flips the resulting directed path. As the underlying graph is cycle-free, the path length is easily shown to be $O(\log n)$ in expectation, and also with high probability for polynomially long update sequences. We prove that this simple random walk paradigm extends to \emph{outerplanar graphs}. Our algorithm maintains constant outdegree with $O(\log n)$ worst-case update time, where the time bound holds in expectation, and also with high probability for polynomially long update sequences. We give a \emph{tight analysis}: outdegree 4 is achievable with $O(\log n)$-length paths, while outdegree 3 incurs $\mathtt{poly}(n)$-length paths. We also extend the argument to $K_{2,t}$-minor-free graphs, for any $t \ge 2$, with the outdegree bound depending only on $t$ and with the same update time guarantees. The locality of [...]

Authors: Karthekeyan Chandrasekaran, Krishna Kalathur

We show that there exists a polynomial-time algorithm to find a stable matching in network hypergraphic preference systems. The key connection that drives the algorithm was discovered by chatting with ChatGPT-5.6 Sol Max. We verified it independently and present the details in our own words.

Authors: Júlia Baligács, Bartłomiej Bosek, Yann Disser, Andreas Emil Feldmann, Grzegorz Gutowski, Katarzyna Kępińska, Paweł Putra, Anna Zych-Pawlewicz

In this paper we study the fractional vertex cover problem on trees in two related models: online and incremental. In the online model, the vertices of the tree are known a priori and the edges arrive one at a time. The goal is to maintain a fractional vertex cover of the tree, i.e., an assignment of fractional weights from [0,1] to the vertices such that the weights of endpoints of every edge sum up to at least one. After each edge arrival, we need to modify the fractional vertex cover to cover the new edge as well. However, we can only increase the values assigned to vertices. The problem was studied before (in the vertex arrival model) by Wang and Wong, who motivated it as a generalization of the ski-rental problem, but also (more importantly) by its close connection to the dual online matching problem. They presented a 1.901-competitive algorithm for general graphs in the vertex arrival model. We present an $\frac{11}{6} \approx 1.83$-competitive algorithm for trees in the more general edge arrival model. In addition, we study the fractional vertex cover problem in an incremental model, where we again seek a fractional vertex cover after every update, but all the updates to the tree are known to the algorithm a priori. In this model, we give a 1.5-competitive algorithm and provide a matching lower bound.

Authors: Enrico Olivucci, Mariia Sobchuk, Sehmimul Hoque, Jeffrey Hnybida, Kyungho W. Kim, Ala Shayeghi, Pooya Ronagh

We prove the first quantum--classical separation for a sampling problem over a continuous domain. For a class of Gibbs states $p\propto e^{-βE}$ on the torus $\mathbb{T}^d$ with smooth ($s$-Gevrey) potential and barrier amplitude $α=e^{βΔ}$, where $Δ= \max E-\min E$, every classical algorithm---querying the value, gradient, or any higher-order derivatives of the log-density---requires $Ω(α)$ queries to sample at constant accuracy in total variation distance, while a quantum algorithm based on quantum singular value thresholding and temperature annealing samples with $\tilde{O}\left(\sqrtα\right)$ queries to an oracle for the gradient. The advantage is quadratic in the barrier amplitude, which becomes exponential in the dimension, $e^{Ω(d)}$, at low temperature. The classical bound is information-theoretic, holding for every classical algorithm with query access to the Gibbs potential and its derivatives at any order.

Authors: Leon Kullmann, Phuoc Lucky Trinh, Leon Kellerhals, Mitja Krebs, André Nichterlein, Stefan Schmid

The classical Minimum Linear Arrangement (MLA) problem has been studied extensively. It is known to be NP-hard and it admits an $O(\sqrt{\log n}\log\log n)$-approximation [Feige and Lee, IPL, 2007]. MLA can be defined as follows as design problem: Given a graph $G$ with vertex set $V(G)$, design a path $H$ on the same vertex set that minimizes the linear arrangement cost $\sum_{uv\in E(G)}\textrm{dist}_H(u,v)$, where $\textrm{dist}_H(u,v)$ indicates the distance of $u$ and $v$ in $H$. We initiate the study of the generalization in which $H$ is allowed to be a caterpillar graph of maximum degree at most $Δ$. Caterpillars are the simplest generalization of paths, having pathwidth one and interpolating between paths and stars via the degree parameter $Δ$. We give an algorithm that lifts any $α$-approximation for MLA to an $(α+3-2/(Δ-1))$-approximation for our problem, thus obtaining an $O(\sqrt{\log n}\log\log n)$-approximation for our more general problem as well. Moreover, we derive a $4$-approximation whenever MLA is polynomial-time solvable, in particular, for trees. Complementing these results, we prove NP-hardness for every constant $Δ\geq 2$, and, in stark contrast to MLA, show it remains NP-hard on trees when $Δ$ is part of the input.

Authors: Christian Bertram, Mads Vestergaard Jensen, Mikkel Thorup, Hanzhi Wang, Shuyi Yan

We study the shortest-path problem on graphs with positive real-valued edge weights. Given a source vertex $s$ and a target vertex $t$, the goal is to calculate the length of the shortest path from $s$ to $t$. We are particularly interested in instances that can be solved in sublinear time. Recently, Haeupler, Hladík, Rozhoň, Tarjan, and Tětek proved that (a version of) the bidirectional Dijkstra's algorithm is instance-optimal on positively weighted multigraphs, both directed and undirected, considering the number of vertices and edges queried by the algorithm. However, multigraphs are not the canonical setting for the shortest-path problem. The problem is typically formulated on simple graphs without loops and parallel edges. They therefore left as an open problem whether bidirectional Dijkstra remains instance-optimal on simple weighted graphs. We answer this question, but for simple graphs, the answer is more complex, depending on the setting. We show that bidirectional Dijkstra is still instance-optimal on simple undirected weighted graphs under the order-oblivious model, where incident edges are given in a random order. In contrast, under the order-dependent model, where incident edges have a given order, we show that bidirectional Dijkstra is not instance-optimal. For simple directed weighted graphs, we show that bidirectional Dijkstra is not instance-optimal under either the order-oblivious or the order-dependent model, being off by a factor of $Θ(m/n)$ in both cases. We further show that no algorithm can have instance-optimality ratio $o(m/n)$ under the order-dependent model, or under the order-oblivious model when $m=O(n\sqrt{n})$. On the positive side, the above results imply that bidirectional Dijkstra is instance-optimal up to logarithmic factors on all sparse directed and undirected graphs satisfying $m/n=\log^{O(1)} n$.

Authors: Taha El Ghazi, Jonas Ellert, Chien-Chung Huang, Tatiana Starikovskaya

Clustering is one of the most fundamental tools in data analysis, allowing large datasets to be summarized by a small number of representative points. Given a metric space $(\mathcal{X}, \mathbb{d})$ and a set $S$ of $n$ points in this space, the continuous $k$-median clustering problem asks to find a set $C$ of $k$ points that minimizes the objective function $\sum_{s\in S} \mathbb{d}(s,C)$. When $\mathcal{X} = Σ^\ell$ is the set of strings of length $\ell$ and $\mathbb{d}$ is the Hamming distance, the continuous $k$-median clustering problem is known to be W[1]-hard when parameterized by $k$. In this work, we present the first $(1+\varepsilon)$-approximation algorithm for this problem with FPT runtime $2^{\mathrm{poly}(\varepsilon^{-1},k)} \cdot n\ell \mathrm{polylog} \; n$. An additional feature of the algorithm is that it can be implemented in streaming, requiring only $\tilde{O}_\varepsilon(\ell k + k^2)$ space. As an auxiliary tool of independent interest, we show the first streaming algorithm for computing an $\varepsilon$-coreset for continuous $k$-median clustering under the Hamming

Authors: Soheil Kolouri

Partial optimal transport compares two measures while leaving part of the mass unmatched, which is what makes it robust to outliers, occlusion, and clutter. The quantity of interest is usually the whole profile - the optimal cost at every transported cardinality - because the right amount to transport is rarely known in advance, and on the real line the PAWL algorithm returns that profile in $O(N\log N)$. Much data is periodic rather than linear: angles, phases, orientations, time of day, hue, and every direction obtained by projecting onto a great circle. On the circle the same problem acquires a global circulation, or equivalently an optimized cut, which the naive exact method handles by running the line algorithm once per support gap, at $O(N^{2}\log N)$. We show that this factor $N$ is unnecessary. The line structure survives in cut-free form, and a free-gap invariant supplies, at every step, a cut at which all previous local updates remain valid line updates. This yields PAWC: an exact $O(N\log N)$ time, $O(N)$ memory algorithm returning all $K+1$ costs, nested active sets and plans in one run, together with a single gap that is simultaneously optimal for every cardinality. Slicing over great circles extends it to $\mathbb{S}^{d-1}$. Empirically the whole profile costs $0.56$ms at $N=4096$ against $1.5$s for a single transported fraction from a general solver; on occluded, cluttered mpeg-7 shapes, holding the descriptor fixed and varying only the cost, it retains $66\%$ of the clean-data retrieval score against $16\%$ for balanced circular OT, and on $\mathbb{S}^{2}$ it halves the fitting error of spherical sliced Wasserstein against contaminated targets, synthetic and real. Code is available at https://github.com/mint-vu/Partial_Wasserstein_on_Circles.

Authors: Soumil Aggarwal, Rohit Vaish, Jatin Yadav

Maximizing Nash welfare over indivisible goods is a central problem in resource allocation. For additive valuations, the best-known approximation factor is roughly $e^{-1/e}\approx0.692$, and the problem is APX-hard. We study Nash welfare maximization under lexicographic valuations, where every good is worth more than the total value of all lower-ranked goods. This large-gap structure makes preferences almost ordinal, which might suggest that the problem becomes easy. We show, however, that the picture is more nuanced: although lexicographic valuations enable stronger algorithmic guarantees, they retain significant computational hardness. Our first main result is a $(1/\sqrt{2}-ε)\approx(0.707-ε)$-approximation algorithm for weighted Nash welfare under lexicographic valuations, improving over the inherited guarantee of roughly $e^{-1/e}$. The algorithm rounds the configuration LP for Nash welfare, for which we show a matching integrality gap of $\sqrt{2}$. Our second main contribution is an exact algorithmic polynomial time framework for ordered lexicographic instances and doubling lexicographic instances. We introduce a domination-based branch-and-prune method for which we prove a mutual-exclusion property between sibling subtrees and use a matrix-based leaf-counting argument to bound the pruned recursion tree by a polynomial when the number of agents is constant. Finally, we show that large gaps do not eliminate hardness, as Nash welfare maximization is NP-hard even for ordered lexicographic valuations, and it is NP-hard to obtain a $0.9996$-approximation even for doubling lexicographic valuations. Thus, lexicographic valuations make Nash welfare maximization easier, but not easy: they admit tighter approximation and exact algorithms in important cases, yet still require intricate techniques and preserve some of the hardness of the general additive setting.

Authors: Liangchen Ge

We study the Lipschitz stability of attention retrieval in hyperbolic spaces. Existing methods lack deterministic guarantees on attention-weight preservation under finite-precision representations. We introduce HCC+, a theoretical framework exploiting three properties of the Poincaré ball: exponential volume growth enabling query-independent boundary truncation; logarithmic covering radius of hyperbolic 1-centers enabling dimension-independent critical-key identification; and a packing bound with constants independent of the embedding dimension. We prove two deterministic guarantees: for exact retrieval, the per-layer attention deviation is bounded by 10\% of its ideal value; for soft attention, the total variation distance decays as $O(1/\sqrt{n})$, the rate of finite-sample variance. As a consequence of the guarding mechanism, the framework achieves a storage reduction factor of $6.1\times$ relative to FP16. We provide the first deterministic, query-independent retrieval certificate in non-Euclidean geometry.

Update: I haven’t joined Anthropic or taken leave from the university.

That means I’ll be teaching as usual this fall, and you can look forward to your regular installment of live lecture blogs. Unlike past years, I’m assigned to my grad seminar in the fall, not the spring. Next semester I’ll be teaching our undergraduate probability class. That is assuming that the University is still here in the spring and hasn’t been put out of business by some new AI super tutor released by my friends across the bay. I mean, it would be the year 2027, and popular forecasts suggest AIs will be able to do everything taught in a CS degree by late 2026. By the end of the spring semester, those same prognosticators predict those AIs will go rogue, and we’ll find ourselves in the reality forecast by James Cameron in his 1984 prophecy, Terminator. Why should I bother dusting off my copy of Bertsekas and Tsitsiklis when the forecasts tell me I should work harder at the gym to prepare myself for robot enslavement in the salt mines?

Actually, you know what would be a good way to prep for the robot apocalypse? Why don’t we spend a semester talking about forecasting and why people are obsessed with being certain about the future?

Answering that question could potentially be a great way to shape a graduate course. We could spend a semester digging into not only how people forecast, but why they forecast. We could split our time in half, looking at the particularities of different domains where people make forecasts, and then looking into the tools they have settled on as mathematical culture.

If you look at the places where forecasts are common, they all have different purposes. A local weather report is very different from a prediction of the end of the world. The former tells you if you should pack an umbrella on the way to work. The latter tells you whether you need to lobby your government to radically change its planned energy buildout. We all believe the evidence supporting forecasts of rain is far more certain and reliable than forecasts of nature’s end. But the costs of being wrong couldn’t be more different.

What impact do these forecasts have? Why do we forecast the weather, and what hidden technology is needed to make these forecasts accurate? Why does Congress demand that we forecast future budgetary consequences of proposed laws, even though we know we can’t predict the actual structural shocks that will render those forecasts moot? Why are people so obsessed with predicting the rise of superintelligent robots? Is it more than a way to justify their greed and obsessive 996 work conditions?

We’d learn a lot from a comparative study. I’d like to look at astronomy, meteorology, climate science, seismology, macroeconomics, government, epidemiology, public opinion research, and millenarianism to see what they have in common and how they differ. Forecasts let people externalize their beliefs about the likelihoods and consequences of various scenarios. Some forecasts are made for mundane planning. Some forecasts are made to literally gamble. Some forecasts communicate possible futures that others might not be considering. Some forecasts are made to be self-fulfilling, to manifest a change in the world the forecaster desires. Some are made to be self-negating, to convince people to act to avoid worst-case scenarios. I’m interested in understanding the threads that link all of these different purposes together.

Though I’m much more interested in the why, the how has some fun tidbits too. The how is about creating certainty about the future by quantifying it. Uncertainty becomes certain once we turn it into an interval, right? In talking about the how-of-forecasting, I could tie together a lot of loose ends in methods that I’ve blogged about over the years.

We might determine the conditions under which pattern recognition (aka machine learning aka AI) becomes a forecast. We could look at how forecasts are evaluated post-hoc with scoring rules and calibration tests and why people think those are good evaluations. We could look at methods for cost-benefit analysis and uncertainty quantification, and how people justify their modeling assumptions to make decisions. We could learn about tools from dynamical systems that move from simple moving averages to complex simulations. We could examine how statistical tools can be applied to extrapolate from the present to the future. And we could see how these sorts of metrics and models tie your hands algorithmically into unsurprising answers.

This sounds like a fun class to me. I predict I’ll teach this class this fall and live blog it here, starting this Thursday. If you’re a Berkeley graduate student whose research depends on forecasts, email me if you’d like to join the course.[footnote: If you do email, please send me a description of your background and why you’re interested.] If you’re not local, I’ll post a syllabus and webpage this week, and I’ll do my best to keep all of the material public. I predict it will be fun.

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By Ben Recht

Authors: Michal Hospodár, Galina Jirásková, Viktor Olejár, Juraj Šebej, György Vaszil

The 17th International Conference on Automata and Formal Languages (AFL 2026) was held in Košice, September 7-10, 2026. It was organized by the Faculty of Science, P. J. Šafárik University in Košice and the Mathematical Institute of the Slovak Academy of Sciences. This volume contains the texts of the three invited lectures and the 18 papers selected by the Program Committee from a total of 27 submissions. The accepted papers, submitted by authors from 14 countries, cover all important areas of automata and formal language theory and their applications such as grammars and acceptors for strings, trees, and graphs, algebraic theories and algorithms for automata and languages, decision problems, relations to complexity theory, picture description and analysis, and other fields.

Authors: Venkatesan Guruswami, Shilun Li, Mihir Singhal

We prove that for every prime power $q$ and every $p \in (0, 1-1/q)$, a random $\mathbb{F}_q$-linear code of rate $1 - h_q(p) - ε$ is $(p, C_{p,q}/ε)$-average-radius list-decodable with probability at least $1 - q^{-Ω(n)}$, i.e., for every center $y \in \mathbb{F}_q^n$, the $C_{p,q}/ε$ codewords closest to $y$ have average fractional Hamming distance at least $p$ from $y$. This extends a similar result for (standard) list-decoding due to Guruswami, Håstad, and Kopparty (2010) to the stronger average-radius guarantee, with the same $O(1/ε)$ list size. For average-radius list-decoding, such a result was previously known only for binary linear codes (Guruswami, Li, Mosheiff, Resch, Silas, and Wootters, 2021) and for general (non-linear) random codes over arbitrary alphabets (Elias, 1991).

Authors: Kirill Osipov

We develop a resource representation for step recursion in which mutable-state width and recursion descent are explicit and independent parameters. A width bound $u$ controls the size of the encoded machine state, while an effective descent $ρ$ determines the available recursion depth $δ_ρ(u)$. For generalized-inverse descents, we derive the depth directly from generator growth and characterize the increasing sequences that can occur as generator orbits. We then connect this depth--width geometry to standard finite-branching computation. Every deterministic bounded-state dynamics is realizable by a single ordinary bounded step recursion over a fixed finite numerical basis. Using deterministic, existential, universal, or alternating aggregation on the same local dynamics yields the corresponding machine semantics. After closure under the width reparameterizations needed to absorb fixed local cost, the resulting language classes are exactly the machine time--space classes on profiles $(δ_ρ(u),u)$. Finally, profile domination quotients effective descents by admissible width reparameterization. Some depth curves collapse, yet polynomial widths support an explicit infinite strict hierarchy between the canonical polynomial- and exponential-depth profiles. Thus descent remains a nonredundant resource coordinate after polynomial width reparameterization; standard complexity classes are calibration points.

Authors: Sijia Dai, Minming Li, Xiaowei Wu, Yong Zhang

The house allocation problem is a classical one-sided matching problem that concerns the assignment of a set of $m$ houses to $n$ agents according to their preferences, where each agent is assigned exactly one house. Among the various objectives studied in this setting, envy-freeness is one of the most widely adopted fairness criteria. As envy-free house allocations do not always exist, we address this challenge by introducing subsidies and aim to compute allocations that achieve envy-freeness with minimum total subsidy. For binary instances, we show that a total subsidy of at most $(n-1)$ suffices to guarantee envy-freeness in house allocation, and this bound is tight. Building on the known NP-hardness for general utilities, we further show that computing an allocation that minimizes the total subsidy is NP-hard, even under binary utilities. However, when there are only a bounded number of types of agents with binary utilities, the problem can be solved in polynomial time. Finally, we present a polynomial time algorithm that computes the minimum subsidy required to achieve envy-freeness for two types of agents with general utilities.

Authors: Zhefan Duan, Huawei Wu

We study the deterministic norm-query complexity of computational problems in finite-dimensional vector spaces over $\mathbb{Q}_p$ equipped with an arbitrary ultrametric norm. For orthogonalization, we prove that no uniform finite query bound depending only on the dimension exists: for every deterministic algorithm that produces an $N$-orthogonal basis for every ultrametric norm $N$, the number of norm queries is unbounded as $N$ varies. We then study the Longest Vector Problem (LVP) for a rank-$m$ $p$-adic lattice. By adapting a brute-force search to the general norm-query setting and eliminating the scalar redundancy among nonzero coefficient vectors modulo $p$, we obtain an algorithm using exactly $(p^m-1)/(p-1)$ norm queries for $m\ge 2$, and prove that no deterministic norm-query algorithm can use fewer queries in the worst case. Finally, we consider the Closest Vector Problem (CVP). Apart from the trivial cases in which no norm query is needed, we prove that the deterministic worst-case norm-query complexity of the CVP is unbounded, even when the lattice and the target vector are fixed.

Authors: Hongmin Li

We study fixed-layout local verification with target-dependent local tests. Let $M$ be a random variable on $\{0,1\}^K$, and let $S$ record the test selected at each coordinate. For each $s\in\operatorname{supp}(S)$, let $F_s$ be the corresponding target fiber and set $D_{\mathrm{fib}}=\max_s\operatorname{VCdim}(F_s)$. We prove $H(M\mid S)\le \log_2\!\left(\sum_{j=0}^{D_{\mathrm{fib}}}\binom Kj\right)$. A fiber that shatters $d$ coordinates yields a weak relaxed locally decodable code with message length $d$ and block length $d+P$ over the original proof alphabet. For a uniform $K$-bit target and fixed proof alphabet, $Q$, and $σ$, the Goldberg--Gur--Saraogi lower bound implies that $I(M;S)\leγK$, for fixed $γ<1$, forces $P=Ω\!\left(K^{1+1/a}/(\log K)^{2+2/a}\right)$, where $a=\lceil Q/σ\rceil$. If $P\le K(\log K)^c$, then $I(M;S)\ge K-O\!\left(K^{a/(a+1)}(\log K)^{3+ac/(a+1)}\right)=K-o(K)$. Any discrete verifier state $T$ determining $S$ satisfies the same information lower bound. Bounded-randomness adaptive branches can be simulated nonadaptively by exposing their decision trees. A branch using at most $r$ random bits and $q$ adaptive proof queries yields a decoder with perfect completeness and at most $1+2^{r+1}\sum_{j

Authors: Prashanth Amireddy, Amik Raj Behera, Srikanth Srinivasan, Madhu Sudan, Sophus Valentin Willumsgaard

We study low-degree testing for group-valued functions over a Boolean slice. Specifically given a degree parameter $d$ and oracle access to a function $f:\{0,1\}^n_{n/2}\to G$ where $\{0,1\}^n_k$ denotes the set of vectors in $\{0,1\}^n$ of Hamming weight $k$ and $G$ is an Abelian group, the low-degree testing problem asks us to distinguish the case where $f$ is a polynomial of degree at most $d$ (with coefficients from $G$) or is $\varepsilon$-far from the set of all such polynomials. Classical works in this area considered functions with domain $\mathbb{F}_q^n$ and range $\mathbb{F}_q$. More recent works have considered the setting where the domain is the Boolean cube [Bafna, Srinivasan, Sudan (Random Struct. Algorithms 2020), Amireddy, Srinivasan, Sudan (RANDOM 2023)], or when the domain is the slice (i.e., $\{0,1\}^n_{k}$) and the range is $\mathbb{F}_2$ [David, Dinur, Goldenberg, Kindler and Shinkar (SIAM J. Comput. 2017), Kalai, Lifshitz, Minzer and Ziegler (FOCS 2024)]. Each of the changes introduces new challenges in designing and analyzing low-degree tests and this happens again in our setting with domain being a slice and range is general. Our main theorem gives a test that makes $O_d(1)$ queries to $f$ and accepts degree-$d$ functions while rejecting functions that are $\varepsilon$-far with probability $Ω(\varepsilon)$. The central proof idea is to reduce this low-degree testing problem to the problem of low-degree testing on the cube. Specifically we show how to randomly embed the $n/2$-dimensional cube $\{0,1\}^{n/2}$ in the $n$-dimensional slice while nearly preserving the proximity of $f$ to the space of degree-$d$ polynomials on this cube. While the embedding is simple and natural, the analysis involves a careful induction with a novel use of a basis of degree-$d$ polynomials on slices (from a work of Anstee, Rónyai and Sali (Graphs and Combinatorics 2002)).

Authors: Bowen Liu, Yizhou Wang, Lingqian Meng

Triangle-based measures, commonly referred to as badness functions, are widely employed to quantify the extent to which a distance matrix deviates from an ideal geometric configuration. Different formulations of these functions may capture distinct facets of local non-uniformity, and their behavior is often influenced by the underlying distance metric chosen for evaluation. In practical settings, although a canonical badness function may be conceptually preferred, factors such as computational cost, algorithmic constraints, or data-specific characteristics frequently necessitate the adoption of modified versions-for instance, approximate forms or alternatives defined under different distance metrics. This gives rise to a central question: to what degree do these variants retain the structural consistency properties of their original counterparts? To address this issue, we develop a systematic correlation-based framework for evaluating structural consistency. As an illustrative instantiation of this framework, we compute badness sequences from a set of representative distance matrices alongside randomly generated triangle configurations, which are designed to cover variants that may arise under diverse practical scenarios. We then assess pairwise similarities among these sequences using four correlation coefficients. The experimental outcomes indicate that certain badness variants exhibit a notably high degree of structural consistency, whereas others reveal complementary behavioral patterns; moreover, the choice of distance metric exerts a considerable influence on the observed trends. These findings offer practical insights for the informed selection of distance metrics and triangle badness function variants in tasks including geometric reconstruction, triangulation, and structural analysis of pairwise distance data.

Authors: Zeping Yi, Yongjun Wanga, Baoshan Wang, Jian Zhang, Songyi Liu

Deploying wireless nodes to maximize coverage area within a given region is an important challenge in wireless sensor networks, UAV path planning, base station placement and other industrial fields. This practical problem can be mathematically equivalent to an optimal circle covering problem. Although theoretical optimal configurations exist for simple cases in mathematics, the NP-hard nature of this problem makes it computationally prohibitive for complex polygons with numerous nodes. Existing approaches are usually designed for regular domains, while those applicable to irregular polygons often suffer from poor initialization, excessive coverage overlap and failure to constrain nodes within the boundary, leading to low coverage efficiency and long runtime. To address these issues, we propose an improved quasi-physical dynamic algorithm (IQPD) for wireless node deployment in arbitrary convex polygons. Our contributions are threefold: (1) proposing a structure-preserving initialization that maps a hexagonal close packing pattern into the target polygon via scaling and affine transformation, ensuring near-optimal initial node distribution; (2) constructing a refined virtual force model by incorporating friction and a radius-expansion optimization mechanism to reduce coverage area overlap; (3) developing a boundary encircling strategy leveraging normal and tangential gradients to reposition nodes deployed outside boundaries after initial optimization. Extensive experimental results demonstrate that our method consistently outperforms other new metaheuristic algorithms across diverse convex polygon shapes, including randomly generated data and real-world scenarios. Our method achieves the highest coverage rate and node utilization rate among all compared algorithms, greatly improving wireless coverage efficiency.

Authors: Nicolás Honorato-Droguett

Given $n$ points on a line or closed cycle and a threshold $r>0$, the connectivity-maintenance problem is to move the points so that every gap between consecutive points is at most $r$, whereas the points-spreading problem requires every gap to be at least $r$. Li and Wang [CCCG 2015; CGT 2025] and Chen, Gu, Li, and Wang [SWAT 2012; DCG 2013] gave $O(n)$-time algorithms for the cyclic versions of min-max points-spreading and min-max connectivity-maintenance, respectively. Ghadiri and Yazdanbod [CCCG 2016] gave an $O(n\log n)$-time algorithm for the linear version of min-sum points-spreading. In this paper, we show that the two problems can be reduced in linear time to each other for both objectives and on both linear and cyclic domains. As an implication, min-sum connectivity-maintenance is solvable in $O(n\log n)$ time on both domains. Finally, we extend the reduction to points on a line with individual thresholds when their initial order is preserved.