arXiv subjects

Grigoris Velegkas

Publications and source records attributed to Grigoris Velegkas.

At least 19 recordsLinked to original sources

arXiv2026

The Power of Recruiting the Smaller Side: Two Additional Traders Suffice in Two-Sided Markets

We study Bulow-Klemperer-style competition complexity in two-sided double auctions with $m$ unit-demand buyers drawn i.i.d. from $F_B$ and $n$ unit-supply sellers drawn i.i.d. from $F_S$. When $m \ge n$ and buyer valuations first-order stochastically dominate seller costs ($F_B \succeq_{\mathrm{FSD}} F_S$), we prove that recruiting just two additional sellers enables Seller Trade Reduction (STR), a prior-independent mechanism, to achieve expected Gains From Trade (GFT) at least the first-best GFT of the original market; symmetrically, when $m \le n$, Buyer Trade Reduction suffices with two additional buyers. This resolves open questions of Babaioff, Goldner, and Gonczarowski (SODA 2020) and Cai, Liaw, Mehta, and Zhao (STOC 2024). Moreover, this uniform bound is optimal: already for $m = n = 1$, no prior-independent mechanism (deterministic or randomized) that is dominant-strategy incentive-compatible (DSIC), individually rational (IR), and weakly budget-balanced (WBB) matches the first-best GFT with one additional seller. We further extend our characterization to $r$-dominance ($F_B \succeq_r F_S$), where $r \in (0, 1]$ and $1 - F_B(x) \ge r(1 - F_S(x))$ for all $x \in \mathbb{R}$ (recovering $F_B \succeq_{\mathrm{FSD}} F_S$ at $r = 1$). When $m \ge n \ge 1$, $c = \lceil 1/r \rceil + 1$ additional sellers suffice for STR to match or exceed the first-best GFT, while every prior-independent DSIC, IR, and WBB mechanism requires at least $\max(2, \lceil 1/r \rceil - 1)$ additional sellers even for $m = n = 1$, pinning down the optimal bound to within two traders. Finally, this $Θ(1/r)$ bound cannot be asymptotically improved even by recruiting on both sides: for $r < 1/64$ and $m = n = \lceil 2/(5r) \rceil$, every prior-independent DSIC, IR, and WBB mechanism requires $c \ge \lceil 1/(64r) \rceil = Ω(1/r)$ additional traders per side.

cs.GT↗

arXiv2026

Query Feedback and Teaching for Stable and Certifiable Language Generation

Language generation in the limit studies whether a learner observing an enumeration of an unknown target language can eventually generate only valid, previously unseen elements. We study stability, introduced by Kalavasis, Mehrotra, and Velegkas (2025), which requires the generator's infinite output set to eventually remain unchanged, and introduce the stronger notion of certifiability, which requires the generator to determine from its observations that its current output is valid. Drawing on teaching sets in learning theory, we introduce a hierarchy of teaching dimensions for generation under three increasingly strict success criteria: correctness, stability, and certifiability. These dimensions measure the minimum number of selected examples whose appearance guarantees the corresponding success condition. We also study stability and certifiability in the membership-query model of Charikar and Pabbaraju (2025). We characterize stable and certifiable generation, both with and without query feedback, and establish their relationships with uniform generation and identification in the limit. In particular, we give a finite-witness characterization of stable generation, analogous to Angluin's tell-tale characterization of identification, resolving a gap left open by Kalavasis, Mehrotra, and Velegkas (2025). We further compare these generation notions with identification in the limit and contrast their teaching requirements on well-studied language families, including regular and erasing pattern languages.

cs.LG↗

arXiv2026

Cogentic: Multi-Agent Orchestration for Automated Proof Discovery

We present Cogentic, a multi-agent harness for automated proof discovery on open research problems. While frontier language models can generate strong mathematical ideas in a single shot, single-shot generation is often insufficient for open problems that require exploring multiple competing conjectures, overcoming subtle technical obstructions, and retaining intermediate progress over a long horizon. Cogentic addresses these challenges through an iterative prove-verify loop in which an orchestrator allocates a population of independent provers across distinct proof directions, subjects their output to adversarial verification by several specialized components, and promotes confirmed intermediate results into a persistent verified ledger that later rounds build on. The harness is designed to be able to solve research-level math and theoretical computer science problems. Using either Gemini 3.1 Pro or an early version of Gemini 4 Argon as the base model, Cogentic produced novel results on open problems across online learning, auction theory, and mechanism design. Each result was independently verified by domain experts and is developed in full in companion papers. We list these results, and new ones as they are verified, at https://sites.google.com/view/cogentic .

cs.AI↗

arXiv2026

Efficiency of Generalized Proportional First-Price Auctions Under Auto-bidding

Auto-bidding is now widely adopted in online advertising platforms, allowing advertisers to specify high-level campaign objectives--such as maximizing total value subject to a return-on-spend (ROS) constraint--rather than manual per-query bids. A central question in algorithmic mechanism design is characterizing the worst-case efficiency loss, or Price of Anarchy (PoA), across auction formats in this prior-free setting. While randomized auctions are known to strictly improve efficiency over deterministic mechanisms for two bidders, two fundamental questions have remained open: (1) what is the optimal PoA for two bidders, and (2) can any mechanism beat the barrier of 2 for general $n \ge 3$ bidders? We resolve both questions using the family of $r$-proportional first-price auctions ($\text{pFPA}_r$), in which each bidder wins with probability proportional to their bid raised to an exponent $r > 0$ and pays their bid upon winning. First, for two bidders, we prove that the standard proportional first-price auction ($r = 1$) achieves a tight $\text{PoA} \le 1.5$, complemented by a matching lower bound showing that no anonymous, monotone mechanism can do better. Second, for general $n \ge 2$ bidders, setting $r = 2n$ achieves $\text{PoA} \le 2 - \frac{1}{4n+1} = 2 - Ω(1/n)$ across all undominated bid profiles, breaking the deterministic barrier of 2 for every finite $n$ and asymptotically matching the known $2 - Θ(1/n)$ lower bound.

cs.GT↗

arXiv2026

Improved Revenue Guarantees for Selling Separately and Bundling

We study how much revenue a seller can lose by restricting attention to selling separately or grand bundling, in the setting of a single additive buyer with independent item values. Although revenue-optimal mechanisms can require lotteries and infinite menus, Babaioff, Immorlica, Lucier, and Weinberg showed that the better of these two simple formats always achieves a constant fraction of optimal revenue. We prove that $\mathrm{OPT} \le 3.52 \max\{\mathrm{SREV}, \mathrm{BREV}\}$, where $\mathrm{SREV}$ and $\mathrm{BREV}$ are the optimal revenues from selling separately and grand bundling, respectively. This improves the previous best-known approximation factor of $5.2$ due to Ma and Simchi-Levi and narrows the gap to the known lower bound of $2$.

cs.GT↗

arXiv2026

Prediction with Expert Advice: Anytime Regret with Many Experts Matches the Fixed-Time Constant

Prediction with expert advice is a fundamental problem in online learning. When the time horizon $T$ is known in advance, the minimax cumulative regret over $n$ experts is asymptotically $\sqrt{\frac{T \ln n}{2}}$. This is achieved by the Multiplicative Weights Update algorithm with a learning rate tuned to $T$, and is known to be tight. If instead the regret bound is required to hold simultaneously at every time $t$, the best known guarantee has been $\sqrt{t \ln n}$---a factor of $\sqrt{2}$ worse---and it has remained unknown whether this factor of $\sqrt{2}$ is necessary. We show that it is not. We give an algorithm, requiring no knowledge of the horizon, whose cumulative regret satisfies $R_t \le \bigl(1 + O(\sqrt{\ln \ln n / \ln n})\bigr)\sqrt{t \ln n / 2}$ simultaneously for every $t \ge 1$.

stat.ML↗

arXiv2026

Efficient Online Inverse Optimization with $O(d)$ Regret

We give a deterministic algorithm for online inverse linear optimization with regret $O(d)$, uniform in the horizon and $O(d^{2})$ time per round. A bound of this order was obtained recently by Dewasurendra, settling a question of Gollapudi et al.\ and of Oki and Sakaue, but by an improper rule that enumerates covers at every scale and costs $T^{Θ(d)}$ a round; ours is the first efficient such bound and the first proper one. We build on the variable-metric framework of Sakaue et al., adding a self-normalized rank-one update, and we replace the $\log\det$ potential by the trace power $\tr(H^{-1/2})$, which is bounded outright and removes the $\ln T$. The bound also holds against an expert that does not optimize, and we give corruption-robust and rank-adaptive variants, and an application to convex minimization.

cs.LG↗

arXiv2026

Forging Self-Funded Marketplaces among Strategic Agents

We introduce the problem of designing mechanisms that incentivize strategic agents to form self-funded marketplaces. In our model, if agent $i$ exerts effort $x_i\in [0,1]$, they incur a cost of $x_i\cdot c_i$ (where $c_i$ is unknown to the mechanism designer) and they generate revenue $x_i\cdot r_i$; crucially, $c_i$ can be greater or smaller than $r_i$. Each effort profile $\mathbf{x}$ yields value $v(\mathbf{x})$ and the objective is to choose an effort vector that maximizes the value while ensuring that every agent $i$ receives a payment $p_i\geq x_i\cdot c_i$ and that $\mathbf{x}$ is budget-balanced, i.e., $\sum_{i} p_i \leq \sum_{i} x_i\cdot r_i$. This problem generalizes the well-studied budget-feasible mechanism design problem, where the requirement is that $\sum_{i} p_i \leq B$ for some predetermined budget $B$. To evaluate the performance of such mechanisms, we first consider the first-best benchmark (the optimal value achievable in the absence of any private information) and show that no truthful auction can achieve a bounded approximation of this benchmark. Also, even in restricted settings, no auction can achieve better than a logarithmic approximation. We complement these results by proposing a class of sequential auctions whose subgame perfect equilibria guarantee a logarithmic approximation of this benchmark. We then introduce an alternative benchmark, the maximin share (MMS), that better captures the thickness of the market and we provide an auction whose subgame perfect equilibria achieve a constant approximation of this benchmark.

cs.GT↗

arXiv2026

From Compensation Design to Budget-Feasible Mechanisms: A Constant Approximation for Subadditive Valuations

Budget-feasible mechanism design is a classic framework introduced by Singer, but there is still a wide gap between existing upper and lower bounds. In this paper, we significantly advance the state of the art. First, without computational constraints, we show that there exists a universally truthful budget-feasible mechanism with the following approximation ratios: - $3$ for monotone submodular valuations and $e+1$ for nonmonotone submodular valuations, improving over $3.798$ and $9.742$, respectively. - $e+1$ for XOS valuations, improving over $28$. In large markets, our approximation can be improved deterministically to $e$. - $2e+1$ for subadditive valuations, improving over $33$. In large markets, our approximation can be improved deterministically to $2e$. Moreover, for subadditive valuations, we obtain a constant-approximation mechanism that runs in polynomial time using demand queries. This improves over the previous best approximation of $O(\log \log n)$, resolving a long-standing open problem going back to Dobzinski, Papadimitriou, and Singer, who conjectured that a constant approximation requires exponentially many demand queries. We obtain these results through a simple and unifying framework based on non-truthful indirect mechanisms, recently coined compensation design. In particular, through a potential argument, we establish constant price-of-stability bounds for compensation design based on marginal-contribution payment rules, which we then translate into truthful direct mechanisms. For subadditive valuations, the core of the argument is a new smoothing lemma showing that every subadditive function can be approximated within a factor of $2$ by a self-bounding function. This is also of independent interest, readily addressing an open question in multiwinner elections by showing the existence of a $2e$-approximate core even under subadditive valuations.

cs.GT↗

arXiv2026

Learning Distributions from Multiple Data Providers

Motivated by learning from heterogeneous and overlapping data providers, we study a stylized model of distribution learning from restricted conditional samples. The goal is to learn an unknown distribution $p$ on a finite domain $[n]$. The learner is given a fixed family of queryable sets $\mathscr{S} \subseteq 2^{[n]}$, and each query to $S \in \mathscr{S}$ returns an independent sample from the conditional distribution $p(\cdot \mid S)$. Learnability is governed by the co-occurrence graph associated with $\mathscr{S}$: two domain elements are adjacent if they appear together in some queryable set. Pointwise consistency is achievable when this graph is connected on the target support. PAC learning requires more: it is possible when the co-occurrence graph is complete. The optimal sample complexity of PAC learning ranges from nearly linear to quadratic. Every query family with complete co-occurrence graph admits sample complexity $\widetilde O(n^2/ε^2)$, and this bound is tight in the worst case. On the other hand, if $[n]$ is queryable then ordinary sampling improves the bound to $Θ(n/ε^2)$, and this cannot be improved further even if every set is queryable. More generally, we identify hierarchical comparabilityas a sufficient structural condition on $\mathscr S$ under which the optimal complexity is nearly linear, $\widetilde Θ(n/ε^2)$, with pairwise query families as a canonical example. Finally, the full range of polynomial rates between linear and quadratic is attainable: for every $α\in (1,2)$, there exists a query family with optimal PAC rate $\widetilde Θ(n^α/ε^2)$.

cs.DS↗

arXiv2026

Compensation Design

We introduce compensation design, the problem of designing payment rules that incentivize high-quality contributions in decentralized environments. Here, a budget-constrained principal with a monotone submodular value function aims to design a payment rule, while agents decide whether to opt in or out depending on their private cost. We show that a simple cost-oblivious and anonymous marginal-contribution payment rule guarantees that pure Nash equilibria always exist and attain a price of anarchy (PoA) of at most $2+o_λ(1)$ in the large-market regime ($λ\to 0$) where each individual cost is at most a $λ$ fraction of the budget. We further show that the factor $2$ is unavoidable among deterministic cost-oblivious rules. Surprisingly, we identify a counterexample showing that a payment rule based on the Shapley value may admit no pure Nash equilibria. We then extend our scope to coarse correlated equilibria. This is further motivated by our intractability result: although a pure Nash equilibrium always exists, computing one is PLS-complete. We establish that coarse correlated equilibria also attain a PoA bound of at most $2+o_λ(1)$, and this guarantee in fact extends even under the payment rule induced by the Shapley value. Moreover, we move beyond monotone submodular value functions and binary actions. First, for (monotone) XOS valuations, we show that no oracle-efficient payment rule can attain a PoA bound of $O(n^{1/2 - ε})$. Second, for submodular but non-monotone valuations, we show that a broad class of natural payment rules fails to guarantee a bounded PoA. Finally, we extend compensation design to the setting where each agent has a combinatorial action set. We provide randomized payment rules with logarithmic PoA guarantees for subadditive values, and matching lower bounds that apply even in the single-agent additive-value setting.

cs.GT↗

arXiv2026

On Language Generation in the Limit with Bounded Memory

We study language generation in the limit under bounded memory. In this task, a learner observes examples from an unknown target language one at a time and must eventually output only new valid examples. Prior work assumes access to the entire history, a strong assumption since realistic algorithms retain limited past information. Classical work in learning theory shows memory constraints dramatically alter learnability; we extend this to language generation. First, we study memoryless generators. Under a mild enumeration restriction, every countable collection of infinite languages remains generable without memory. Without this restriction, we exactly characterize when memoryless generation is possible. For finite collections, we characterize the optimal minimax density achievable by memoryless generators -- the best density guaranteed against any collection of a given size. This combinatorial bound relies on Sperner's theorem and symmetric chain decompositions. We further show that a sliding window of the last $W$ examples does not improve this worst-case density, whereas allowing it to store $b$ adaptively chosen past examples improves the achievable density for every $b \geq 1$. Finally, we revisit identification in the limit, where the learner must converge to a single correct hypothesis for the target language. We focus on its incremental variant, where the learner remembers only its previous guess. Here, although exact identification fails on a collection of just three languages, a mild relaxation requiring convergence to an ``approximate'' version of the target is achievable for every finite collection. These results show bounded memory affects these tasks differently: generation remains achievable for every countable collection, while density and identification are confined to finite collections, with guarantees weakening as the collection grows.

cs.DS↗

arXiv2026

What is Learnable in Valiant's Theory of the Learnable?

Valiant's 1984 paper is widely credited with introducing the PAC learning model, but it, in fact, introduced a different model: unlike PAC learning, the learner receives only positives, may issue membership queries, and must output a hypothesis with no false positives. Prior work characterized variants, including the case without queries. We revisit Valiant's original model and ask: *Which classes are learnable in it?* For every finite domain, including Valiant's Boolean-hypercube setting, we show that a class is learnable if and only if every realizable positive sample can be certified by a poly-size adaptive query-compression scheme. This is a new variant of sample compression where the learner certifies samples via a short interaction with the membership oracle. Our characterization shows that learnability in Valiant's model is strictly sandwiched between learnability in the PAC model and the variant of Valiant's model without membership queries. This is one of the rare cases where introducing membership queries changes the set of learnable classes, and not just the sample or computational complexity. Next, we study the natural extension of the model to arbitrary domains. While we do not obtain an exact characterization, our techniques readily generalize and show that the same strict sandwiching persists. Finally, we show that $d$-dimensional halfspaces, which are not learnable without queries, are learnable with queries: we give a $\mathrm{poly}(d) \tilde{O}(1/ε)$ sample and $\mathrm{poly}(d) \mathrm{polylog}(1/ε)$ query algorithm, and prove that at least $Ω(d)$ samples or queries are necessary. To our knowledge, this is the first algorithm for halfspaces in Valiant's model. Together, these results uncover a surprisingly rich theory behind Valiant's original notion of learnability and introduce ideas that may be of independent interest in learning theory.

stat.ML↗

arXiv2026

On the Learning Curves of Revenue Maximization

Learning curves are a fundamental primitive in supervised learning, describing how an algorithm's performance improves with more data and providing a quantitative measure of its generalization ability. Formally, a learning curve plots the decay of an algorithm's error for a fixed underlying distribution as a function of the number of training samples. Prior work on revenue-maximizing learning algorithms, starting with the seminal work of Cole and Roughgarden [STOC, 2014], adopts a distribution-free perspective, which parallels the PAC learning framework in learning theory. This approach evaluates performance against the hardest possible sequence of valuation distributions, one for each sample size, effectively defining the upper envelope of learning curves over all possible distributions, thus leading to error bounds that do not capture the shape of the learning curves. In this work we initiate the study of learning curves for revenue maximization and provide a near-complete characterization of their rate of decay in the basic setting of a single item and a single buyer. In the absence of any restriction on the valuation distribution, we show that there exists a Bayes-consistent algorithm, meaning that its learning curve converges to zero for any arbitrary valuation distribution as the number of samples $n \to \infty$. However, this convergence must be arbitrarily slow, even if the optimal revenue is finite. In contrast, if the optimal revenue is achieved by a finite price, then the optimal rate of decay is roughly $1/\sqrt{n}$. Finally, for distributions supported on discrete sets of values, we show that learning curves decay almost exponentially fast, a rate unattainable under the PAC framework.

cs.LG↗

arXiv2026

Differentially Private Language Generation and Identification in the Limit

We initiate the study of language generation in the limit, a model recently introduced by Kleinberg and Mullainathan [KM24], under the constraint of differential privacy. We consider the continual release model, where a generator must eventually output a stream of valid strings while protecting the privacy of the entire input sequence. Our first main result is that for countable collections of languages, privacy comes at no qualitative cost: we provide an $\varepsilon$-differentially-private algorithm that generates in the limit from any countable collection. This stands in contrast to many learning settings where privacy renders learnability impossible. However, privacy does impose a quantitative cost: there are finite collections of size $k$ for which uniform private generation requires $Ω(k/\varepsilon)$ samples, whereas just one sample suffices non-privately. We then turn to the harder problem of language identification in the limit. Here, we show that privacy creates fundamental barriers. We prove that no $\varepsilon$-DP algorithm can identify a collection containing two languages with an infinite intersection and a finite set difference, a condition far stronger than the classical non-private characterization of identification. Next, we turn to the stochastic setting where the sample strings are sampled i.i.d. from a distribution (instead of being generated by an adversary). Here, we show that private identification is possible if and only if the collection is identifiable in the adversarial model. Together, our results establish new dimensions along which generation and identification differ and, for identification, a separation between adversarial and stochastic settings induced by privacy constraints.

stat.ML↗

arXiv2025

On Agnostic PAC Learning in the Small Error Regime

Binary classification in the classic PAC model exhibits a curious phenomenon: Empirical Risk Minimization (ERM) learners are suboptimal in the realizable case yet optimal in the agnostic case. Roughly speaking, this owes itself to the fact that non-realizable distributions $\mathcal{D}$ are simply more difficult to learn than realizable distributions -- even when one discounts a learner's error by $\mathrm{err}(h^*_{\mathcal{D}})$, the error of the best hypothesis in $\mathcal{H}$ for $\mathcal{D}$. Thus, optimal agnostic learners are permitted to incur excess error on (easier-to-learn) distributions $\mathcal{D}$ for which $τ= \mathrm{err}(h^*_{\mathcal{D}})$ is small. Recent work of Hanneke, Larsen, and Zhivotovskiy (FOCS `24) addresses this shortcoming by including $τ$ itself as a parameter in the agnostic error term. In this more fine-grained model, they demonstrate tightness of the error lower bound $τ+ Ω\left(\sqrt{\frac{τ(d + \log(1 / δ))}{m}} + \frac{d + \log(1 / δ)}{m} \right)$ in a regime where $τ> d/m$, and leave open the question of whether there may be a higher lower bound when $τ\approx d/m$, with $d$ denoting $\mathrm{VC}(\mathcal{H})$. In this work, we resolve this question by exhibiting a learner which achieves error $c \cdot τ+ O \left(\sqrt{\frac{τ(d + \log(1 / δ))}{m}} + \frac{d + \log(1 / δ)}{m} \right)$ for a constant $c \leq 2.1$, thus matching the lower bound when $τ\approx d/m$. Further, our learner is computationally efficient and is based upon careful aggregations of ERM classifiers, making progress on two other questions of Hanneke, Larsen, and Zhivotovskiy (FOCS `24). We leave open the interesting question of whether our approach can be refined to lower the constant from 2.1 to 1, which would completely settle the complexity of agnostic learning.

cs.LG↗

arXiv2025

Language Generation with Infinite Contamination

We study language generation in the limit, where an algorithm observes an adversarial enumeration of strings from an unknown target language $K$ and must eventually generate new, unseen strings from $K$. Kleinberg and Mullainathan [KM24] proved that generation is achievable in surprisingly general settings. But their generator suffers from ``mode collapse,'' producing from an ever-smaller subset of the target. To address this, Kleinberg and Wei [KW25] require the generator's output to be ``dense'' in the target language. They showed that generation with density, surprisingly, remains achievable at the same generality. Both results assume perfect data: no noisy insertions and no omissions. This raises a central question: how much contamination can generation tolerate? Recent works made partial progress on this question by studying (non-dense) generation with either finite amounts of noise (but no omissions) or omissions (but no noise). We characterize robustness under contaminated enumerations: 1. Generation under Contamination: Language generation in the limit is achievable for all countable collections iff the fraction of contaminated examples converges to zero. When this fails, we characterize which collections are generable. 2. Dense Generation under Contamination: Dense generation is strictly less robust to contamination than generation. As a byproduct, we resolve an open question of Raman and Raman [ICML25] by showing that generation is possible with only membership oracle access under finitely many contaminated examples. Finally, we introduce a beyond-worst-case model inspired by curriculum learning and prove that dense generation is achievable even with infinite contamination provided the fraction of contaminated examples converges to zero. This suggests curriculum learning may be crucial for learning from noisy web data.

stat.ML↗

arXiv2025

Pointwise Lipschitz Continuous Graph Algorithms

In many real-world applications, it is undesirable to drastically change the problem solution after a small perturbation in the input, as unstable outputs can lead to costly transaction fees, privacy and security concerns, reduced user trust, and lack of replicability. Despite the widespread application of graph algorithms, many classical algorithms are not robust to small input disturbances. Towards addressing this issue, we study the pointwise Lipschitz continuity of graph algorithms, a notion of stability introduced by Kumabe and Yoshida [KY23, FOCS'23] and further studied in related settings [KY24, ICALP'24], [KY25, SODA'25], [GKY25, ESA'25]. Our main result is a linear programming (LP) based minimum $S$-$T$ cut algorithm with a provably optimal Lipschitz constant, as witnessed by an accompanying lower bound. As a direct corollary, we give the first dynamic minimum $S$-$T$ cut algorithm with non-trivial recourse bound. At the core of our techniques is a novel framework for analyzing the Lipschitz constant of regularized LP relaxations. Our framework crucially unlocks the use of weighted regularizers, which could not be analyzed through previous methods, and leads to polynomial improvements in the Lipschitz constant compared to what is achievable through previous techniques. To demonstrate the flexibility of our methods, we also design an LP-based $b$-matching algorithm that improves on the state-of-the-art [KY23] Lipschitz constant in certain input regimes when $b\equiv 1$. Moreover, our algorithm cleanly extends to the general case when $b\geq 1$, whereas [KY23] is specialized to the case of $b\equiv 1$.

cs.DS↗