algorithm developers

Abstract: We study antiferromagnetic spin systems on random regular graphs through the lens of the phase transitions characterized by the complex zeros of the partition function. Our motivating examples include the hardcore model and the antiferromagnetic Potts model (including proper $q$-colorings at $β=0$). For both models on random $d$-regular graphs, we establish, with high probability over the graph, zero-free regions throughout the interior of the replica-symmetric regime. Asymptotically, this regime extends up to tree occupation density $$α< α_{\mathrm{RSB}}(d) = (2\log d - 2\log\log d + 2 - 2\log 2 + o_d(1))/d$$ for the hardcore model and up to degree $$d < d_{\mathrm{RSB}}(q) = (2q-1)\log q - 2\log 2 + o_q(1)$$ for proper $q$-colorings. Leveraging the standard algorithmic implications of zero-freeness, we obtain deterministic polynomial-time algorithms that multiplicatively approximate the corresponding partition functions at fixed rational parameters throughout this regime, with high probability over the graph. Our algorithmic guarantees significantly improve upon prior work and extend beyond the parameter ranges reached by previously known search algorithms. To establish our results, we first develop reinforced small subgraph conditioning, which resolves short-cycle fluctuations in the complex plane by lifting classical small subgraph conditioning on random regular graphs to random holomorphic functions. Second, we overcome the standard second-moment barrier caused by the exponential blowup of the bare second moment inside the replica-symmetric phase, by showing that replica symmetry implies a nonpositive Bethe functional and upgrading planted overlap concentration to an exponentially small bound, which enables a typical-graph truncation that controls complex second moments up to the replica symmetry breaking threshold.

Abstract: Let $\mathcal O_K$ be the ring of integers of a number field $K$. A finite subset $\mathcal H=\{η_1,\ldots,η_k\}\subset\mathcal O_K$ is said to be \emph{admissible} if for every prime ideal $\mathfrak p$ in $\mathcal O_K$, the reductions of the $η_i$ modulo $\mathfrak p$ do not represent every residue class. The diameter of a finite subset $S$ of $\mathcal O_K$ is defined to be \[\mathrm{diam}(S):= \max_{η_i,η_j\in S:η_i\neqη_j}|\mathcal N(η_i-η_j)|\] where $\mathcal N$ denotes the norm down to the rational field. This paper computationally investigates admissible subsets of rings of integers in imaginary quadratic fields of class number 1 (such as the Gaussian integers or the Eisenstein integers) that are large relative to their diameter. The motivation is twofold. Firstly, there is the improvement of results analogous to the question of an upper bound for the least gap between large rational primes, which has been one of the great number theoretic successes of the twenty-first century. A second motivation is that there is the opportunity to disprove a variation of the second Hardy-Littlewood conjecture. The calculations in the paper provide new records on the smallest distance between primes in the fields in question, and explicit counter-examples to a generalisation of the second Hardy-Littlewood conjecture in all bar one of the fields.

Abstract: The Equal Subset Sum problem asks, given n positive integers bounded by M, to find two distinct non-empty subsets with the same sum. We study this problem in the regime $M = 2^{o(n)}$, where the pigeonhole principle guarantees that a solution exists, and the main challenge is to find one efficiently. A straightforward dynamic-programming algorithm runs in time poly(n).$M$. After a sequence of improvements, the current best algorithm, due to Jin, Williams, and Zhang (ESA 2025), achieves running time poly(n).$M^{1/3}$. In this work, we give the first poly(n).$M^{o(1)}$-time algorithm for Equal Subset Sum in the regime $M = 2^{o(n)}$. We complement this algorithmic result with a tight conditional lower bound showing that any substantial improvement would require a major breakthrough in approximation algorithms for lattice problems, building on recent techniques of Vafa and Vaikuntanathan (STOC 2025). Finally, we show that our techniques yield an asymptotically faster algorithm for the regime when $M = Θ(2^{n})$: a poly(n)$\cdot 2^{n/4}$-time algorithm in the average case.

Abstract: We study the inverse problem of recovering unknown parameters governing the dynamics and rewards of Mean Field Games (MFGs) from partial observations of population trajectories, made stochastic by common noise. The observer seeks to recover a consistent parameter $θ$ via Bayesian inference, facing the challenge that evaluating the likelihood requires solving a new Nash equilibrium (NE) for each candidate $θ$. Due to common noise, the mean-field evolution is stochastic and equilibrium policies are population dependent, which makes solving for each parameter computationally expensive. To overcome this, we introduce the Bayesian-Extended Mean Field Game (BE-MFG), whose single NE simultaneously encodes equilibrium policies for all parameters, enabling efficient learning of the conditional density of mean-field trajectories. We formalize parameter consistency and identifiability in MFGs with common noise and investigate conditions under which consistent parameter recovery is guaranteed. On three synthetic environments of various complexity, we recover the ground-truth parameter accurately from a small number of partial, stochastic observations, outperforming an adaptation of inverse reinforcement learning for MFGs. A fourth experiment, on public electricity market data, suggests that the approach can also be applied to real-world data.

Abstract: The conclusions of an experimental study of algorithms are only as good as the instances they are run on. Benchmark libraries for well-studied problems took decades to curate, while evaluations of new problems often fall back on random instances, which are known to be misleading: they may say more about the random generator than about the algorithmic question at hand. We treat instance generation as an optimization problem: parameterize the instance space, take the solver's running time as the objective, and let a black-box optimizer search for hard instances. We develop a widely applicable workflow built on the Nevergrad optimization platform and demonstrate it on four NP-hard problems: satisfiability, traveling salesperson, Hamiltonian cycle, and non-permutation flow-shop scheduling. The search needs only the solver, an instance encoder, and a compute budget. Our evolved instances beat the hardest instance of a budget-matched random draw in all four case studies, by an order of magnitude or more in three. They are often competitive with long-curated benchmark sets: our SAT champion is harder (for a three-solver portfolio) than 78% of the 498 SAT Competition formulas in its size class; our 20-city TSP champions take several times longer than the hardest measured classical TSPLIB instance (229 cities); and our flow-shop champion takes 238.6 seconds of CP Optimizer time to solve to proven optimality (versus 4.5 seconds for the hardest of 100,000 random draws) while denying CP-SAT an optimality proof. The optimizer even recovers known structure from running time alone: it locates a SAT clause/variable ratio near the phase transition and finds structurally different champions for the two scheduling engines.

Abstract: We give a randomized algorithm that, for an $m\times n$ matrix whose columns have Euclidean norm at most one, finds a signing with discrepancy below $99$ in expected $\widetilde O(L_{\mathrm{in}}+n^ω)$ bit operations, where $L_{\mathrm{in}}$ is the input encoding length and $ω$ is the exponent of matrix multiplication. The previous fastest algorithm with a constant discrepancy bound, due to Li, is deterministic and uses $O(mn+n^{ω+2}\log^3 n)$ arithmetic operations. We build on the spectral signing framework of Guo, Fang, and Lu and of Li, which rounds a single fractional coloring while one spectral potential controls every row. Li's potential tracks a top eigenvalue, and the gap below that eigenvalue, which can be of order $1/n$, limits the length of every step. We replace it with a resolvent potential that is smooth without any gap condition, and we draw step signs at random, so that the potential only needs to be concave while progress is measured separately. Our key insight is to reuse a single subspace of directions throughout a batch of steps, thereby using only one spectral computation, so that the more expensive linear algebraic operations are paid for only polylogarithmically often at each scale. Within a batch, we sample directions from a Gaussian completion matrix whose inverse we maintain under rank-one row replacements. Finally, we use a finite-precision analysis to prove stability of the algorithm and attain the bit-complexity bound.

Abstract: We give a deterministic data structure for range weighted longest increasing subsequence (LIS) queries on $n$ values with positive weights. Preprocessing takes $O(n\log^2 n)$ time and space. Value queries take $O(\log n)$ time, and an optimal subsequence of $k$ elements can be reported in $O(k + \log n)$ time. This resolves an open question on truly subquadratic preprocessing with polylogarithmic query time. The key is a single-crossing property of envelopes for value-separated endpoint sets, enabling $O(\log n)$-time persistent merges. The bounds assume constant-time exact addition and comparison of weights.

Abstract: We study learning-augmented streaming algorithms for estimating the value of Boolean Maximum Constraint Satisfaction Problems (Max-CSPs). Specifically, we consider streaming algorithms equipped with an $\varepsilon$-accurate oracle: for each variable, the oracle outputs a label in $\{-1,1\}$ that agrees with a fixed optimal assignment with probability $1/2+\varepsilon$, and disagrees otherwise. Prior work by Dong, Peng, and Vakilian [ITCS 2025] showed that, for the special case of Max-Cut, such an oracle enables a $(1/2+Ω(\varepsilon^2))$-approximation using $\operatorname{poly}(1/\varepsilon)$ words of space in insertion-only streams, and $\operatorname{poly}(1/\varepsilon,\log n)$ words of space in dynamic streams. We substantially generalize and strengthen their result, provided that a highly accurate estimate of $\varepsilon$ is available. For any Boolean Max-$2$CSP and any $η>0$ independent of $\varepsilon$, we obtain, with high constant probability, a single-pass $(1-η)$-approximation using $\operatorname{poly}(1/\varepsilon,1/η)$ words of space in insertion-only streams, and $\operatorname{poly}(1/\varepsilon,1/η,\log n)$ words of space in dynamic streams, where $n$ is the number of variables. We further extend our techniques to general Boolean Max-$k$CSPs for $k\ge 3$, with slightly worse space complexity.

Abstract: The Game of Cycles, invented by Francis Su, is a two-player game in which the players take turns drawing arrows on the edges of a plane graph, never creating a sink or a source. On a tree only the sink-source rule matters, and the player who makes the last move wins. Several authors conjectured that on a tree the second player wins exactly when the number of markable edges is even. We show that this parity conjecture is false, and we explain why. The explanation is a hidden game of Nim: a path segment that is blocked by an arrow at one end and free at the other behaves exactly like a nim heap whose size is its length. Combined with a calculus for cutting and gluing trees at vertices next to leaves, this shows that the conjecture fails as soon as a single tree has Grundy value at least 2. The smallest counterexamples have 15 vertices. Trees with Grundy values 2 and 4 first appear at 17 vertices, and every value up to 63 occurs, which answers a question of Fokkink and Zandee. We also refute a related conjecture of Zandee. We then look closely at spiders. The spiders S(2,3,z) violate the conjecture for every z >= 14. The Grundy values of S(1,2j,z) are given by an exact formula governed by binary carries. For 99 pairs (x,y), the value of S(x,y,z) is eventually the parity of z. On the positive side, the conjecture holds for all trees in which every thread has at most two edges, and this bound is sharp. Along the way we prove a twin-branch reduction, give an algorithm that is polynomial for spiders with a fixed number of legs, and collect the questions that we could not answer. The boldest asks whether every non-negative integer is the Grundy value of some tree.

Abstract: Choosing a few robust algorithms or schedules can require evaluating every candidate under many adverse scenarios, even though an agent only needs to identify the winners. A predictor may suggest those winners, but cannot rule out a surprise in an unevaluated scenario. We introduce Certified Adaptive Selection (CAS), which returns the exact top-$k$ set as soon as remaining uncertainty cannot change it. CAS keeps guaranteed lower and upper bounds on Expected Shortfall (ES), the average loss in the worst scenarios. Its stopping test compares the worst admissible selected candidate with the best admissible excluded one; for information-disjoint Cartesian intervals this one-cut test is necessary and sufficient. Prediction has a separate role: selecting the next outcome to evaluate, never authorizing the decision. For empirical ES with exact unit-cost queries, we compute in polynomial time the fewest evaluations that could have proved the realized decision in hindsight. Our prediction-guided agent CAS-PG is exact for every predictor, including adversarial ones. On two public algorithm-selection matrices, CAS-PG certifies the exact top three after 630 of 3150 and 981 of 2200 replayed queries, or $1.14$ and $1.18$ times that hindsight minimum. Without historical outcomes, its transcript-only variant requires $1.28$ and $1.37$ times the minimum, outperforming the tested non-predictive certified policies. Separately, a prediction-free CAS policy avoids $93.0\%$ of exact recourse solves in synthetic energy scheduling. One certificate thus supports several acquisition policies and separately proved risk/cost extensions: prediction changes the work, not the justified choice.

Abstract: A parallel sampler can choose at random which coordinates to generate together. For a target $p$ and the output law $q_U$ of a schedule $U$ drawn afresh for each output, we compare the schedule-averaged divergence $\mathbb{E}_U D_{\mathrm{KL}}(p\|q_U)$ with the output-law divergence $D_{\mathrm{KL}}(p\|\mathbb{E}_U q_U)$, in nats. Rounds commit independent draws from exact conditional marginals frozen at the round's start, each output's schedule is drawn before sampling from a finite mixture of schedules, and every scheduled round is charged. For the uniform distribution on binary strings of even length $n$ with $n/2$ ones, the least expected number of rounds at a fixed tolerance $\varepsilon>0$ is $Θ_\varepsilon((\log n)^2)$ under the first criterion but $(e^{-\varepsilon}+o(1))\log_2\ln n$ under the second, and the double-logarithmic order is attainable with the same round count on every run. For this target, a count-state recursion gives the exact least output divergence at every round cap over all causal randomized policies. The separation extends to weighted fixed-count targets with bounded weight ratios and quota fractions bounded away from zero and one, without permutation symmetry, and to the full-support Beta$(1,1)$--Bernoulli target, where the sharp expected cost under the output-law criterion is $(e^{-\varepsilon}/2+o(1))\log_2\log_2n$, attained with rare $n$-round runs. Prior work implies the order of the schedule-averaged cost on the balanced target and its $O_\varepsilon((\log n)^2)$ upper bounds on both extensions.

Abstract: Continuous random node features increase the expressivity of message-passing neural networks (MPNNs), but existing theoretical analyses are not fine-grained enough to capture their limitations, particularly in approximating functions across graph sizes or under a fixed number of random bits. We develop a population-level framework for size-uniform approximation. For MPNNs with a graph-size-independent Lipschitz bound in their initial features, including standard mean, max, and order-normalized sum architectures, we show that continuous random features can be quantized to finitely many colors with arbitrarily small uniform error in expected output. This reduction lets us transfer limitations on finite-color distinguishability to limitations on size-uniform approximation with continuous random features. We use this framework to prove that these MPNNs are not size-uniformly universal, analyze the expressivity gains from colors and independent trials, and establish incomparability between random-color MPNNs and higher-order Weisfeiler-Leman methods. We also give finite-color MPNN constructions for tree subgraph isomorphism via color coding and a local MST task.

Abstract: In this note, we extend the analysis underlying a recent matrix-multiplication result by OpenAI to rectangular products and prove that $ω(1,k,1)\le 2$ for $0\le k\le \frac{1}{2}$ and $ω(1,k,1)\le 1+k+\frac{1}{4k}$ for $k\ge \frac{1}{2}$. In particular, $ω(1,\frac{1}{2},1)=2$ and the dual exponent satisfies $α\ge \frac{1}{2}$. We use the shared-leg entropy inequality and polynomial-multiplication degenerations from that work, retaining two-leg symmetry and the orientation of each sector. Logarithmic averaging produces homogeneous auxiliary profiles. Their powered versions have a common asymptotic slope, and bounding their intercepts gives the spectral constraint $b\le 4a(1-a)$. This yields the rectangular curve by tensor-spectrum duality. As an application, Zwick's algorithm for all-pairs shortest paths in directed unweighted graphs runs in $O(n^{2.5})$ time. Combining the rectangular bound with the $(\min,+)$-product improvement of Alman and Vassilevska Williams further gives $O(n^{2.4999})$ running time.

Abstract: We present a deterministic reduction from min-sum subset convolution to min-plus matrix product. We show that if the min-plus product of two $D\times D$ matrices with $β$-bit integer entries can be computed in $D^{3-δ}\operatorname{poly}(β,\log D)$ time for a fixed rational $0<δ<1$, then min-sum subset convolution on an $n$-element universe can be solved in $(2+2^{-δ})^n 2^{O(\sqrt n\log(n+1))}\operatorname{poly}(n,β)$ time. Instantiating this reduction with the recent breakthrough on subcubic min-plus matrix product by Alman and Vassilevska Williams gives a Las Vegas algorithm with expected running time $O^*(2.9987^n)$ and a deterministic algorithm with running time $O^*(2.9997^n)$, strictly breaking the longstanding $3^n$ computational barrier. Notably, these speedups translate directly to database query optimization, yielding the same expected and deterministic running-time bounds for join ordering under the $C_{\mathrm{out}}$ cost function.

Abstract: Bayesian optimization (BO) is widely used as a standard approach for expensive black-box optimization. However, BO algorithms often involve parameters that must be specified in advance, and their performance can strongly depend on these choices. We propose a framework for optimizing such parameters using sample paths drawn from a Gaussian process (GP) inferred from the information available at the start of BO. We use cumulative regret as the performance metric for a BO algorithm. By running the BO algorithm on the generated sample paths, we obtain an empirical estimate of its expected cumulative regret for a given parameter configuration. Optimizing this estimate allows us to identify parameter configurations that, given the currently available information, are expected to achieve low cumulative regret. Since this parameter optimization is itself a black-box optimization problem, we employ another BO procedure to solve it, which we refer to as outer BO. Through experiments, we demonstrate that the proposed framework can effectively select parameter configurations that achieve strong performance among a range of candidate configurations.

Abstract: Reusing the scores that select a Best-of-$N$ winner can overstate its expected reward. We study evaluation from a fixed matrix of $K$ independent scores per candidate for a policy that selects using $J$ fresh scores. A single estimator based only on this matrix is exactly unbiased for expected judge reward under every independent, stable collection of candidate-specific score laws if and only if $J<K$, for every pool size $M\ge N\ge2$. At $J=K-1$, the selector deepens as $K$ grows. For independent Gaussian scores with common variance and fixed $M\ge N\ge2$, the unbiased minimax risk in this regime is of order $σ^2/\sqrt K$, attained by Holdout; allowing bias improves the rate to $σ^2/K$. For two candidates, we derive the minimum-variance unbiased estimator at known variance and the sharp asymptotic unbiased minimax constant $1/(π\sqrt2)$, which Holdout attains without knowing the variance. The cyclic average over subsets and ties can be computed in $O(MK\log M)$ operations. At fixed selector depth, cyclic evaluation of bounded scores has $O(K^{-1})$ risk uniformly in pool size. The impossibility result concerns the fixed matrix: one additional fresh winner score permits unbiased evaluation of the all-$K$ policy.

Abstract: We study the complexity of Nash equilibrium computation in discrete Colonel Blotto games. For two-player general-sum Colonel Blotto with monotonic piecewise-constant battlefield payoffs, we show that computing an inverse-polynomial approximate Nash equilibrium is PPAD-hard, even with only two battlefields and a constant number of pieces per battlefield, thereby resolving an open question of [KPF+25]. Allowing more battlefields, the hardness persists when every battlefield payoff is $2$-piecewise constant with respect to either player's allocation. A general PPAD-membership theorem for multiplayer Blotto with arbitrary local reward functions then implies PPAD-completeness for both hardness results. Motivated by fixed-rank bimatrix games, we then identify a tractable frontier within two-player general-sum Colonel Blotto. If the social payoff has rank-$r$ structure, then, for every fixed $r$, an $\varepsilon$-Nash equilibrium can be computed in time polynomial in $B_1,B_2,k$, and $1/\varepsilon$. Thus constant-rank two-player Colonel Blotto remains tractable despite its exponentially large pure strategy spaces. Finally, we study multiplayer winner-takes-all Colonel Blotto with player-specific battlefield values and uncover a sharp tie-breaking frontier. If every highest bidder receives her full battlefield value, an exact pure Nash equilibrium can be computed in polynomial time for arbitrary binary-encoded budgets. In contrast, under ordinary equal splitting among highest bidders, we prove that computing an inverse-polynomial-accuracy Nash equilibrium is PPAD-hard even with only five players. This resolves an open question posed in concurrent work by Bichler and Ghosh [BG26], who established PPAD-hardness in the same player-specific equal-splitting setting when the number of players grows with the instance and asked whether hardness persists for a constant number of players.

Abstract: Large Language Models (LLMs) have been used for novel discovery of algorithms, theorems, drugs, and other tasks through the use of harnesses that prompt an LLM to iteratively optimize an objective. In this work, we study the relationship between the population of previous iterates and eventual discovery success. We generalize past work on harness design to develop a suite of 12 harnesses called 'Modular' and characterize their performance across 5 diverse discovery tasks, finding that discovery success is brittle and sensitive to harness design. We uncover mode collapse, characterized by a dramatic drop in the diversity of iterates, as a common failure mode. We find that popular state-of-the-art harnesses and diversity-inducing harness interventions, which aim to prolong this collapse, yield inconsistent gains. Our results instead uncover that the performance of early discoveries is predictive of eventual success. We therefore propose a universally applicable intervention that performs an initial stage of parallel exploration in order to initialize subsequent iterative optimization. Our method provides consistent gains across many harnesses and target applications, confirming the importance of initialization in LLM-driven discovery.

Abstract: Dynamic Time Warping (DTW) is a classical similarity measure for strings and time series that allows local stretching. Given non-empty strings $S,T$ over an alphabet $Σ$ and a cost function $δ:Σ^2\to\mathbb{R}_{\ge0}$, $DTW_δ(S,T)$ is the minimum total cost of equal-length expansions of $S$ and $T$ obtained by duplicating characters. For strings of length at most $n$, DTW is computable in $O(n^2)$ time, and this is conditionally optimal under the Orthogonal Vectors Hypothesis (OVH). We study the low-distance regime, where an integer $k$ upper-bounds $DTW_δ(S,T)$, assuming $δ(a,a)=0$ and $δ(a,b)\ge1$ for $a\ne b$. For several classical similarity measures, this regime admits $O(n+\operatorname{poly}(k))$ algorithms, whereas for DTW with metric costs the best known bound is $O(nk)$. We show that this dependence is essentially optimal: assuming OVH, computing DTW requires $n^{1-o(1)}k$ time even for the discrete mismatch-cost function, which assigns cost $1$ to every mismatch. The lower bound applies to the whole spectrum of thresholds $k$ between constant and linear in $n$. Our reduction from Orthogonal Vectors encodes vector coordinates in the lengths of equal-character runs. The resulting instances are very structured: collapsing runs to single characters reveals long substrings with short periods. We complement the lower bound with a $\tilde O(n+\operatorname{poly}(k))$-time algorithm whenever, after collapsing runs in the inputs, every substring with period $O(k)$ has length $\operatorname{poly}(k)$. Finally, we extend this lower bound to DTW pattern matching, which asks whether any non-empty substring of a length-$n$ text has DTW distance at most $k$ from a length-$m$ pattern. We prove that the classic $O(nm)$-time dynamic-programming algorithm is near-optimal under OVH, even when $k=O(\log n)$.

Abstract: A majority coloring of a graph $G = (V,E)$ is a coloring of $V(G)$ such that, for each vertex $v$, the number of neighbors of $v$ with the same color as $v$ is at most $deg(v)/2$. A strong majority coloring is a coloring of $V(G)$ such that, for each vertex $v$, every monochromatic subset of $N(v)$ has size at most $deg(v)/2$. The (strong) majority coloring game is a two-player Maker-Breaker-type game, in which two players Alice and Bob color the vertices of a graph $G$ alternately, maintaining the (strong) majority condition. The least number of colors such that Alice has a winning strategy in such a game is called the (strong) majority game chromatic number of the graph $G$, denoted $μ_g(G)$ (or $\mathrm{Maj}_g(G)$ for the strong version). For the majority coloring game, we prove that $μ_g(G) \le 3$ under the following cases: $G$ is a $2$-caterpillar, $G$ is a rooted tree with all leaves at depth $k \le 4$, and $G$ is a subdivision of some graph. The latter resolves a problem posed by Bosek--Grytczuk--Jakóbczak in 2019, who also asked whether $μ_g(T) \le 3$ for every tree $T$. For the latter question, we discuss various difficulties that arise when natural strategies are attempted by Alice to win the majority coloring game on trees. We include a comparison with the marking game and relaxed coloring game on trees, and with the majority coloring game on locally finite acyclic graphs $G$ with $δ(G) > 1$. For the strong majority coloring game, we compute $\mathrm{Maj}_g(C_n)$ exactly for each cycle $C_n$, $n \ge 3$. We also initiate the study of the computational complexity of the strong majority coloring game; specifically, we prove that the decision version of the Strong Majority Game Chromatic Number problem is PSPACE-complete. We also show that the Strong Majority 2-Coloring problem is NP-complete on Eulerian graphs.

Abstract: Community-detection methods search over possible partitions of a network, but the structure of this partition space is rarely examined directly. This paper introduces two complementary tools that make~$\mathcal{P}$ analytically and visually accessible. First, a canonical labelling scheme based on the Restricted Growth Sequence~(RGS) is adopted, assigning each partition a unique, permutation-invariant identifier and eliminating the need for pairwise similarity measures such as the Normalized Mutual Information. Second, granularity~$Γ$ is defined and combined with modularity~$Q$ to produce a two-dimensional projection: the Partition Space Map~(PSM). Within the $(Γ,Q)$ plane, only a discrete set of integer values of~$Γ$ is attainable, and upper and lower bounds on~$Q$ are derived directly from the deviation matrix. These tools are demonstrated through complete enumeration of~$\mathcal{P}$ for small benchmark graphs, revealing the fine structure of modularity degeneracy and the combinatorial constraints that shape the feasible region. The framework provides an exact, assumption-free reference for evaluating heuristic community detection algorithms and lays the groundwork for geometry-aware exploration of partition space in larger networks.

Abstract: Four decades on, Håstad's classical $2^{Ω(n^{1/(d-1)})}$ lower bound for depth-$d$ circuits computing Parity remains the best known $\mathrm{AC}^0$ circuit lower bound for any explicit function. Majority has long been a compelling candidate for stronger lower bounds: the most natural circuits computing it are substantially larger than those for Parity and have repeatedly been conjectured to be optimal. We present a simple construction, found by GPT-6 Astra, of depth-$d$ circuits of size $2^{O(n^{1/(d-1)})}$ for any symmetric function. This result settles the asymptotic $\mathrm{AC}^0$ circuit complexity of Majority, matching Håstad's lower bound.

Abstract: We consider the fair division problem with indivisible goods and study the following decision problem: given a fair division instance, does there exist an allocation that is envy-free and efficient? We consider two efficiency criteria: Pareto-optimality and social welfare optimality. We provide a complete landscape on the computational complexity of this decision problem, with the number of agents ranging from $2$ to $\infty$, both additive valuations and general valuations, and the more restricted class of $k$-ary valuation functions (where an item's marginal value is restricted to $\{0,1,\ldots,k-1\}$ for some constant $k\geq2$). One interesting observation is that many versions of the above-mentioned decision problems fall into the ``gap'' between the first and the second levels of the polynomial hierarchy. Specifically, assuming the polynomial hierarchy does not collapse to the first level (i.e., assuming $\text{NP}\neq\text{coNP}$), these problems are in $(Σ_2^{\text{p}}\capΠ_2^{\text{p}})\setminus(\text{NP}\cup\text{coNP})$. In particular, we provide a fine-grained complexity analysis across different parameter regimes, including the number of agents and the choice of valuation models. Depending on different parameters, many problems admit different complexity classifications, ranging from the intermediate classes $Θ_2^{\text{p}}$ and $Δ_2^{\text{p}}$ between the two levels to $Σ_2^{\text{p}}$-completeness. Finally, De Keijzer et al. show the $Σ_2^{\text{p}}$-completeness of the decision problem when considering Pareto-optimality as the efficiency criterion with additive valuations. Our main results extend this result to more restricted settings, such as instances with a constant number of agents or $3$-ary valuation functions, which resolves the open problem given by Bouveret and Lang.

Abstract: We give a deterministic algorithm for exact modular subset sum that, for every modulus m, computes all reachable residues and one requested witness in O(m) time and O(m) auxiliary words. The input is a compact list of distinct residues with multiplicities, and the word-RAM supports constant-time modular arithmetic. The algorithm represents reachable residues as intervals along cycles of repeated addition, charging work on partial cycles to newly reached residues. Processing prime factors in increasing order keeps the cost of rebuilding and changing cycles linear. A classical theorem on subset sums of distinct invertible residues bounds the number of boundary lists by O(m^(3/4)); together they contain O(m) interval endpoints. Comparison sorting the short lists and radix sorting the long ones then takes O(m) total time. The algorithm is fast in practice, using arrays and interval lists rather than heavy data structures.

Abstract: We establish a strongly sublinear counterpart of a recent result of Chudnovsky, E S, and Lokshtanov (arXiv 2025) on treewidth and tree-independence number. Namely, we prove that a hereditary graph class has strongly sublinear tree-independence number if and only if, for every fixed clique bound, its graphs of bounded clique number have strongly sublinear treewidth. In fact, this is part of a broader equivalence theorem. For hereditary classes, these conditions are also equivalent to having clique-dependent polynomial expansion, to admitting balanced separators whose size is bounded by $Kω(G)^s |V(G)|^{1-β}$ for fixed $K,s,β>0$, and to admitting balanced clique-based separators of strongly sublinear size (equivalently, weight). Thus, we show that all these properties, which arose independently in the study of subexponential-time exact algorithms and polynomial-time approximation schemes, in fact describe the same hereditary graph classes. As a consequence of our equivalence theorem, we also show that every hereditary class $\mathcal C$ with strongly sublinear tree-independence number admits a subexponential-time algorithm that, given $G\in\mathcal C$, computes a tree decomposition of $G$ with strongly sublinear independence number.

Abstract: Kannan's algorithm, as analyzed by Hanrot and Stehlé in 2007, solves the exact Euclidean shortest vector problem in polynomial space and $n^{\frac{n}{2e}+o(n)}$ time. In the classical setting with polynomial space, we obtain the first improvement on this bound via a randomized algorithm that runs in $n^{\frac{n}{4e}+o(n)}$ time. The main idea is to represent a fixed shortest vector in many ways as a difference of samples, thereby enabling the low-space collision search of Lyu and Zhu (SODA 2023) to replace exhaustive enumeration in the original analysis.

Abstract: In previous papers, we began the study of sequence prediction algorithms adapted to stringological word complexity measures. In particular, we defined a complexity measure called Arithmetic Repetition Complexity (ARC) which admits a polynomial-time prediction algorithm with a mistake bound quasilinear in the complexity. Here, we show a weaker complexity measure related to ARC that admits an especially efficient prediction algorithm: an algorithm that runs in quasilinear time and polylog space for appropriate highly-structured sequences. The complexity measure is defined via a restricted class of "zipline programs" (a variant of straight-line programs), which we call layered. We thus get a less expressive measure with a more efficient algorithm (compared to our results for ARC), demonstrating a possible tradeoff.

Abstract: AD-k stands for Alternating Differences through order k, and it is a property of set functions denoting that the first order difference of the set function is nonnegative (a.k.a. monotnocity), the second order difference is nonpositive (submodularity), and so on with signs alternating through order k. Chen et al. [1] conjectured that in an influence diffusion model called the general threshold model originally defined by Kempe et al. [2], if every local influence function is AD-k, then the global influence spread function is also AD-k, for any (possibly cyclic) directed graph and any k. This paper provides a complete proof showing that the conjecture is true. The proof utilizes Mobius inversion and decision tree partition method and extends the probability distribution of node triggering sets into a generalized algebraic structure allowing negative weights for triggering sets. The extension to negative-weighted triggering sets may be of independent interest.

Abstract: We develop a certified continuation framework for equilibrium computation and for training deep equilibrium networks (DEQs), with training formulated as interpolation to accuracy $2^{-b}$. For inference, compact input homotopy selects a unique branch from a supplied start root, and a rounded Newton tracker follows it under certified boundary, conditioning, derivative, and tube-radius bounds. For training, we augment local-plus-low-rank recurrence with programmable dormant bilinear rank-one channels. Loaded Tikhonov solves diagnose a failed interpolation pass without spectral decomposition; an output-preserving repair aligned with the pass residual supplies the required direction. Training requires certified gate realization and column stability on each pass region, well-posed inference, and finite-update error budgets. With polynomial geometric, encoding, precision, and complete backend budgets, both certified inference and training have bit cost $O(\operatorname{poly}(L+b))$, where $L$ is the encoded instance length. The trainer uses $O(b+\ell)$ passes and reserve channels from an initial residual bounded by $2^\ell$. These guarantees concern a certified promise class. Lean 4 verifies the quantitative core and concrete inference backend; numerical comparisons illustrate the loaded mechanism.

Abstract: Motivated by recent work on tree independence number, we study the path independence number of a graph $G$: the minimum integer $k$ such that there is a path decomposition of $G$ where each bag induces a graph with independence number at most $k$. We show that every graph excluding both an induced forest minor and an induced star has bounded path independence number. This characterises when a graph class that excludes an induced star has bounded path independence number while also partially resolving a conjecture of Dallard, Krnc, Kwon, Milani{č}, Munaro, Štorgel and Wiederrecht (2024). Furthermore, we show that graphs excluding both an apex-forest induced minor and an induced star have bounded tree independence number. As a consequence, for every fixed apex-forest $H$ and integer $t$, there is a polynomial-time algorithm to test whether a $K_{1,t}$-induced-subgraph-free graph contains $H$ as an induced minor. Moreover, it follows that the Maximum Weight Independent Set problem, as well as several other NP-hard problems, can be solved in polynomial-time on $K_{1,t}$-induced-subgraph-free graphs that exclude $H$ as an induced minor.

Abstract: For a graph $F$, the Turán number $\operatorname{ex}(n,F)$ is the maximum number of edges in an $n$-vertex graph containing no copy of $F$. Determining the Turán numbers of even cycles is a central problem in extremal graph theory and remains open in general. For $C_6$, the best previous upper bound was due to Füredi, Naor, and Verstraëte [Advances in Mathematics, 2006], who proved that, for sufficiently large positive integer $n$, $$ \operatorname{ex}(n,C_6) \leq λn^{4/3}+O(n)<0.6272 n^{4/3}, $$ where $λ$ is the real root of $ 16λ^3-4λ^2+λ-3=0$. We improve this bound by showing that, for sufficiently large positive integer $n$, $$ \operatorname{ex}(n,C_6) \leq αn^{4/3}+O(n)<0.6144 n^{4/3}, $$ where $α$ is the unique real root of $ 4 α^{3} (3/2)^{1-1/(2α)} =1$ in the interval $(1/2,2/3)$.

Abstract: In online inverse linear optimization, the learner predicts a weight at each round, observes the optimal action of the agent, and updates its prediction. In the general setting, the gap of $\log T$ between the regret upper bound $O(d \log T)$ and the lower bound $Ω(d)$ is unresolved (here $T$ is the total number of rounds and $d$ is the dimension). When the action set is M-convex, the regret is known to be bounded by $O(d \log d)$, but the method attaining it computes a center of gravity at every round. This paper therefore proposes Small-Gradient Skipping (SGS), a mechanism that skips the update at rounds without a mistake in the case where the correct action is uniformly separated from the other candidates, and applies it to online gradient descent, the online Newton step, and MetaGrad. The number of mistakes is then bounded, for all three, by a quantity independent of $T$; and for the online Newton step and for MetaGrad with SGS, the dimension dependence of the regret becomes $O(d^2)$ when the forward problem is an integer linear program, that is, the factor $\log T$ is removed. Moreover, when the action set is M-convex, the regret is bounded efficiently without computing a center of gravity.