Wen Fang, Xianxian Li, Jun Liu, Jie Luo, Yongxin Tong, Ke Xu
We study self-reference and solution independence in Boolean $K$-SAT as a finite analogue of Gödel incompleteness. Independence is defined through factorial moments of the solution count. Constant-width random formulas fail this criterion due to correlations among overlapping assignments. At a covering threshold, logarithmic-width random CNFs have a Poisson solution count, with positive probabilities of unsatisfiability and unique satisfiability. Replacing a redundant clause by one falsified by the unique solution yields an unsatisfiable companion preserving the unsigned incidence graph. For every permitted local view of a good uniquely satisfiable formula, an unsatisfiable companion preserves that view. Irreducibility defeats deterministic and bounded-error evaluators using $N^c$ clause queries, for fixed $0<c<1$. It establishes a local information obstruction, not an unrestricted time lower bound. For fixed $δ>0$, transcripts of $N^{1-δ}$ queries leave $N-o(N)$ bits of witness entropy. On typical support graphs, expansion yields linear Resolution width, linear PC/PCR and SOS degree, and exponential proof size. Bounds hold for semantic Cutting Planes under a partition condition and for Positivstellensatz with bounded product width, uniformly over unsatisfiable support-preserving signings. Structural evidence and the limits of acceleration strategies motivate the self-referential $K$-SAT exhaustive-search hypothesis (SRK-ESH): no uniform deterministic algorithm deciding the promise family achieves a fixed proportional saving in the exponent. This dual-evidence criterion combines intrinsic hardness mechanisms with algorithmic evidence. We interpret this hypothesis as a quantitative projection of the Gödel-style local--global gap. The information obstruction constrains quantum local observation and statistical learning from bounded local information.
Binh-Minh Bui-Xuan, Florent Krasnopol, Bruno Monasson, Nathalie Sznajder
Temporal graphs are graphs where the presence or properties of their vertices and edges change over time. When time is discrete, a temporal graph can be defined as a sequence of static graphs over a discrete time span, called lifetime, or as a single graph where each edge is associated with a specific set of time instants where the edge is alive. For static graphs, Courcelle's Theorem asserts that any graph problem expressible in monadic second-order logic can be solved in linear time on graphs of bounded tree-width. We propose the first adaptation of Courcelle's Theorem for monadic second-order logic on temporal graphs that does not explicitly rely on a parameter proportional to the lifetime, or defined as the maximum number of time-edges incident with any vertex which in the worst case is higher than the lifetime. We then introduce the notion of derivative over a sliding time window of a chosen size, and define the tree-width and twin-width of the temporal graph's derivative. We exemplify its usefulness with meta-theorems with respect to a temporal variant of first-order logic. The resulting logic expresses a wide range of temporal graph problems including a version of temporal cliques, an important notion when querying time series databases for community structures.
Coflows capture the collective communication requirements of data-parallel applications, and scheduling them well is essential to the performance of modern data centers. We study non-splitting coflow scheduling in a heterogeneous parallel network whose $m$ network cores may be Electronic Packet Switches (EPS), not-all-stop Optical Circuit Switches (OCS), or all-stop OCS, where every flow must be transmitted entirely through a single core. Previous approximation algorithms for this setting have guarantees that grow linearly with the number of cores $m$ or with the maximum flow degree $τ$ of a port, and therefore become weak as data centers deploy more cores. The source of this dependence is the rounding step: rounding each flow to the core holding its largest fractional value can inflate the load of a port by a factor of $m$. We replace this step with an iterative relaxation rounding scheme that handles the input-port and output-port capacity constraints simultaneously. For minimizing the makespan, this yields approximation ratios of $\min\{τ, 3\}$, $2\min\{τ, 3\}$, and $4\min\{τ, 3\}$ for pure EPS, pure not-all-stop OCS, and pure all-stop OCS networks, respectively. For minimizing the total weighted coflow completion time, we obtain approximation ratios of $8\min\{τ, 4\}$, $16\min\{τ, 4\}$, and $32\min\{τ, 4\}$ for the same three architectures. In a hybrid network, the guarantee is governed by the weakest architecture present, so the overall ratios are at most $4\min\{τ,3\}\le 12$ and $32\min\{τ,4\}\le 128$, respectively. Unlike all previous guarantees for non-splitting coflow scheduling in heterogeneous parallel networks, these bounds are absolute constants that do not grow with the number of cores $m$ or with the flow degree $τ$. We also show that the integrality gap of the underlying linear program is at least $2$.
Given a string $T[1..n]$, its substring complexity is defined as $δ=\max_k d_k/k$, where $d_k$ is the number of its distinct substrings of length $k$ in $T$. We show how to compute it in $O(n+r\log r)$ time, using only $O(r)$ words of space, where $r$ denotes the number of runs in the Burrows-Wheeler transform.
We study multidimensional online generalized assignment under nonstationary arrivals, motivated by resource allocation for large language model agent services. Providers may commit model and tool allowances to agent tasks upon admission, reserving budgets for their internal execution loops. Each request is rejected or accepted with an irrevocable assignment to one compatible server, earning reward and consuming a vector of its nonreplenishing budgets. Arrivals are independent with known, potentially time-varying type distributions. Scalar reductions can discard feasible combinations of requests with complementary resource requirements. To maximize expected reward, we develop Vector-Aware Best-Fit, which combines linear-programming-guided candidate routing with calibrated acceptance while retaining vector feasibility. Relative to the expected clairvoyant offline optimum, its guarantee improves on scalar reduction in the multidimensional case and has the best possible inverse-dimensional order. With an appropriate acceptance parameter, the proposed policy attains the optimal competitive ratio on all-heavy instances, where every feasible assignment consumes more than half of at least one resource budget. We further develop two heuristics that retain its performance guarantee. Computational results show that the heuristics perform well against benchmarks in most tested settings, especially when capacity is scarce relative to demand.
Order graphs were introduced by Arroyuelo et al.\ (TODS 2024) as a generalization of the ring index, to support worst-case optimal joins on relations of arity $d$ while storing as few columns as possible: the index stores one column per edge, so its size is proportional to the number of edges of an order graph. It was known that the minimum number of edges lies between $Ω(2^d\sqrt d)$ and $O(2^d d)$, and the lower bound had been claimed to be $42$ for $d=6$. We first exhibit an order graph with only $40$ edges for $d=6$, and show that the proof of that lower bound holds only for unions of cycles, whereas order graphs can be smaller than the smallest set of cycles. We give a corrected statement and proof that holds for every order graph: there must be at least $\max_m\binom dm\lceil m/2\rceil$ edges. This bound is $40$ for $d=6$, so our $40$-edge graph is optimal. We then settle the asymptotic question: the minimum size is $Θ(2^d\sqrt d)$. This is obtained via an explicit construction, based on symmetric chain decompositions of the Boolean lattice, with at most $(d-1)\binom{d}{\lfloor d/2\rfloor}+2^d$ edges. The upper bound is within a factor $4+O(d^{-1/2})$ of the lower bound, and both numerically and asymptotically better than trie switching.
We study online allocation with increasing returns, where an uncertain number of customers arrive sequentially and can be irrevocably assigned to any product with remaining capacity. Each assignment earns a base reward plus a bonus that is nondecreasing in the number of previous assignments to the same product. Prior work has focused primarily on fixed or diminishing returns. Under increasing returns, the central challenge is that concentrating assignments can unlock large future rewards, while a different product may be preferable if arrivals stop early. We show that classical approaches, including greedy and natural LP-based rounding algorithms, can perform arbitrarily poorly, even with homogeneous bonuses. We evaluate performance through competitive ratios relative to the hindsight offline optimum. For homogeneous bonuses, we establish a tight competitive ratio of $1/2$. For arbitrary heterogeneous bonuses, however, no online algorithm can guarantee a competitive ratio greater than $1/m$, where $m$ is the number of products. By imposing discrete concavity on the bonus functions, we limit the concentration of rewards in later assignments and restore constant guarantees. Under this structure, we develop a re-solving framework that repeatedly selects a planning horizon, constructs a feasible allocation plan from the current state, and executes its assignments in a specific order. For arbitrary arrival distributions, we design distribution-oblivious deterministic and randomized algorithms with competitive ratios of $0.2537$ and $0.3864$, respectively. For known increasing-failure-rate (IFR) arrival distributions, we exploit distributional information to obtain a deterministic $1/2$-competitive algorithm.
Online sorting is a problem that has gained attention recently, and is a natural translation of sorting into an online setting. The problem at hand is to place elements of an online sequence immediately and irrevocably with the goal of producing an array that is as close to sorted as possible upon inserting all elements. Recent work by Bercea et al. [ESA 2026] introduced the incorporation of learned predictions into the online sorting domain. They show lower bounds for the consistency and robustness of any algorithm, including randomization, then provide a deterministic algorithm that is $O(n/r)$-consistent and $O(r)$-robust, which are tight bounds. However, this algorithm requires prior knowledge of the optimal offline cost of the full input sequence before all elements have been revealed. We resolve this by providing an algorithm with matching bounds that does not require knowledge of the optimal cost, then extend these results to the more general problem of online metric TSP, in line with the prior work.
We present the first local FPT algorithm for finding important separators: Given an undirected graph $G$ with $n$ vertices and $m$ edges, vertices $s,t$ and parameters $k$ and $ν$, we can enumerate all $s$-$t$ important separators of size at most $k$ and volume at most $ν$ in time $\tilde{O}(ν2^{O(k)})$. Known approaches spend either $Ω(m)$ time (not local) or $ν^{Ω(k)}$ time (not FPT). Using this local algorithm, we show the following applications. (1) The first is an improved construction of unbreakable decomposition, a powerful generalization of tree decompositions. For any $k$, we can compute a $(5k,k)$-unbreakable decomposition with adhesion $4k$ in $\tilde{O}(k^{O(k)}m)$ time. This strictly subsumes the result of Anand, Lee, Li, Long, and Saranurak (SODA 2025) and is faster than the recent $O(k^{O(k^2)}m)$-time algorithm by Korhonen (STOC 2025) when $k = ω\Bigl(\sqrt{\frac{\log \log n}{\log \log \log n}}\Bigr)$. Our algorithm is also simpler than that of Korhonen (STOC 2025). (2) We give a near-linear time algorithm for single-sink important separators. For any sink $t$, we can list all $v$-$t$ important separators of size at most $k$ across all $v \in V(G)$ in time $\tilde{O}(2^{O(k)}m)$. Known approaches trivially compute $v$-$t$ important separators for each $v$, taking $Ω(4^k k mn)$ time. (3) Using the above result on single-sink important separators, we speed up undirected shadow removal and give the first almost-linear time FPT algorithm for Edge Multicut.
In this paper, we investigate the low-depth circuit complexity of Group Isomorphism in the multiplication (Cayley) table model. We prove the first circuit lower bounds for Group Isomorphism: namely, we show that every family of depth-$2$ Boolean circuits deciding Group Isomorphism requires quasipolynomial-size. (We also extend this to a limited class of depth-3 circuits, namely $\textsf{CNF}$-of-$\textsf{XOR}$s.) We complement this with upper bounds of uniform depth-$2\frac{1}{2}$ circuits of quasipolynomial-size: depth 3 circuits with poly-logarithmic bottom fan-in. A sequence of previous results from 1970-2025 progressively reduced the circuit depth from polynomial to $3\frac{1}{2}$; all of these results relied on the generator-enumerator strategy and, in fact, applied more generally to quasigroups. In contrast, our depth-$2\frac{1}{2}$ construction follows a fundamentally different strategy that exploits structure more specific to groups. We guess a composition series for each group, together with generators for its terms and the isomorphism types of its composition factors. We then inductively verify that the corresponding extensions at each level of the two composition series are compatible. A central part in this approach brings to bear the extensive work on the Short Presentation Conjecture, in tandem with the algorithmic theory of group extensions and cohomology.
Benjamin Bergougnoux, Vera Chekan, Giannos Stamoulis
For a graph $G$, the parameter treedepth measures the minimum depth among all forests $F$, called elimination forests, such that $G$ is a subgraph of the ancestor-descendant closure of $F$. We introduce a logic, called neighborhood operator logic with acyclicity, connectivity and clique constraints ($\mathsf{NEO}_2[\mathsf{FRec}]+\mathsf{ACK}$ for short), that captures all NP-hard problems like Independent Set or Hamiltonian Cycle that are known to be tractable in time $2^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$ and space $n^{\mathcal{O}(1)}$ on $n$-vertex graphs provided with elimination forests of depth $k$. We provide a model checking algorithm for $\mathsf{NEO}_2[\mathsf{FRec}]+\mathsf{ACK}$ with such complexity that unifies and extends these results. For $\mathsf{NEO}_2[\mathsf{FRec}]+\mathsf{k}$, the fragment of the above logic that does not use acyclicity and connectivity constraints, we get a strengthening of this result, where the space complexity is reduced to $\mathcal{O}(k\log(n))$. With a similar mechanism as the distance neighborhood logic introduced in [Bergougnoux, Dreier and Jaffke, SODA 2023], the logic $\mathsf{NEO}_2[\mathsf{FRec}]+\mathsf{ACK}$ is an extension of the fully-existential $\mathsf{MSO}_2$ with predicates for (1) querying generalizations of the neighborhoods of vertex sets, (2) verifying the connectivity and acyclicity of vertex and edge sets, and (3) verifying that a vertex set induces a clique. Our results provide $2^{\mathcal{O}(k)}n^{\mathcal{O}(1)}$ time and $n^{\mathcal{O}(1)}$ space algorithms for problems for which the existence of such algorithms was previously unknown. In particular, $\mathsf{NEO}_2[\mathsf{FRec}]$ captures CNF-SAT via the incidence graphs associated to CNF formulas, and it also captures several modulo counting problems like Odd Dominating Set.
We study episodic finite-horizon Bayesian bandits, or meta-bandits, with independent arms. Each episode draws fresh arm qualities from unknown recurring priors. Observations arrive after known delays, and plays obey hard per-episode caps and total budgets. A play earns reward and informs both the current quality and the recurring prior: the learner must explore, use, and generalize. We formulate a common planning optimization over learning trajectories. Its LP relaxation values experiments through current reward, resource consumption and continuation: the value of future decisions from the learning state produced. Dual prices separate the arms into local programs and certify every admissible policy, including hierarchical Thompson sampling. Execution enforces the hard constraints while preserving the promised learning transitions or accounting for their loss. The framework also represents risk-sensitive objectives and prices purchased observations and buyout options. Delayed reports create planned restlessness: an arm's information changes while it is not played, through reports of its own earlier plays arriving on a known calendar. For stationary priors, finite reward alphabets, bounded report lag and nonbinding budgets, posterior sampling over the recurring priors bounds the price of not knowing them by $O(\sqrt{N})$ over $N$ episodes, with other parameters fixed. For one undelayed $T$-step episode with arms capped at $T/X$ plays each, we obtain the LP-relative approximation factor $1+(2πX)^{-1/2}+O(X^{-1})$, improving previous factors, with a matching lower bound for the leading term.
We (the author and the AI systems that did the heavy lifting) show that the quantum query complexity of minimizing a convex function over a convex subset of $\mathbb{R}^n$ with evaluation and membership queries is $\tildeΩ(n)$, nearly matching the best known upper bound. In particular, we show this even for quadratic minimization, which is equivalent to inverting an $n \times n$ real matrix using matrix-vector queries. We also show linear or nearly linear lower bounds on the quantum query complexity of computing the trace, the sign of the determinant, and the magnitude of the determinant of a real matrix in the matrix-vector query model. We use a novel quantum lower bound technique, the determinantal witness method, based on identifying a witness whose Fourier transform vanishes on low-rank matrices and that correlates well with the function being computed.
The min-sum $k$-clustering problem asks to partition an input of $n$ points into $k$ clusters so as to minimize the total pairwise distance within clusters. It is a natural alternative to center-based objectives when the data is not well captured by a small set of representative points. We study for this problem two standard sketching methods, dimension reduction and coresets, which represent the input more compactly while preserving the min-sum cost of every $k$-partition up to $1\pm\varepsilon$ factor. Our first contribution addresses dimension reduction in Euclidean space. The Johnson-Lindenstrauss (JL) Lemma readily implies that a random linear map to target dimension $t=O(\varepsilon^{-2}\log n)$ preserves, with high probability, the min-sum cost of all $k$-partitions. We show that target dimension $t=O(\varepsilon^{-2}(ddim\cdot\log(1/\varepsilon)+\log k+\log\log n))$ suffices for Euclidean inputs of doubling dimension at most $ddim$, and moreover this bound is nearly optimal. We also show that target dimension $t=O(\varepsilon^{-2}(\log(1/\varepsilon)+\log k+\logΔ))$ suffices for Euclidean inputs of aspect ratio at most $Δ$. Our second contribution is to design the first $\varepsilon$-coreset for metric min-sum $k$-clustering. It is actually a movement-based coreset, has size $(1/\varepsilon)^{O(ddim)} k\log n$ for inputs of doubling dimension at most $ddim$, and can be constructed in randomized near-linear-time. We further use these coresets to design streaming algorithms for the problem.
We study unconstrained maximization of a fixed nonnegative submodular function when the ground set changes by insertions and deletions. A uniformly random subset gives a $1/4$-approximation with constantly many value-oracle queries per update. We obtain improvements over this bound for arbitrary oblivious update sequences, without knowing future deletions. For value estimation, we give a $(\sqrt2-1-\varepsilon)$-approximation using $O_{\varepsilon,δ}(\log^2(N+1))$ expected amortized oracle queries, with failure probability at most $δ$ at each fixed time. Here $N$ bounds the number of active elements. The algorithm maintains Poisson samples of the positive mass of random chain lower bounds. We prove that the factor $\sqrt2-1$ is tight for this certificate, even when polynomially many independent orders are used. For every fixed integer $K\ge2$, we maintain a feasible $(1/4+Ω(K^{-2}))$-approximate set using $\widetilde O_K(N^{1/K})$ amortized queries. Thus every fixed exponent $γ>0$ permits a fixed improvement over $1/4$ with $\widetilde O_γ(N^γ)$ update queries. In the square-root regime, a stronger algorithm attains $ρ_{\rm sq}-\varepsilon$, where $ρ_{\rm sq}>0.3177$. The maintained sets are feasible after every update, and their approximation guarantees hold in expectation. All bounds count oracle queries; the value estimator may query previously seen inactive elements, whereas the set algorithms use only currently active elements.
Perfect $L_2$ sampling is a powerful primitive for turnstile data streams, allowing algorithms to extract an index with probability proportional to its squared frequency, with applications to dynamic join size estimation, query optimization, heavy-hitters, sampling from self-joins, and real-time monitoring of evolving datasets. The setting $p=2$ is also the largest $p$ for which $L_p$ sampling can be achieved in polylogarithmic space. However, existing space-optimal perfect samplers rely on explicit polynomial duplication, requiring inefficient polynomial update times per stream arrival. Resolving a open question left by Swartworth, Woodruff, and Zhou [FOCS 2025], we present a perfect $L_2$ sampler that achieves an expected $\text{polylog}(n)$ update time while maintaining an optimal $\tilde{\mathcal{O}}(\log^3 n)$ bits of space.
Differential privacy (DP) has become the standard for privacy-preserving data analysis, yet its practical implementation in federated or distributed settings often faces a significant utility gap compared to the central model. While existing frameworks, including local, shuffle, and secure aggregation models, offer vetted primitives, they can incur high error for important tasks such as selection. Our first contribution is to show that this is inherent in the aggregation model: any private selection algorithm relying on the standard noisy aggregation primitive requires $Ω(\sqrt{d}/\varepsilon)$ samples. To address this limitation, we propose adding a new primitive that implements a version of the Sparse Vector Technique (SVT) and argue that it has simple implementations under various trust models. We demonstrate its power by developing an $\varepsilon$-differentially private algorithm for the selection problem. For selection in dimension $d$, our algorithm achieves an additive error of $O\left(\frac{\log(dn)}{\varepsilon}\right)$, matching that of the central model when $n$ is polynomially bounded in $d$. Furthermore, we show that this can be done using a near-optimal expected number $O(d\log^*d)$ of SVT queries. Together with existing lower bounds for the shuffle model, this result establishes an exponential separation between the SVT model and shuffle or aggregation-based models. We demonstrate the model's broad applicability to diverse tasks such as $F_p$ loss minimization, heavy-hitter identification, and sparse mean estimation.
We present a randomized algorithm for maintaining a $(Δ+1)$ vertex coloring of a fully dynamic graph against an adaptive adversary with $\tilde{O}(\sqrt{n})$ amortized update time. Our algorithm improves upon the recent randomized algorithms of Behnezhad, Rajaraman, and Wasim (SODA 2025) with $\tilde{O}(n^{8/9})$ and Flin and Halldórsson (ICALP 2025) with $\tilde{O}(n^{2/3})$ amortized update time. Our result (nearly) matches a fundamental limit of $O(\sqrt{n})$ update time identified by Flin and Halldórsson and settles their main open question. The previous algorithms' main improvement came from efficient coloring of the dense parts of the graph. Our approach complements these prior work by handling the sparse parts more efficiently, using ideas inspired by the recently developed dynamic entropy compression method. A key new aspect of our work is to tailor this method specifically to dynamic algorithms---as opposed to original local search algorithms---and achieve update times that can be sublinear in the number of involved constraints per update.
Arnold Filtser, Gramoz Goranci, Ali Momeni, Neel Patel, Maximilian Probst Gutenberg
We study the fully-dynamic all-pairs minimum cut problem on planar graphs. Given an $n$-vertex undirected planar graph $G=(V,E)$ undergoing edge insertions and deletions, the goal is to efficiently process these updates while supporting $s$-$t$ minimum cut queries for any pair of vertices $s$ and $t$. We give a $(1+\varepsilon)$-approximate dynamic algorithm that supports edge updates and minimum cut queries in $n^{o(1)}$ time, for any $1/\text{poly}(\log n) < \varepsilon < 1$. For general graphs, previous results either are limited to partially dynamic settings or achieve only logarithmic-factor approximations. The main conceptual and technical contribution is to show that a suitable adaptation of the planar emulator of Chang, Krauthgamer, and Tan~[STOC'22] satisfies the stronger pattern emulator introduced by Chen and Tan~[ICALP'25]. This leads almost immediately to a nearly linear-time construction of pattern emulators, which, when combined with the recent fully dynamic $(1+\varepsilon)$-approximate all-pairs shortest paths algorithm of Filtser, Goranci, Patel, and Probst Gutenberg~[FOCS'24], yields our result.
We study the canonical \textsf{Maximum Clique} and \textsf{Maximum Independent Set} problems in the one-pass edge-arrival graph streaming setting. Here, the edges of some input graph $G = (V,E)$ are presented one at a time (possibly including deletions), before an algorithm needs to produce either a large clique or independent set at the end of the stream, with the focus being on space complexity. We are interested in finding $β$-approximate solutions, for any $β\geq 1$. Previous work gave an algorithm using $\tilde{O}\left(n^2/β^2\right)$ bits of space, together with a corresponding $\tildeΩ\left(n^2/β^2\right)$ two-party communication lower bound [Halldórsson et al., ICALP'12], seeming to resolve the problem. However, their algorithm crucially relies on randomness, and the best known deterministic algorithm remains a folklore derandomisation using $O\left(n^2/β\right)$ bits of space, leaving a (deterministic) gap of size $\tilde{O}(β)$. We make progress towards this gap with a strong lower bound for a restricted range of $β$: whenever $β\leq \sqrt{n}/36\log n$, any deterministic algorithm for either problem must use $Ω\left(\frac{n^2}{β\cdot\log n}\right)$ bits of space. Our proof is via a two-party one-way communication lower bound, and highlights the power of randomness when approaching either of these problems.
An increasingly important requirement for clustering algorithms is that decisions and outcomes need to be clearly understandable for human users. We address this requirement for the Correlation Clustering (CC) problem, in case the objects to be clustered are accompanied by labels. In CC one seeks to partition a set of objects into clusters based on pairwise similarity and dissimilarity information. Our proposed model only considers clusters that can be explained by a label tree, which is essentially a decision tree that partitions the objects according to labels. The goal is to find an explainable partition of the data that minimizes the CC objective function. We formalize this problem from a theoretical viewpoint, and prove several hardness of approximation results, also when constraining the size of the tree. Our proposed greedy method finds locally optimal splits, and works recursively in a top-down approach. To efficiently compute the best split, we carefully maintain certain counters. An experimental evaluation demonstrates its effectiveness. The proposed algorithm processes graphs with over million edges in a few minutes, can find the ground truth in synthetic data, and is able to produce interpretable clusters in real world networks.
Order maintenance asks for a data structure that keeps a sequence of elements under insertions and deletions and tells which of two elements comes first. We give an algorithm, assembled from published components, that performs each operation in worst-case constant time, and verify it in Lean 4. The algorithm is a program for a random-access machine deeply embedded in Lean, whose interpreter counts every instruction. We prove that the program answers every query as a simple reference specification does, that each operation takes at most a fixed number of steps, independent of the number of elements and of the history, and that after $t$ operations no value it has computed has more than $O(\log t)$ bits, so it runs unchanged on a machine with words of that size. A verified translation that preserves step counts exactly connects the high-level proofs to the machine program, so the interpreter is the only cost model to trust. An unverified native implementation shows the algorithm's practical cost.
Population-scale analysis of phased genetic variants requires both allele-carrier retrieval and ordered haplotype context. We examine H1, an adaptive allele-haplotype incidence-row index, and H2, a canonical edge-path representation derived from a fixed, nonoverlapping variant catalog. We prove conditional reconstruction of abstract H2 walks and distinguish this property from unrestricted equivalence to genomic variation graphs. A completed local notebook experiment used a chromosome 1 interval (20-22 Mb), 200 diploid individuals, 24,921 SNV/indel records, and 45 SVTYPE-annotated records. For 24,966 extracted nonreference rows across 400 haplotypes, the serialized H1 incidence-row files occupied 0.872 MiB with a simple adaptive layout and 1.126 MiB with a custom Roaring-like layout, versus 1.190 MiB for a bitmap-payload baseline; a whole-file Zstandard archive of the adaptive layout occupied 0.207 MiB. For a separately constructed nonoverlapping SNV projection containing 22,727 sites, indexed H2 edge sequences occupied 75,130,882 bytes in a raw format and 5,156,418 bytes with context-Huffman encoding (14.57-fold reduction). Carrier-equivalence and graph-adjacency assertions passed in the notebook, but full crosscallset phasing, nucleotide-level SV reconstruction, and comparison with production indexes remain unresolved. The measurements establish a restricted implementation feasibility result, not superior end-to-end pangenome compression.
We investigate a simple method to compute an unbiased estimate of the number of nodes of a directed acyclic graph (DAG) by following a random directed path. This method is due to Leonard Pitt from 1987, and it generalizes a method of D. E. Knuth from the 1960s for estimating the size of a rooted tree. A slight variation makes it possible to estimate the number of source-sink paths. We extend the algorithm to use many "agents" that walk cooperatively through the graph. This extension unifies and generalizes several methods that have been proposed for approximately counting the nodes of a tree.
Seungbum Jo, Geunho Kim, Kunihiko Sadakane, Srinivasa Rao Satti
Given a 1D array over a total order and a query position in the array, a next larger value query returns the nearest position to the right whose value is larger than the value at the query position. Next smaller value queries and previous larger/smaller value queries are defined analogously. These queries are closely related to the $k$-th range minimum/maximum problem, which asks for the position of the $k$-th leftmost minimum/maximum value when the query range contains multiple minimum/maximum values. In this paper, we consider the encoding problem for answering any arbitrary subfamily of these four queries, where the queries must be answered without accessing the input array. We give a unified approach for obtaining space lower bounds for such encodings. We also obtain data structures for the query subfamilies that match the lower bounds up to lower-order additive terms, while supporting the queries in constant time. Our results generalize and improve several previous results on this problem [Gawrychowski and Nicholson, ICALP-2015; Jo and Satti, TCS-2016; Tsur, IPL-2019; Jo and Kim, Algorithmica-2025]. In addition, these results give tight bounds on the encoding complexity of all subfamilies of these queries for (1) permutations, (2) arrays without consecutive equal elements, and (3) general arrays.
The low-degree method and its associated lower bounds are widely used to guide algorithm design and to provide evidence of computational hardness in average-case inference, high-dimensional statistics, random optimization, and related problems. This led to the low-degree conjecture, which predicts that when the low-degree advantage between a planted distribution and a uniform null distribution remains bounded, no efficient distinguisher can succeed after independent noise, provided that the planted distribution has permutation symmetry. Several works have produced counterexamples to variants of this conjecture or to versions for algorithms with higher time complexity, but the conjecture remained open in its standard binary, polynomial-time formulation. We disprove the polynomial-time low-degree conjecture by giving a family of examples. For every fixed integer $r\geq3$, we construct a permutation-invariant distribution $\mathbb{P}_n$ on simple graphs, with $\mathbb{Q}_n=G(n,1/2)$, such that every marginal of $\mathbb{P}_n$ on at most $D_n=Θ((\log n)^{r-1})$ edges is uniform. Therefore, the low-degree advantage is zero through degree $D_n$. Nevertheless, after every edge is independently resampled at a fixed positive rate, a deterministic rank test strongly distinguishes the resulting distribution from $\mathbb{Q}_n$ in linear time in the graph input size. The construction chooses a subspace of a Reed--Muller code whose nonzero polynomials have small absolute bias, selects points whose evaluation vectors have no short linear dependencies, and evaluates a random alternating bilinear form on pairs of these vectors. Our result shows that low-degree indistinguishability, a uniform null distribution, permutation invariance, and independent resampling do not by themselves imply polynomial-time hardness, and suggests that a valid general conjecture must impose an additional condition.
Binary Iterative Hard Thresholding (BIHT) is a simple, yet effective, greedy method for recovering a sparse vector from one-bit sign measurements. In its original form, BIHT performs a ``gradient-descent'' step, followed by hard thresholding. A convergence analysis of this algorithm was left open in the introductory work of Jacques, Laska, Boufounos, and Baraniuk [IEEE Trans. Inform. Theory (2013)] and has remained unresolved for over a decade, with subsequent sharp analyses studying a normalized variant instead, that additionally projects every iterate onto the unit sphere. This paper resolves that gap and characterizes when per-iterate normalization is algorithmically necessary. In the noiseless setting, we prove a universal, sample-optimal convergence theorem for the original BIHT algorithm. Specifically, with $\widetilde O(s/ε)$ measurements, a deterministic, computable finite-time iterate has directional error at most $ε$, simultaneously for every $s$-sparse unit vector. This matches the optimal sample dependence achieved by normalized BIHT in prior work. Under sign corruptions, we prove a sharp separation. If at most a $τ$ fraction of signs are flipped adversarially, then BIHT, without per-iterate normalization, still reaches the robust error floor at an early iterate with a matching $\widetilde O(s/ε)$ sample complexity rate as its normalized variant. This recovery, however, is not stable. We prove a scalar lower bound showing that any nontrivial corruption pattern, even one that involves only one flipped sign together with one clean sign, forces the iterates to oscillate indefinitely. Consequently, no general last-iterate convergence theorem can hold for BIHT under sign corruptions, while its normalized surrogate provably escapes this instance.
Xiaoyu Leng, Hongchao Qin, Rong-Hua Li, Guoren Wang
Hypergraphs capture higher-order interactions involving multiple entities. Existing decomposition models, such as $k$-core and nbr-$k$-core, rely on degree or neighborhood constraints and may overlook fine-grained density variations. We propose a $(k,δ)$-dense subhypergraph model based on capped hyperedge contributions. For each parameter pair, the unique dense set is the inclusionwise smallest maximizer of a penalized capped hyperedge contribution. These dense sets form a nested hierarchy characterized by egalitarian $δ$-orientations and hypergraph reachability. We develop three algorithms for dense subhypergraph search and two for complete density decomposition. Both decomposition algorithms use integral flow balancing: Flow-Decomp extracts density levels through strongly connected component condensation, while Reverse-Decomp uses direct reverse propagation. Once an egalitarian orientation is obtained, both recover all density levels in linear time. We further derive sandwich bounds that restrict computation to equivalent residual subproblems and develop incremental algorithms for maintaining the hierarchy under hyperedge insertions and deletions. Experiments on nine real-world hypergraphs, including datasets with over 15 million vertices, compare our methods with four established decomposition models. The results show that Reverse-Decomp improves decomposition efficiency, incremental maintenance outperforms recomputation in the tested settings, and the proposed model captures density structures overlooked by degree-based decompositions.
We consider the tensor principal components analysis (tensor PCA) problem on the sphere, in which we are asked to recover an unknown vector in $\mathcal{S}_N=\{x\in\mathbb{R}^N:\lVert x\rVert^2=N\}$ from a noisy observation of its $k$-th tensor power, for a fixed $k\ge3$. Tensor PCA exhibits a well-studied information-computation gap: global algorithms such as spectral methods and sum-of-squares work above an algorithmic threshold, below which we know matching lower bounds (against these and some other restricted models of computation). In contrast, local algorithms such as Markov chains were only known to succeed from a random initialization at a polynomially larger signal-to-noise ratio, and were believed to be inferior to global algorithms for tensor PCA. Continuing the program initiated in Part I [arXiv:2610.16705], we show that spherical Langevin dynamics, when supplemented with annealing and ergodic averaging, achieves polynomial-time strong recovery above the algorithmic threshold from a uniformly random initialization. Together with Part I, our results demonstrate that local Markov chains can succeed all the way down to the algorithmic threshold from an uninformative cold start.