Papers

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

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

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

Consider the canonical universal hash family $h(x)= ((ax+b)\text{ mod } p)\text{ mod } m$, where $a,b$ are chosen uniformly from $\mathbb Z_p$, which we call linear hashing, being used to hash $n$ elements into $m=\Theta(n)$ buckets. For any universal family, the expected size of the largest bucket is at least $\Omega(\log n / \log\log n)$ and at most $O(\sqrt{n})$. The only improvement upon these trivial bounds for linear hashing is a 2019 upper bound of $\tilde{O}(n^{1/3})$ by Knudsen. We show that for any $p$ sufficiently larger than $n$, there is a set of $n$ keys whose expected maximum load is $n^{\Omega(1/\log\log n)}$, proving linear hashing does not have a polylogarithmic maximum load. We extend the same bounds to the classical multiply-shift hash family of Dietzfelbinger, Hagerup, Katajainen, and Penttonen. Our main contribution is an equivalence between the maximum load problem to a density variant of arithmetic Kakeya sets. We then complete the lower bound using a construction of Green and Ruzsa of a small set containing long arithmetic progressions with every difference in a prescribed range. Surprisingly, our equivalence also implies that any substantial improvement over Knudsen's upper bound would imply new results about standard arithmetic Kakeya sets. More precisely, an $O(n^{1/3-\varepsilon})$ upper bound would improve known bounds for unions of complete integer arithmetic progressions, while an $n^{o(1)}$ upper bound would imply Bourgain's arithmetic-progression criterion.

The Bitner–Ehrlich–Reingold algorithm generates the binary reflected Gray code with constant work per codeword, using a focus-pointer array. Its compact update is easy to state but gives little indication of why such pointers should exist. This note reconstructs a geometric route to the algorithm. The sequence of flipped bit positions is the ruler sequence, OEIS A007814. We realize its finite prefixes as in-order traversals of recursively expanding trees, decorate each activation with its nearest ancestor to the right, contract all stack operations between successive outputs into successor jumps, and prepare the bounded level-indexed stack by predicting its future slot values. This gives and proves a branch-based loopless generator. The construction grew from an earlier informal public exposition by the author. Viewing horizontal position in the tree as time then explains each stack-slot write as a prediction for the next activation at the same level. Finally, the two possible future roles—an inherited continuation for a right child and a default continuation for a left child—are scheduled together, leading to the two assignments of the published focus-pointer algorithm. The account is a reconstructed derivation, not a claim about the historical reasoning of Bitner, Ehrlich, or Reingold.

In this brief note, we close the asymptotic gap between known upper and lower bounds for fractional online matching under edge arrivals. We prove that the optimal competitive ratio for this problem is $1/2+\Theta(1/n)$. The algorithm was suggested and analyzed by OpenAI's ChatGPT Sol based on a single prompt. The presentation was streamlined over a few hours, based on a back and forth discussion with the author, who assumes responsibility for any errors.

The classical duplication episode clustering (EC) model introduced by Guigó et al. in the 1990s provides a foundational approach for inferring genomic duplication events crucial to understanding genome evolution. This model clusters single gene duplications from a collection of gene trees at locations in the species tree to minimize the total number of such locations, called duplication episodes. Here, we introduce NetEC, a novel extension of this problem to phylogenetic networks. To solve NetEC, we first develop a polynomial-time dynamic programming (DP) algorithm for testing whether a given set of network nodes can serve as episode locations. We then propose a main inference algorithm that utilizes this DP component to optimize the episode count; while the feasibility test runs in polynomial time, the full optimization has exponential worst-case complexity, and an optional heuristic mode is provided for larger instances. We also propose an extended episode analysis procedure that identifies additional genomic duplication candidates below reticulation nodes, complementing the main algorithm by resolving potential upward clustering of duplications induced by reticulation. We evaluate our method on simulated data and on an empirical Pandanales dataset comprising over 29,000 gene trees, demonstrating exact and accurate inference of genomic duplication events even in the presence of multiple reticulations.

Current approaches to quantum search fail to deeply exploit extant structure in the underlying Hilbert space. Decomposing the search by this structure empowers new strategies and formulations for quantum search and algorithm design. We present a new decomposition technique acting directly on this structure by recursively decomposing the Hilbert space and constructing the search operator from reflections over the resulting partition. When initial and target states factorise over this partition, dynamics reduce to a single rotation in a two-dimensional plane at each level, with angle given by a scalar recurrence. This recurrence avoids error accumulation from separately bounding success probabilities at each level, yielding an exact state description enabling treatment of the recursion as a whole. We obtain the target state deterministically and derive oracle and non-oracle costs independently of the search setting. For unstructured search, our approach attains the simultaneously optimal $\Theta(\sqrt{N})$ oracle and non-oracle gate counts. For spatial search on $d$-dimension grids, it recovers the $O(\sqrt{N})$ time for $d\geq3$ and the $O\bigl(\sqrt{N}(\log N)^{3/2}\bigr)$ bound of Aaronson and Ambainis for $d=2$. The exact description of the recursion extends over our decomposition to new subdivision structures and provides a new approach for applying and analysing recursion in quantum algorithm design.

Quantum states with conserved total spins, or spin eigenfunctions, are important for studying quantum chemistry and quantum manybody physics problems. A typical class of spin eigenfunctions are Dicke states, which attain maximal spins. While we already have many efficient quantum algorithms to prepare Dicke states, it is not yet clear if we could do so for arbitrary spin eigenfunctions deterministically. Generalizing Bärtschi and Eidenbenz's elegant algorithms for Dicke state preparation, we successfully prepare arbitrary spin eigenfunctions characterized by branching paths and binary spin trees. As a byproduct, we also develop the corresponding classical algorithms to reconstruct all these spin states.

A graph $G$ is apex if $G$ has a vertex $v$ such that $G-v$ is planar. We prove that every $2$-connected apex cubic graph is three-edge-colorable. This result gives the final piece of the proof for the well-known Tutte's three-edge-coloring conjecture from 1966 \cite{tutte}. The proof, as well as the result, generalizes that of the Four Color Theorem, which requires computer checks. As in the previous proof of the Four Color Theorem, the proof is constructive. More precisely, given a $2$-connected apex cubic graph $G$ on $n$ vertices, our reducibility and discharging procedure yields a three-edge-coloring of $G$ in $O(n^2)$ time. As an additional reproducibility check for our computer checks, independent implementations reconstructed from the detailed pseudocode (given in the appendix) using generative AI systems reproduced the required computational results. These reconstructions are not part of the mathematical justification of the theorem, but provide additional evidence for the reproducibility of the computations.

We study spectral density estimation for the normalized adjacency matrix of an unweighted graph under local access model. Previously, Cohen-Steiner et al. [KDD 2018] proposed an algorithm for $\varepsilon$-approximate spectral density estimation in the Wasserstein-1 distance, using $2^{O(1/\varepsilon)}$ local queries to the graph. In this paper, we prove that every constant-success estimator with Wasserstein–$1$ error at most $\eps$ requires $2^{\Omega(1/\eps)}$ queries, showing that the Cohen-Steiner algorithm is optimal up to constant in the exponent. This resolves the open problem left by previous researches Jin et al. [COLT 2023] and Peng et al. [COLT 2026]. We then turn to quantum local access model. We give an $\widetilde O(\eps^{-3})$-query algorithm estimating the spectral density with Wasserstein-1 error at most $\eps$. Finally, we prove a $\widetilde\Omega(\eps^{-4/3})$ quantum lower bound when the graph is sufficiently large. As a result, quantum local access model changes the dependence on $\eps$ from exponential to polynomial.

The uniform car-parking process is the one-dimensional random sequential adsorption of unit cars on a segment of finite length $s$: cars arrive at uniformly random positions and park wherever they fit, until no gap admits another. This paper develops the exact finite-$s$ theory. The joint density of the parked positions is resolved into jamming cells, on each of which it is a rational function, and evaluated by a subset recursion in $O(2^n n)$ operations; the marginal and gap order statistics are obtained as hyperlogarithms whose weight is fixed by the number of coordinates integrated out; and the absorption count and the aggregate quantities are treated through the integral equation descending from Rényi.

We study one-sided online bipartite matching with recourse. In this setting, one side of a bipartite graph is known in advance, while vertices on the other side arrive online together with their incident edges. After each arrival, the algorithm must maintain a maximum-cardinality matching while minimizing the total number of reallocations, also known as the recourse budget. Despite extensive work, the exact recourse complexity of the problem remains unsettled: the best lower bound is $\Omega(n \log n)$, whereas the best upper bound is $\mathcal{O}(n \log^2 n)$, where $n$ denotes the number of online vertices. Tight upper bounds of $\mathcal{O}(n \log n)$ are known only for restricted graph classes, such as forests. The best known upper bounds are attained by a very simple and natural algorithm SAP, which after each arrival applies a shortest augmenting path, and it is conjectured to be optimal. All known upper bound analyses of this algorithm do not depend on the particular maximum matching maintained by the algorithm. Consequently, they also apply to a more difficult problem, which we call the malicious matching setting: after each arrival, the maintained matching is replaced by a worst-case maximum matching for the next step. This led to the conjecture that the malicious setting still admits an $\mathcal{O}(n \log n)$ recourse bound, in line with the conjectured optimal complexity of the original model. Our main result is an $\Omega(n \log^2 n)$ lower bound for the malicious matching setting, thus disproving the conjecture. Together with the previous upper bound, this settles the asymptotic recourse complexity of the malicious variant of the problem. We complement our lower bound with an upper bound of $\mathcal{O}(n \log n)$ for expander graphs.

Metric decompositions are a fundamental tool in the design of algorithms involving distances. We study fast algorithms for sampling from probabilistic metric decompositions of $n$-point sets in $\ell_\infty$ and $\ell_2$ spaces of high dimension $d$. For $\ell_\infty$, we design a padded-decomposition algorithm that runs in time $\tilde{O}(nd^2)$, which is near-linear in $n$, and achieves padding parameter $\tilde{O}(\log n)$. Our algorithm constructs a new sparse neighborhood cover that is based on geometric properties of $\ell_\infty$ [Indyk, JCSS'01], and utilizes recent reductions between covers and decompositions [Conroy and Filtser, STOC'25]. For $\ell_2$, we design a separating-decomposition algorithm that achieves near optimal separation $\tilde{O}(\sqrt{\log n})$ in almost-linear time $n^{1+o(1)}$. Our bounds improve over known algorithms with similar running time by a factor $\Omega(\sqrt{\log n})$, and the techniques have additional applications to spanners and nearest-neighbor search.

We prove a strict space separation between static and ordinary dynamic approximate membership at every fixed error rate. For each fixed $\varepsilon\in(0,1)$, a capacity-$n$ ordinary dynamic filter over a universe of size $u$, with zero false negatives, pointwise false-positive probability at most $\varepsilon$, arbitrary history dependence, a free public random tape, and at most $H$ bits of persistent state, satisfies \[ H\ge \bigl(\log_2(1/\varepsilon)+a_\varepsilon^{\rm c}\bigr)n-o(n), \] under only $u/n\to\infty$. The constant $a_\varepsilon^{\rm c}$ is an explicit variational threshold obtained by preserving the dependence between the parent accepted mass and the successor reservoir. The structural step is a common-continuation transport lemma. A joint posterior KL bound gives a branch-specific survivor support; the same legal delete–insert word transports that support to one successor state, forcing an accepted reservoir. We then keep the parent outside mass $1-X$ in the conditional-entropy argument instead of replacing it by $1-\varepsilon$. This yields a two-variable analytic envelope, with no selected thresholds, dyadic witnesses, or numerical assumptions.

The 3SUM problem asks, given sets $A,B,C$ of integers, whether there exist $a\in A$ and $b\in B$ whose sum belongs to $C$. In the preprocessed variant with unknown $C$, one preprocesses sets $A$ and $B$, each of size $n$, and subsequently answers a query specified by subsets $A'\subseteq A$, $B'\subseteq B$ and a target set $C'$ of size $O(n)$, by solving the 3SUM instance $(A',B',C')$. Kirkpatrick, Kuszmaul, Mathialagan, and Vassilevska Williams [ICALP 2026] gave the first algorithm with subquadratic space for this problem, achieving $\tilde{O}(n^{3/2+\epsilon})$ query time using $\tilde{O}(n^{2-2\epsilon/3})$ space, for every $\epsilon\in[0,1/2]$. Their algorithm employs separate mechanisms for heavy and light targets, and for each heavy target it stores explicitly the list of pairs $(a,b)$ summing to it; these lists dominate the space bound. We present a unified construction that uses a single mechanism for all queries. Instead of storing these lists of pairs, we recover them on demand by leveraging the Fiat–Naor data structure [SICOMP 1999] to invert the function $(a,b)\mapsto (a+b\bmod p)$. This simplification improves the space bound to $\tilde{O}(n^{\max(2-\epsilon, 11/6-\epsilon/3)})$, while maintaining the same query time. Moreover, our construction is the first to achieve subquadratic space while supporting adaptively chosen queries.

Let $k_{1,\varepsilon}(n)$ be the smallest number of real linear measurements needed by a randomized oblivious sketch that estimates the nuclear norm of every fixed real $n\times n$ matrix within a factor $1\pm\varepsilon$, with probability at least $2/3$. For every fixed $0<\varepsilon<1$, we prove \[ \frac{n^2}{(\log n)^{A_\varepsilon}} \le k_{1,\varepsilon}(n) \le C_\varepsilon \frac{n^2\{\log\log(e^e n)\}^2}{\log(e n)}. \] Previously, the best unrestricted bounds for general linear sketches of the Schatten–1 norm were $\Omega(n)$ and the trivial $O(n^2)$ upper bound (Li, Nguyen, Woodruff'19), leaving a polynomial gap. Our bounds close that gap up to polylogarithmic factors and give a nontrivial logarithmic saving below the $n^2$-measurement storage bound. The result extends much further. Write $k_{p,\varepsilon}(n)$ for the analogous sketch dimension for the Schatten–$p$ norm. For every fixed finite $p>0$ that is not a positive even integer, there are positive constants $A_{p,\varepsilon},C_{p,\varepsilon},c_p$ such that \[ \frac{n^2}{(\log n)^{A_{p,\varepsilon}}} \le k_{p,\varepsilon}(n) \le C_{p,\varepsilon}\frac{n^2}{(\log n)^{c_p}}, \] so $k_{p,\varepsilon}(n)=n^{2-o(1)}$ throughout the non-even regime. Together with the known tight bounds $\Theta_{p,\varepsilon}(n^{2-4/p})$ for positive even $p$ and $\Theta_\varepsilon(n^2)$ for $p=\infty$ (Li, Woodruff'16), our results close the remaining polynomial gap across the Schatten family and complete, up to polylogarithmic factors, the polynomial-order classification of general linear sketches for all Schatten-$p$ norms.

In many practical deployments of differential privacy, queries do not arrive all at once. We study online differentially private query answering under a finite zero-concentrated differential privacy (zCDP) contract. In this setting, queries arrive sequentially, carry different accuracy thresholds, and may overlap with information already released. We formulate this setting as residual privacy budgeting: for each arriving query, the mechanism first credits reusable support from previous DP outputs and then spends new budget only on the remaining support required to satisfy the current threshold. The controller separates feasible cases, where the minimal residual support is allocated exactly, from scarcity cases, where a weighted shortfall-conservation optimiser assigns limited support according to query difficulty. We define the weight using the Query Influence Factor (QIF), a diagnostic signal for query difficulty and instability rather than query importance. For scalar Gaussian exact reuse, inverse-variance fusion justifies additive support. We prove zCDP composition, residual minimality, 1-competitiveness against the offline optimum in the feasible regime, and avoidable expenditure for allocators that ignore released history. A scarcity impossibility result shows that no online allocator can guarantee a competitive ratio better than 1/n in threshold satisfaction, contextualising the QIF scarcity layer as a design choice for an inherently hard online problem.

We prove that for every fixed inverse temperature $\beta < 1 / 2$, with high probability over the disorder, the single-site Glauber dynamics for the $n$-spin Sherrington-Kirkpatrick model mixes from every initial configuration to within total variation distance $\varepsilon$ in $O_{\beta}\left(n \log\left(n / \varepsilon\right)\right)$ steps. The bound holds uniformly over all external fields and is optimal up to constants depending only on $\beta$. The main ingredient is a deterministic criterion for optimal-order Poincaré inequalities in general Ising models, established via the integrated Bakry-Émery criterion together with a new two-spin estimate. A standard application of the localization-scheme framework of Chen and Eldan then upgrades the Poincaré inequality to a modified log-Sobolev inequality, yielding the optimal mixing-time bound. The main ideas underlying the proof of the Poincaré inequality were generated by GPT-5.6 Sol Ultra.

Kuszmaul, Liang, and Zhou (SODA 2026) ask whether succinct constant-time dynamic fusion nodes exist when the number of stored keys is polylogarithmic in the universe size. We give a negative answer for always-correct structures. For n^8 <= U, log_2 U >= 2^70, and redundancy 0 <= R < n, a dynamic dictionary requires at least 2^-26 log_2(1+n/(R+1)) expected-amortized cell probes per operation. The model permits fixed layouts of packed cells of at most one word each, including the short-spill convention used by succinct word-RAM structures, and covers zero-error Las Vegas algorithms with fresh per-invocation randomness and almost-sure termination. The proof repairs a conditioning defect in the inherited communication argument by placing pointwise probe caps inside the consistency event, then extends the lower bound to large universes through a scale-adaptive entropy parameter. Consequently, when U=2^w and n=ceil(w^c) for any fixed c>0, no always-correct predecessor structure can use log_2 binom(U,n)+o(n) persistent mutable bits and support constant-time operations. Constant expected-amortized time requires Omega(n) redundant bits. Lean 4 checks the complete packed-memory and fresh-random indexed models, hard distribution, communication bounds, separator, nested-forest accounting, deterministic and Las Vegas lower bounds, strict-predecessor reduction, and redundancy corollaries.

A longstanding limitation of known techniques for parallel derandomization was that they incurred at least polylogarithmic overhead in work. For instance, for fundamental and frequently used problems such as maximal independent set, maximal matching, and $(\Delta+1)$-coloring, where $\Delta$ denotes the maximum degree of the graph, the best-known deterministic parallel algorithms with polylogarithmic depth used $\Omega((m+n)\mathrm{poly}(\log n))$ work on $n$-vertex, $m$-edge graphs; see, e.g., Luby [FOCS '88]. Consequently, at least $\mathrm{poly}(\log n)$ processors were needed for these algorithms to outperform straightforward single-processor algorithms. Recently, Ghaffari and Grunau [FOCS '25] introduced a new parallel derandomization method that substantially reduced the overhead from $\mathrm{poly}(\log n)$ to $\mathrm{poly}(\log\log n)$, achieving work bounds of $O((m+n)\mathrm{poly}(\log\log n))$. In this paper, we settle this line of research by obtaining linear work bounds of $O(m+n)$, thereby achieving truly work-efficient parallel derandomization.

We present an algorithm with a natural geometric interpretation for computing the determinant of a matrix $B\in\mathbb{Z}^{d\times d}$. It improves upon the current fastest deterministic algorithms by a factor of $d^{\omega(1)+1-\omega(2)}\approx d^{0.1213}$, where $\omega(k)$ denotes the exponent required for multiplying a $d\times d$ matrix with a $d\times d^k$ matrix. Our approach builds on a recent result of Klein and Reuter (STOC 2025), who introduced a novel algorithmic idea for lattice basis computation that can be viewed as extending the Euclidean algorithm from $\mathbb{Z}$ to $\mathbb{Z}^d$. By adapting their techniques, we compute the determinant with the same bit complexity as applying the generalized Euclidean algorithm to an input matrix $A\in\mathbb{Z}^{d\times 2d}$ with $\|A\| = \|B\|$, namely $\tilde{O}(d^{\omega(2)}\log\|B\|)$. Prior to this work, the fastest deterministic algorithm for computing the determinant required $\tilde{O}(d^{\omega(1)+1}\log\|B\|)$ bit operations.

Fine and Wilf [Proc. Amer. Math. Soc. 1965] showed that any string of length at least $p+q-d$ with periods $p$ and $q$ also has period $d=\gcd(p,q)$. For parameterized strings, Apostolico and Giancarlo [Discrete Appl. Math. 2008] proved an analogue with length bound $p+q$, assuming that the two induced bijections commute. Ideguchi et al. [SPIRE 2023] removed this assumption and gave the bound $p+q+\min(p,q)(\sigma-1)$, where $\sigma$ is the number of distinct letters. This was later improved by Hamai et al. [SPIRE 2024] to $p+q+\min(p,q)(\sigma-2)$, which was used to bound the number of non-equivalent parameterized squares. In this paper, we establish the optimal Fine–Wilf type bound for parameterized strings. Namely, if a string $s$ containing $\sigma$ distinct letters has parameterized periods $p$ and $q$ and satisfies $|s| \ge p+q+(\sigma-3)d+1$, where $d=\gcd(p,q)$, then $d$ is also a parameterized period of $s$. We also give matching lower-bound instances, proving that our bound is optimal for any $\sigma \geq 2$.

We study the rounding of a supplied half-integral feasible solution of the root-assignment bidirected cut relaxation for Steiner Forest (Forest-BCR). Byrka, Grandoni, and Traub [IPCO 2025] proved a $16/9$ guarantee for a recursive framework that normalizes the LP point, selects a vertex set of maximum projected LP density, buys a minimum spanning tree on that set, contracts it, and recurses. We prove that the same framework has guarantee $8/5$. The new analysis keeps the orientation and the root label of each projected half-unit of LP mass. In a simple projection, the cut constraints at a terminal of degree two determine the root-assignment vector of every demand incident with it, and half-integrality leaves only two possibilities: a unit assignment to one root, which forces excess outdegree inside that root's support, or a split assignment to two roots, which forces overlap between their supports. For every connected component $C$ of the split-root graph this yields $ \beta_C\ \ge\ \frac{L_C}{2}$, where $\beta_C$ is the circuit rank of the union of the root supports in $C$ and $L_C$ is the number of its vertices of degree two in the full projection. Balancing the density certificate obtained from this inequality against the ordinary degree sum gives a vertex set of density at least $5/8$, and the inherited contraction lemma turns that into the $8/5$ rounding. For every $q\ge3$ we also construct a normalized half-integral point whose maximum projected density is exactly $5q/[2(4q-1)]$, so the universal projected-density bound is asymptotically tight.

In the Shortest Common Superstring problem (SCS), one is given a finite set of strings and is asked to find a shortest string containing every input string as a substring. Its best known approximation ratio is $2.466$, whereas the currently strongest upper bound on the approximation guarantee of the maximum-overlap greedy algorithm is $3.396$ (Englert, Matsakis, and Veselý, 2023), though it is conjectured to be $2$. We improve both approximation guarantees: SCS admits a $\frac{7}{3}$ approximation and the approximation guarantee of the greedy algorithm is at most $3$. The main technical ingredient of our proof is a certain inequality for optimum cycle covers of an overlap graph associated with the input strings. Every previous improvement of greedy's worst-case guarantee and the two recent record guarantees for general SCS are driven by it. We improve this inequality by pushing it to its limit: for a particular coefficient of this inequality, we show a new upper bound and prove that it cannot be improved further.

We give a catalytic logspace algorithm for the problem of graph spectral sparsification. Given an undirected graph $G$ on $n$ vertices and $\varepsilon>0$, our algorithm outputs an $\varepsilon$-spectral sparsifier of $G$ with $O(n\varepsilon^{-2}\log n)$ edges, matching the effective resistance sampling of Spielman and Srivastava (STOC 2008). This gives a new, natural problem in catalytic logspace that is not known to be in deterministic $\mathbf{NC}$ or $\mathbf{SC}$. Our main contribution is an entirely new technique in the compress–or–random paradigm for catalytic logspace that we believe will have further applications. We first analyze effective-resistance sparsification using a pessimistic estimator that can itself be computed in catalytic logspace. The estimator is motivated by the viewpoint of graph quasirandomness and immediately gives a simple, deterministic greedy algorithm for graph sparsification. Subsequently, we show that such a pessimistic estimator can be transformed into an algorithm that performs an in-place compression of a string with bad potential. Our algorithm is based on using the potential function to define a measure over strings, and implementing arithmetic coding using this measure in-place. This compression technique is substantially distinct from all prior tools in the field of catalytic computation.

Consider the following problem of learning an unknown linear order on $n$ items. In each round, the learner guesses a complete ordering of the items and receives either confirmation that the guess is correct or a counterexample: a pair of items in the wrong order. The goal is to identify the unknown order using as few queries as possible. We study this problem when up to $k$ of the returned counterexamples may be untruthful, where $k$ is not known in advance. We determine the optimal query complexity up to constant factors: \[ \Theta(n\log n + nk). \] Thus, while the noiseless complexity matches the classical complexity of sorting, each untruthful counterexample incurs an additional cost of order $n$. The upper bound is based on a geometric representation of permutations and Grünbaum's theorem, while the lower bound combines sorting arguments with a Condorcet-type construction. We also study the case where the target ranking has a low-dimensional geometric representation: each item is represented by a point in $\mathbb{R}^d$, and the ranking is obtained by projecting the points onto an unknown direction. For these classes we give an upper bound of $O(d^2\log n+dk)$ and a lower bound of $\Omega(d\log n+dk)$, leaving a factor of $d$ gap in the noiseless term.

We show that border rank is equal to rank for Kruskal tensors. We also give an analogous Kruskal condition for alternating tensors, which certifies uniqueness of skew rank decompositions. Furthermore, we show that border skew rank is equal to skew rank for alternating Kruskal tensors, and we give an algorithm to find the minimum skew rank decomposition of alternating Kruskal tensors.

Modern machine learning compilers select tensor memory layouts to minimize execution cost under hardware constraints. Layout selection is global: an operator may be fastest under one layout while its consumers prefer another, and aligning these preferences requires explicit layout conversions that can hurt model performance. Despite its practical importance, layout selection lacks a formal basis, so current compilers rely on ad-hoc heuristics. This paper presents the first formal study of layout selection in machine learning compilers. We formulate the problem as combinatorial optimization over dataflow graphs, minimizing the sum of operator execution costs and the per-tensor cost of these conversions. Our theoretical analysis shows that optimal layout selection is computationally hard, even for programs containing only matrix multiplications over two-dimensional tensors. We design an optimal polynomial-time algorithm for dataflow graphs of bounded treewidth. For general instances, we give a weighted MaxSAT encoding that an off-the-shelf solver can optimize. The formulation unifies several existing layout optimization strategies, including XLA's layout assignment, partition dimension selection in systolic array compilers, and layout planning in mobile GPU optimizers. We implement the formalization in a production compiler for an AI accelerator and measure the execution time of the compiled models under greedy heuristics, the compiler's rule-based strategy, and an optimal solver. Simple heuristics degrade execution time by up to $5\times$ on some workloads. Where the compiler's cost model is accurate, the solver matches or beats the rule-based strategy. On workloads with complex data movement it falls behind, and since the solver minimizes the stated objective exactly, that gap isolates cost-model error from search quality, showing where compiler effort actually pays off.

Calibration requires probabilistic reports to be conditionally unbiased and reliably interpretable as probabilities. A calibration measure assigns numerical error to miscalibrated reports. Haghtalab et al. (2024) proposed an approximately truthful calibration measure for online prediction, leaving open whether exact truthfulness is compatible with completeness and soundness. We resolve this question negatively for sequential binary prediction: exact truthfulness is incompatible with completeness and soundness, even for independent outcomes. We then show that this impossibility is specific to exact truthfulness. We give two general reductions from a base calibration measure, producing additively and multiplicatively approximately truthful calibration measures, respectively. Applying the multiplicative reduction, for every $0 < \varepsilon < 1$ we construct a sound and complete calibration measure that is $(1+\exp(-T^{(1-\varepsilon)/2}/2))$-multiplicatively truthful. This improves the approximate-truthfulness guarantee of Haghtalab et al. (2024).