Papers

Rényi entropy estimation has been comprehensively investigated by Acharya, Orlitsky, Suresh and Tyagi (SODA 2015; IEEE Trans. Inf. Theory 2017) and consequent works, whereas only the sample complexity of Rényi entropy estimation of integer order has been settled. In this paper, we settle the sample complexity of Rényi entropy estimation of noninteger order, thereby completing the complexity picture of Rényi entropy estimation. Specifically, we show that for any noninteger $\alpha > 0$, it is sufficient and necessary to use \[ \Theta\!\left(\frac{d^{\max\{1/\alpha,1\}}}{\varepsilon^{1/\alpha}\log(d)} + \frac{d^{|1-1/\alpha|}}{\varepsilon^2}\right) \] samples to estimate the Rényi entropy of order $\alpha$ of an unknown discrete distribution over an alphabet of size $d$ to within additive error $\varepsilon$. For the upper bound, we reduce the bias using a refined polynomial approximation estimator for large probabilities. For the lower bound, we employ a different hard instance equipped with a new moment matching construction. The constructive moment matching has constant bounded high-order moments, while attaining a fixed ratio between the $\alpha$-th moments, which is of independent interest.

Min-Plus Convolution is a central problem in fine-grained complexity, and the associated Min-Plus Convolution Hypothesis forms the basis for a wide range of conditional lower bounds for fundamental problems. It is closely connected to the APSP and 3SUM Hypotheses, and in fact implies both, making it a unifying hypothesis for two of the main pillars of the area. In this work we establish several strong results related to Min-Plus Convolution. We design a universe reduction, showing, under a plausible additive combinatorics assumption, that the Min-Plus Convolution Hypothesis is equivalent to the Strong Min-Plus Convolution Hypothesis. We also obtain tight conditional lower bounds for multiple long-standing problems, including Min-Max Convolution and Bounded Monotone Min-Plus Convolution. Our approach is inspired by Fischer's recent equivalence between several variants of APSP [STOC '26], but extending that technique to the arithmetic setting requires overcoming deep obstacles. To this end, we develop a novel additive structure theorem that can be viewed as a higher-order substitute of the Balog-Szemerédi-Gowers (BSG) theorem, allowing us to extract strong additive structure even from weakly structured sets. Building on this structural result, we show that certain structured 3SUM instances (namely, sets with low rank) can be solved in truly subquadratic time. This algorithm forms the main algorithmic ingredient in our reductions. Besides, it generalizes all previously known truly subquadratic-time special cases of 3SUM, and is therefore of independent interest.

An interval graph is the intersection graph of a family of intervals on the real line. Interval Completion asks whether a given graph can be transformed into an interval graph by adding at most $k$ edges. Although the problem is fixed-parameter tractable when parameterized by $k$, whether it admits a polynomial kernel has long been an open question. We resolve this question by giving the first polynomial kernel for Interval Completion. Our main contribution is a parameter-preserving polynomial-time reduction from Interval Completion to Odd Cycle Transversal (OCT). Combining this reduction with the known randomized and deterministic polynomial kernels for OCT and a polynomial-time reduction back to Interval Completion, we obtain a randomized kernel with $\widetilde O(k^{18})$ vertices and $\widetilde O(k^{36})$ edges, and a deterministic kernel with $O(k^{36})$ vertices and $O(k^{72})$ edges. Here, $\widetilde O$ suppresses polylogarithmic factors in $k$.

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 $\beta$-bit integer entries can be computed in $D^{3-\delta}\operatorname{poly}(\beta,\log D)$ time for a fixed rational $0<\delta<1$, then min-sum subset convolution on an $n$-element universe can be solved in $(2+2^{-\delta})^n 2^{O(\sqrt n\log(n+1))}\operatorname{poly}(n,\beta)$ 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.

In the cow-path problem, a cow must find a goal lying at an unknown distance on one of $w$ paths connected only at the origin, and performance is measured by competitive ratio. Kao, Reif and Tate designed an efficient randomized algorithm in which the cow visits the paths in a fixed cyclic order. They proved the algorithm is optimal for $w=2$, and subsequently Kao, Ma, Sipser and Yin proved its optimality for all $w$, with a claim that no algorithm does better than the best cyclic one. This note provides a detailed proof of that claim.

We show that if the 3SUM problem on integer-valued inputs can be solved in truly subquadratic time, then it can also be solved in truly subquadratic time on real-valued inputs. This answers an open problem posed by Chan, Vassilevska Williams, and Xu [STOC 2022], and constitutes the first such tight real-to-integer self-reduction in fine-grained complexity. Our proof relies on a surprising combination of the Frank–Tardos theorem from optimization with Freiman-type theorems from additive combinatorics.

Sparse attention mechanisms estimate attention over $n$ tokens using a small subset of keys. Many existing approaches use maximum inner product search (MIPS) to retrieve the heaviest keys, which motivates the following question: given black-box access to a MIPS oracle, how many keys must be retrieved to output an $\varepsilon$-accurate attention estimate? We answer this question by unifying prior approaches through the framework of priority sampling. With a single MIPS index, we show that $\Theta(\sqrt{n}/\varepsilon)$ retrieved keys are both sufficient and necessary. With $\Theta(\log n)$ indices, we give an algorithm that retrieves only $O(\log n+1/\varepsilon^2)$ keys and prove that this is near-optimal. More generally, we design algorithms that establish a smooth tradeoff between the number of MIPS indices and number of retrieved keys. We then show that if we allow augmentation of keys and queries, we can bypass the above lower bounds: there exists a simple priority-sampling estimator using a single MIPS index and $O(1/\varepsilon^2)$ retrieved keys. When integrated into LLM inference, our algorithms outperform top-$k$ and sampling approaches used in prior work and yield attention approximation that scales favorably to long contexts.

The undirected multiple-unicast conjecture [LL04] asserts that network coding offers no throughput advantage over multicommodity flow. We refute this conjecture by constructing a deterministic linear network code over $\mathbb{F}_9$ on a 182-vertex bipartite subgraph of the point-line incidence graph of $\mathrm{PG}(2,9)$. The construction supports $157$ independent unicast sessions at common coding rate at least $1$, while every fractional multicommodity flow has common rate at most $147/157$. By the amplification theorem of~[BGS17], this yields a family of undirected multiple-unicast instances with coding gap $\Omega((\log n)^\varepsilon)$ for some $\varepsilon>0$. We also introduce a nondeterministic model of network coding based on locally verifiable certificates, which guides our construction and may be of independent interest. Building on the high-girth graph and error-correcting code framework of [BH25], we use GPT-6 to find a nondeterministic counterexample based on a new choice of Reed–Solomon local codes, and then convert this example into a causal code using an edge orientation and local search.

Finding heavy nodes in a tree—those whose counts exceed a given threshold—is a building block for analysis and learning over structured data. Achieving record-level differential privacy (DP) without sacrificing accuracy is challenging because each record contributes to counts along an entire root-to-leaf path, allowing privacy costs to accumulate across levels. Existing methods account for the multiple threshold comparisons for each record incur additive error margins of $\Omega_{\varepsilon,\delta}(\log h)$ or $\Omega_{\varepsilon,\delta}(\sqrt{\log h})$ for tree height $h$. We introduce BetweenCut, an $(\varepsilon,\delta)$-DP algorithm with an additive error margin of $O_{\varepsilon,\delta}(\log\log h)$, improving the existing bounds for deep trees. This error holds simultaneously for all nodes and is independent of the input database size.

We give a deterministic polynomial-time $1.6908$-approximation for Maximum Weighted $3$-Set Packing, breaking the $\sqrt3$ locality-gap barrier of squared-weight local search. The approximation ratio for this problem progressed from Berman's $2$ [Ber00] to Neuwohner's $2-\frac{1}{63{,}700{,}992}+\epsilon$ [Neu21]. Thiery and Ward then obtained $1.786$ [TW23], while Thiery subsequently improved the bound to $1.761+\epsilon$ and finally to $\sqrt3 \approx 1.732051$ through a layered exchange analysis [Thi23]. Thiery also proved that $\sqrt3$ is a locality-gap lower bound for the squared-weight objective even with exchanges of arbitrary size. Our algorithm performs in two phases and combines two objectives. Phase~I computes a bounded-exchange local optimum for the squared-weight potential and analyzes it through Thiery's layered framework, while strengthening the terminal analysis by preserving internal tree-edge slack for nonsingleton components and exploiting the incidence structure of $3$-sets for final singletons. This yields an augmented structural inequality with residual positive claw gain under the original objective. Phase~II switches to the original objective and recovers sufficient residual gain through an auxiliary weighted $9$-Set Packing instance. A covering argument transfers the structural bound through the high-girth lift used only in the analysis.

We give a deterministic polynomial-time algorithm for the Bilu–Linial signing problem on bipartite graphs. For every finite simple bipartite graph of maximum degree at most an integer $\Delta\ge3$, the algorithm assigns signs to its edges so that the signed adjacency matrix has operator norm strictly less than $2\sqrt{\Delta-1}$. Our algorithm builds on the randomized recursive repair framework of Jadbabaie, Saberi, and Sra~\cite{JSS26}, with deterministic rules for sign selection and vertex deletion.

We give an $n^{0.3+\varepsilon}$-approximation algorithm for the Steiner $k$-Forest problem, for any constant $\varepsilon>0$. As a function of $n$, this improves over the $O(\min\{\sqrt{n},\sqrt{k}\})$-approximation of Gupta et al. [ESA'07, TALG'10] which has stood for nearly two decades for the general case, as well as the later $n^{0.448}$-approximation of Dinitz et. al [APPROX-RANDOM'14, TALG'17] for the uniform weight case. On the other hand, we show that, due to lower bounds on the Densest $k$-Subgraph problem, the $O(\sqrt k)$-approximation for Steiner $k$-Forest likely cannot be improved. Specifically, we show that for any sufficiently small $\varepsilon>0$, an $O(k^{1/2-\varepsilon})$-approximation for Steiner $k$-Forest would surpass known degree-$n^{\Omega(\varepsilon^2)}$ Sum-of-Squares integrality gaps for Densest k-Subgraph and refute the corresponding dense-versus-random conjecture.

We study sampling from the Gibbs distribution of the Sherrington-Kirkpatrick (SK) model with Glauber dynamics. For every fixed inverse temperature $0\leq\beta<1$ and every fixed $M,D>0$, we give a polynomial-time simulated annealing algorithm whose output distribution is within $n^{-M}$ total-variation distance of the Gibbs distribution, with probability at least $1-n^{-D}$ over the interaction matrix. The algorithm starts from uniform product spins and uses Glauber dynamics along an increasing inverse-temperature schedule. The same approach also yields partition-function estimates with relative error $n^{-M}$. The key ingredient is a quantitative weak Poincaré inequality. Building on the stochastic-localization approach to weak Poincaré inequalities [Davies, Lee, Sandhu, and Shi, arXiv:2607.08160, 2026], we use Gaussian isoperimetry to directly compare exact stochastic localization paths. Our comparison tolerates failures of local Lipschitz continuity of the posterior mean by controlling their accumulated cost. The local Lipschitzness comes from approximating the posterior means by stationary points of the Thouless-Anderson-Palmer (TAP) free energy in locally strongly convex regions located by approximate message passing, a paradigm introduced by algorithmic stochastic localization [El Alaoui, Montanari, and Sellke, 2025; Celentano, 2024]. The approximation errors are roughly characterized by the validity of the TAP gradient equations, but in a different coordinate. To show that the errors rarely accumulate too much, we need a strong concentration control. A direct argument would require solving a variant of an open problem by Talagrand [2010, Research Problem 1.7.9]. We bypass this open problem with an intermediate conditioning step and transfer the control back by establishing the stability of the TAP gradient under deletion of coordinates.

Bentert, Fomin, Golovach, Korhonen, Lochet, Panolan, Ramanujan, Saurabh, and Simonov (SODA 2025) initiated the parameterized study of Edge-Disjoint Shortest Cycle Packing: given a weighted graph $G$ and an integer $k$, decide whether $G$ contains $k$ edge-disjoint cycles of minimum weight. They showed that the problem admits an algorithm running in time $n^{O(k^6)}$ and asked whether it is fixed-parameter tractable or $W[1]$-hard parameterized by $k$. We resolve this question by proving that Edge-Disjoint Shortest Cycle Packing is $W[1]$-hard parameterized by $k$, even on unweighted subcubic graphs. The same lower bound also applies to the vertex-disjoint variant. For planar graphs, they provides a construction of a kernel with $O(k^2)$ vertices and an algorithm running in time $k^{O(k)} \cdot n^{O(1)}$, and explicitly asked whether the problem admits a single-exponential algorithm of running time $2^{O(k)} \cdot n^{O(1)}$. Rather than addressing this question in isolation, we introduce a more general framework, Diverse Shortest Cycle Coverage, which asks for $k$ shortest cycles that may overlap in a controlled way while maximizing the total weight of covered edges. This framework simultaneously captures edge-disjoint shortest cycle packing, the problem of finding diverse shortest cycles, and the problem of maximizing edge coverage by shortest cycles. Our main algorithmic result shows that Diverse Shortest Cycle Coverage can be solved on planar graphs in time $2^{O(k)} \cdot n^{O(1)}$, thereby giving a single-exponential algorithm for Edge-Disjoint Shortest Cycle Packing as a special case. The key idea of our algorithm is a structural analysis of the Laminar Shortest Cycles Tree, a tree-like decomposition that reveals a laminar interaction pattern among shortest cycles in planar graphs and enables an efficient dynamic programming algorithm.

For every fixed proper minor-closed class $\mathscr C$ and every $\epsilon>0$, we give a deterministic LOCAL algorithm that returns a dominating set of size at most $(2a(\mathscr C)+1+\epsilon)\gamma_f(G)$ on every $G\in\mathscr C$. Here $a(\mathscr C)$ is the supremum edge-to-vertex ratio in $\mathscr C$, and $\gamma_f(G)$ is the fractional domination number. The class also admits a deterministic $(1+\epsilon)$-approximation for fractional dominating set and a randomized algorithm that always returns a dominating set and has expected size at most $(1+\epsilon)\gamma(G)$. In each case, the number of rounds depends only on $\mathscr C$ and $\epsilon$. None of these algorithms requires the number of vertices or the maximum degree as part of the input. For planar graphs, this gives the deterministic guarantee $(7+\epsilon)\gamma_f(G)$. Together with the lower bound of Hilke, Lenzen and Suomela, it determines the infimum of the deterministic constant-round approximation ratios for planar minimum dominating set as $7$, settling a question that had remained open since their work. The corresponding infima, measured against the integral optimum, are $7$ for graphs of Euler genus at most any fixed $g\ge0$, $2t-3$ for $K_t$-minor-free graphs with $3\le t\le9$, and $2r+1$ for graphs of treewidth or pathwidth at most any fixed $r\ge1$. We also prove that, for every integer $r\ge1$, no deterministic constant-round LOCAL algorithm achieves an approximation ratio below $2r+1$ on the $r$-th powers of paths, even when every vertex knows the number of vertices. This gives a new proof that the limiting constants are optimal for planar graphs, graphs of bounded treewidth or pathwidth, and $K_t$-minor-free graphs with $3\le t\le9$. For triangle-free planar graphs, the corresponding infimum is $5$.

When an envy-free allocation of indivisible goods does not exist, monetary transfers can restore envy-freeness. Existing work on fair division with subsidies, however, typically assumes that these payments are provided by an external source, an assumption that may be unrealistic in many applications. We address this limitation by allowing only monetary transfers between agents, with each agent's payments constrained by their individual budget. We show that while it is polynomial-time tractable to determine whether a given allocation can be made envy-free by payments under individual budgets, the general problem of computing such an allocation from scratch is NP-hard, even when agents have relatively large budgets. Motivated by this intractability, we conduct a parameterized complexity analysis, establishing fixed-parameter tractability with respect to the number of goods or to the joint parameter number of agents and good types, and we provide efficient algorithms in several special cases. For explicitly listed items, our type-based algorithm answers the envy-freeness part of an open question of T. T. Nguyen and J. Rothe, "Complexity Results and Exact Algorithms for Fair Division of Indivisible Items: A Survey."

A graph $G$ is $k$-degenerate if there exists an ordering $v_1, v_2, \dots, v_n$ of its vertices such that each vertex $v_i$ has at most $k$ neighbors $v_j$ in $G$ with $j < i$. A well-ordered $k$-degenerate graph is a labeled graph on the vertex set $\{1, 2, \dots, n\}$ in which every vertex $i$ has at most $k$ neighbors among $1, 2, \dots, i-1$. We present the first simple algorithms that generate, rank, and unrank cyclic pivot Gray codes for well-ordered $k$-degenerate graphs, where consecutive graphs differ by the addition, removal, or pivoting of a single edge. Our algorithm generates each well-ordered $k$-degenerate graph in constant amortized time per graph, using $O(n^2)$ space, while ranking and unranking take $O(n^2)$ time and $O(n^2)$ space.

Dijkstra's algorithm is traditionally classified as a greedy algorithm. We argue that this classification misrepresents the mechanism by which the algorithm determines shortest paths. We present a direct global-to-local proof of correctness. At each iteration, the algorithm identifies a shortest path over the entire set of paths whose endpoints remain unsolved. This global minimum is necessarily a shortest path to its own endpoint. More fundamentally, the algorithm uses already determined shortest paths to establish one additional shortest path at each iteration. We identify this progression as dynamic programming. A three-step exclusion proof shows how the full path space is reduced to frontier candidates, how relaxation removes inferior alternatives, and why minimum extraction yields the global minimum. The choice is thus determined by a global comparison, not by an optimistic local greedy choice. We conclude that the algorithm is more accurately characterized by dynamic programming and deterministic path exclusion than by the traditional greedy classification.

We present a new asymptotic matrix multiplication construction yielding $\omega<2.37115924$, and therefore an algorithm that multiplies two $n\times n$ matrices in $O(n^{2.37115924})$ arithmetic operations over the rationals, reals, or complex numbers. The construction improves the bound $2.371177$ of Dupont et al. within the combination-loss framework for the Coppersmith–Winograd tensor. Our approach combines a shared recursive parameter family with a conditioned optimization procedure that addresses two obstacles: weak gradients at low-mass constituents and competing bottlenecks in the retained exponent. A mass- and probability-dependent rescaling supports quasi-Newton optimization in $212\,820$ coordinates, while a six-control correction balances the three extraction roles at each relevant stage. Positive rational entropy-dual factors and an independent ordinary-string verifier turn the resulting construction into an exact certificate. The verified upper-bound expression lies in $[2.3711592385931075066123,\,2.3711592385931075066124]$. The complete witness and an offline verifier using only the Python standard library accompany the paper.

We prove conditional lower bounds for approximating dynamic time warping (DTW) over the uniform metric on three symbols. Let $N, M$ be the two string lengths and $n, m$ their respective numbers of runs. Under the Gap Block Disjointness (GBD) hypothesis, for every fixed $\kappa \in (0,1/6)$ and $\delta \in (0,1)$, no deterministic algorithm $N^{\kappa\delta}$-approximates DTW on two explicit strings of length $N$ in $O(N^{2-\delta})$ time. This matches the known tradeoff between approximation and running-time exponents for deterministic algorithms within any fixed factor greater than six in the approximation exponent. For run-length encoded strings, the Orthogonal Vectors Hypothesis (OVH) rules out any constant-factor approximation in $\widetilde{O}((nm)^{1-\delta})$ time for every constant $\delta>0$. More generally, for every fixed $\eta \in (0,1)$, it rules out $(N+M)^{1-\eta}$-approximation in the same running time. These bounds extend to every fixed metric with at least three points, including absolute distance on $\{0,1,2\}$. Both reductions use long runs to force equal-symbol matches in low-cost alignments. For explicit strings, we combine Boolean gadgets with deterministic gap amplification to obtain a polynomial approximation gap. For run-length encoded strings, an encoding of regular-expression membership gives accepting instances an alignment whose cost does not increase as selected runs grow. Rejecting instances have distance at least the chosen run length, while the number of runs stays fixed.

Standard diffusion samplers generate samples through repeated evaluations of a learned score function. Parallel sampling methods seek to accelerate generation by trading additional evaluations for fewer sequential rounds. This raises the question of how much sequential dependence is unavoidable, even when many score queries can be made simultaneously. We establish the first polynomial parallel-round lower bounds for diffusion sampling with approximate scores. Specifically, we prove (1) a $\widetilde{\Omega}(d^{1/3})$-round lower bound for sampling smooth, near-isotropic Gaussian mixtures in $R^d$, and (2) an $\Omega(d)$-round lower bound for uniform sampling from anisotropic axis-aligned boxes contained in the unit ball. Both bounds hold for arbitrary randomized algorithms making polynomially many queries per round at arbitrary locations and noise levels, with inverse-polynomial score error and constant total variation accuracy. The linear bound is tight for our box family. Our constructions use fixed approximate score oracles that enforce sequential access to hidden information while satisfying the accuracy guarantee at every noise level.

Huffman coding is one of the oldest and most fundamental problems in computer science. Given a string of length $\TextLength$ over a general alphabet, the goal is to assign a binary codeword to each character so that no codeword is a prefix of another and the total encoded length of the string is minimized. Huffman coding is widely used in practical compression systems, including file compression. As modern datasets continue to grow, it is natural to study whether a Huffman code can be constructed efficiently in the massively parallel computation (\MPC) model. The celebrated Huffman coding algorithm admits two straightforward \MPC implementations: for any constant $\epsilon\in(0,1)$, one uses $O(\TextLength^\epsilon)$ memory per machine but requires $\Theta(\log \TextLength)$ rounds, while the other runs in $O(1)$ rounds but requires $\Theta(\sqrt{\TextLength})$ memory per machine. We give the first nontrivial \MPC algorithm for Huffman coding that bypasses both limitations. For every constant $\epsilon>0$, our algorithm uses $O_\epsilon(\log\log \TextLength)$ rounds and $\softO(\TextLength^\epsilon)$ memory per machine, while its total memory and total computation are $\softO(\TextLength)$. This provides a rare example in which an exact solution to a problem whose classical algorithm is sequential in nature can be obtained in a sublogarithmic number of \MPC rounds.

In neutral-atom quantum computers, atoms are moved to target positions along paths of empty positions, and a target position may be reserved for one species of atom. Motivated by this task, we introduce Single-Move Labeled Token Routing: every source and every target vertex of a graph is assigned a set of labels, and tokens occupy the sources. A solution consists of a matching that assigns each source to a compatible target (one whose label set intersects its own), a route for each matched pair, and a movement order in which, when a token is moved, its route contains no other token. The problem is known to be polynomial-time solvable when every source is compatible with every target, and $\mathsf{NP}$-complete on grid graphs when each source is compatible with exactly one target. We prove that the latter case remains $\mathsf{NP}$-complete on grids and on planar graphs of maximum degree four even when some solution has pairwise edge-disjoint routes. On trees, the problem is known to be $\mathsf{NP}$-complete even for maximum degree three. We study trees through the solution edge multiplicity, the largest number of routes of a solution sharing an edge, and the candidate edge multiplicity, the largest number of compatible pairs whose paths share an edge. We prove that on trees of maximum degree three, the problem is $\mathsf{W}[1]$-hard parameterized by a bound on the solution edge multiplicity, even when a movement order is given, and that on trees of unbounded degree, it is $\mathsf{NP}$-complete even when the candidate edge multiplicity is at most eight. We show that on trees the problem is fixed-parameter tractable parameterized by the maximum degree together with the candidate edge multiplicity, and also by the candidate vertex multiplicity, the same count at vertices. Unless $\mathsf{P}=\mathsf{NP}$, neither the maximum degree nor the candidate edge multiplicity can be omitted.

For the online combinatorial allocation problem with subadditive valuations, Correa and Cristi (STOC 2023) proved the existence of a $6$-competitive online algorithm, improving on the previous best $O(\log\!\log m)$-competitive online algorithm due to Dütting, Kesselheim, and Lucier (FOCS 2020), where $m$ is the number of items. However, Correa and Cristi's result is existential, and it was left open whether a constant competitive ratio is attainable via an efficient online algorithm that uses a polynomial number of demand oracle queries. In this work, we answer this affirmatively, giving an expected-polynomial-time $(6 + \epsilon)$-competitive online algorithm for any constant $\epsilon > 0$. Our techniques also recover, in the offline setting, the $(2 + \epsilon)$-approximation result of Feige (STOC, 2006). Finally, when the buyers' valuations are drawn from identical distributions, we exploit symmetry to obtain an improved $(60/11+\epsilon)$-competitive algorithm. Starting with the natural configuration LP relaxation for the problem, our main technical contribution is a recursive Bundle Score Generator (BSG) that resolves item conflicts by assigning correlated scores to the items requested by each buyer. Unlike the Random Score Generator whose existence is shown by Correa and Cristi, our BSG is efficiently computable in expected polynomial time. Moreover, it satisfies a stochastic dominance property that is sufficient to recover both Feige's offline result and Correa and Cristi's online result.

Let f:2^N –> \cZ^+ be a polymatroid (an integer-valued non-decreasing submodular set function with f(emptyset) = 0). A k-polymatroid satisfies that f(e) <= k for all e in N. We call a subset S of N independent if f(S) equals the sum of f(e) over the elements e of S and f(e) > 0 for all e in S. Finding a maximum-size independent set in a 2-polymatroid has been studied and polynomial-time algorithms are known for linear polymatroids. For k >= 3, the problem is NP-hard, and an approximation algorithm with ratio approaching 2/k is known and is obtained by swapping as long as possible a "large" subset from the current solution by a set with one more element. Here we give a simple analysis of the more particular two-for-one repeated swapping heuristic, obtaining a (weaker) 2/(k+1)-approximation.

In online inverse linear optimization, a learner recommends an action and then observes the choice of an expert who maximizes a fixed, unknown linear objective on $\mathbb{R}^{d}$; the goal is to learn to optimize this objective without observing it. Sakaue recently obtained the optimal regret $O(\sqrt d)$ with a randomized algorithm making $(dT)^{O(d)}$ linear optimizations per round, and asked whether it can be attained in polynomial time. We answer positively: our deterministic algorithm has regret $O(\sqrt d)$ for every horizon $T$ and runs in time polynomial in $d$ and $T$. It is a variant of the variable-metric algorithms of Sakaue et al.\ and Cai et al., in which a metric update is revoked once the query point moves far enough from where the update was made.

We prove that the probability that Gaussian elimination with partial pivoting on $n \times n$ random matrices has growth $\rho$ is at least inverse quasi-polynomial: $\Omega(\exp(-c \log^2 (\rho)\log(n)))$ for some constant $c > 0$. This lower bound breaks standard conjectures in the smoothed and average-case analysis of Gaussian elimination. To the best of our knowledge, it is the first non-trivial lower bound on the probability of large growth for Gaussian random matrices.

When optimizing computing architecture for specific computationally intensive tasks, such as evaluating or training a neural network, it is important to identify what computational subtasks will offer the greatest decrease in cost (e.g., wall clock time or energy usage). This problem is known as Hardware-Software (HS) Partitioning, and it has a variety of formulations, many of which are NP-Hard. In this paper, we will define a family of HS Partitioning formulations which admit an exact fixed parameter tractable algorithm based on the directed pathwidth of the given task graph and show that this family of problems contains multiple existing formulations such as makespan minimization. Finally, we examine task graphs with small directed pathwidth that arise in real world applications and show that our algorithm can provide a speed up of up to 200x over a comparable linear programming approach utilizing the Gurobi ILP Library.

The Gram-Schmidt Walk is a randomized vector-balancing algorithm whose subgaussian guarantees support applications in discrepancy, experimental design, and data compression; however, these theoretical guarantees are established in exact arithmetic, whereas implementations must approximate least-squares directions, boundary updates, and sampling probabilities in finite precision. This is important because small numerical errors can change which coordinates freeze and thereby alter the subsequent trajectory. We analyze the concentration of the perturbed Gram-Schmidt walk directly under bounded, potentially biased and history-dependent errors. For $n$ input vectors of Euclidean norm at most one, we obtain a modified MGF bound depending on key error sources which recovers the original result as the error goes to zero. We also construct a full-column-rank instance in which bounded update errors produce bias of order $\min\{n^2\varepsilon,n\}$, showing that updates accumulate error unavoidably under this model. Finally, we validate our findings in a variety of settings by ablating on the bit precision and problem size.

We construct explicit n x n x n tensors of border rank at least 3n-o(n), improving the previous record of (2 + $\varepsilon$)n due to (Landsberg and Michalek 2025). We also prove that the linear flattening method cannot be used to establish lower bounds on border rank beyond 2n-1. When n is odd, we prove that this bound is achieved by Koszul flattenings. When n is even, a result of (Landsberg 2015) shows that Koszul flattenings can achieve 2n-2, leaving an open gap of size one. Our 2n-1 bound improves the best known 6n-4 linear flattening barrier due to (Garg et al. 2019) and (Buczyński 2026). Combined, these results show that our 3n-o(n) construction, as well as the construction of Landsberg and Michalek, provide explicit examples of tensors with higher border rank than any linear flattening can achieve.