Papers

Operational requirements developed with the City of Braunschweig frame municipal micromobility planning under geofenced exclusions, mandatory retained sites, spacing rules, and area-level caps. Each policy edit requires a new feasible plan; full-set greedy takes tens of seconds per alternative at city scale. We present CLIPPER (Constraint-exact Low-latency Iterative Planning with Pooled Evaluation and Replay). It forms bounded candidate pools but recomputes exact current gains and checks every active constraint before selection. Coverage from each candidate alone sets the initial order. Offline full-set scans measure gains omitted by the pool; online, a conservative bound triggers expansion or audit. CLIPPER-F gives each proposal group the same number of candidate slots. Across Braunschweig, Munich, and Berlin, its mean coverage over complete chains stays within 0.245 percentage points of full-set greedy under the same policy, with 13.6–28.9 times lower mean rollout time. CLIPPER-A instead distributes one shared candidate budget across the groups. Under its coverage-prioritized policy, it uses 9–15% of full-set greedy's rollout time under the same policy, with mean gaps of 1.82 percentage points in Braunschweig, 0.12 in Munich, and 0.27 in Berlin. Together, CLIPPER enables rapid, replayable comparison of recorded city-scale planning states while enforcing every encoded model constraint.

The permanent of an $n\times n$ $0/1$ matrix $A$ equals the number of perfect matchings in the bipartite graph with edges defined by $A$. Jerrum, Sinclair, and Vigoda (2004) presented an FPRAS for approximating the permanent of any nonnegative matrix using a novel simulated-annealing algorithm. The running time was improved by Bezáková, Štefankovič, Vazirani, and Vigoda (2008) to $O(n^7\log^4 n)$ for $0/1$ matrices, for any fixed approximation and success parameters. We present the first asymptotic improvement over this running time bound, obtaining an $O(n^6\log^5 n)$-time algorithm. As in the previous works, our algorithm extends to arbitrary nonnegative matrices. The analysis of Bezáková et al. yields an $O(n^4)$ relaxation time bound for the JSV Markov chain on perfect and near-perfect matchings with ideal hole weights, under which each hole pattern (the unmatched vertices, if any) is equally likely in the stationary distribution. We introduce a restricted Poincaré inequality for the partition into hole patterns and prove an $O(n^3)$ bound on the corresponding restricted relaxation time. Our proof uses a coupled multicommodity flow argument inspired by a recent transport-flow argument of Chen et al.~(2025) for the Jerrum-Sinclair chain on all matchings.

Randomized sketch-and-solve algorithms accelerate overconstrained $\ell_2$ regression by replacing the input with a smaller problem. Standard subspace embeddings guarantee that the cost of the regression is nearly preserved, but coordinate-wise accuracy of the solution is more delicate: we want the solution vector itself to be close to the optimal solution in $\ell_\infty$ norm. In particular, we want to find a vector $x'\in \mathbb{R}^d$ such that $\|x'-x^*\|_\infty\leq \frac{\epsilon}{\sqrt d}\cdot \|Ax^\star-b\|_2\cdot \|A^\dagger\|_{\rm op}$. Price, Song and Woodruff initiated the study of this problem and showed that the subsampled randomized Hadamard transform (SRHT) with $O(\epsilon^{-2} d^{1+\Theta(\sqrt{\log\log n/\log d})})$ rows achieves this guarantee. A subsequent work of Song, Ye, Yin and Zhang claimed to improve the row count to $O(\epsilon^{-2}d\log^3 n)$. Unfortunately, their proof relies on an independence assumption that does not hold in general, and we exhibit an explicit instance on which it fails. To achieve a truly nearly-linear-in-$d$ row count, we introduce a new fast, dense randomized transform, which combines a randomized Hadamard flattening, a random permutation, and balanced, disjoint Gaussian pooling. Conditioned on the Hadamard-and-permutation stage, the sketched problem becomes an exact Gaussian regression in which the noise is independent of the entire sketched design; this conditional independence is exactly what the earlier argument was missing. Our sketch yields the $\ell_\infty$ guarantee with $m=O(\epsilon^{-2}d\log d)$ rows, uses one Hadamard pass with a padded internal dimension $N=\widetilde{O}(n+\epsilon^{-2}d^3)$, and is efficient to apply: the sketched pair $(SA, Sb)$ can be computed in $O(Nd\log N)=\widetilde{O}(nd+\epsilon^{-2}d^4)$ time.

Updating a graph by inserting or replacing nodes while preserving semantics and reusing existing structure is a recurring computational problem. In population genetics, this problem arises in the genotype representation graph (GRG), a directed acyclic graph that losslessly encodes phased genetic variation across hundreds of thousands of samples by sharing subgraph structure for individual mutations. In a GRG, each mutation's carrier set is implicitly encoded as the set of leaf nodes reachable from the node it is assigned to. Updating a mutation is therefore a structural editing problem, and current approaches remap mutations individually. This paper introduces a batched mutation-remapping algorithm that replaces independent reuse-aware traversals with a single shared reverse-topological pass, identifying reuse candidates for an entire batch at once. The pass propagates compact bit-parallel per-mutation state and uses an adaptive sparse/dense carrier set representation spanning rare-to-common variant densities. Batching is the memory-scalable complement to split-based parallelism, which instead replicates graph and traversal state per worker. Our remapping is evaluated on a controlled update workload and on end-to-end allele polarization, a bulk carrier set update that is common in population genetic analysis. Our approach is up to 10.5$\times$ faster than independent remapping while preserving exact carrier-set semantics.

The problem of finding a fair consensus ranking is an active research topic in the domain of fair rank aggregation and has been well studied; however, existing studies predominantly consider both the input rankings and the output ranking to be permutations where elements are always strictly ordered. In practice, however, a ranking with ties is far more common. This study bridges this gap by presenting a post-processing approach to determine the closest fair consensus ranking when an unfair tie-aware consensus ranking is provided. It proposes an exact algorithm and a fast heuristic to achieve this.

The time-dependent traveling salesman problem with time windows (TDTSPTW) generalizes the well-known traveling salesman problem with time windows by accounting the effects of congestion on travel times. In this paper, we develop an exact framework for the TDTSPTW with a makespan objective that extends the range of instances solvable to optimality under loose time windows while remaining effective across all levels of time-window tightness. Our framework relies on a dynamic-programming labeling algorithm and combines column generation, ng-memory augmentation, and exact search, using completion bounds for state-space sparsification, variable fixing, and exact search pruning. Embedded within a branch-and-price method, the framework solves all instances with up to 45 customers in a benchmark comprising more than 10,000 instances, including all instances without time windows with up to 50 customers.

We consider the buy-at-bulk facility location problem (BBFL), a problem combining the classic facility location problem with buy-at-bulk network design, which finds motivation in telecommunication networks. In it, we are given a graph with edge lengths, opening costs and demands for each vertex, and a monotone and subadditive capacity-cost function, and our task is to open facilities on a subset of the vertices and route the demand from each vertex to these facilities. The cost of a solution (which we want to minimize) is given by the opening costs of the chosen facilities, plus the cost on each edge, which is given by its length times the cost of providing enough capacity for the demands through the edge, given by the capacity-cost function. A common variant of the problem, the $k$-cable facility location problem (kCFL), considers the case where capacity is provided by buying copies of given cable types, each with a certain capacity and cost. We study BBFL on tree instances and show, for the unit-demand and splittable variants, that the problem admits a PTAS (a $(1+\epsilon)$-approximation for any $\epsilon > 0$). We also consider kCFL in the new setting of cable-unsplittable demands, where the demand of a vertex cannot be split among multiple cables. We show that the problem is NP-hard to approximate to a factor better than $3/2$ on stars, and then provide an algorithm for tree instances that outputs a solution with optimal cost, but which exceeds the capacity on each cable by a factor of $1+\epsilon$. As a consequence, we show that the problem has a $2$-approximation algorithm on trees.

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 complement this with upper bounds of uniform depth-$2\frac{1}{2}$ circuits of quasipolynomial-size. 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.

We study strong coresets for $\ell_p$ subspace approximation. Given a matrix $A\in\mathbb{R}^{n\times d}$, the goal is to sample and rescale a small number of its rows to obtain $SA$ such that $\left\|SA(I-P_F)\right\|_{p,2}^p=(1\pm\varepsilon)\left\|A(I-P_F)\right\|_{p,2}^p$ simultaneously for every subspace $F\subseteq\mathbb{R}^d$ of dimension at most $k$, where $P_F$ is the orthogonal projector onto $F$. Woodruff and Yasuda [WY25] (FOCS 2025) obtained coreset sizes $\widetilde{O}_p(k\varepsilon^{-4/p})$ for $1\leq p<2$ and $\widetilde{O}_p(k^{p/2}\varepsilon^{-p})$ for $p>2$. We improve these bounds to $\widetilde{O}_p(k\varepsilon^{-2})$ and $\widetilde{O}_p(k^{p/2}\varepsilon^{-2})$, respectively. For $1\leq p<2$, our algorithm runs in $\widetilde{O}_p(\mathrm{nnz}(A)+d^\omega+k\varepsilon^{-2})$ time. The resulting coreset size matches the sampling lower bound [LWW21] up to logarithmic factors when $k+1\geq C\log(1/\varepsilon)$ for an absolute constant $C$. For $p>2$, our algorithm runs in $\widetilde{O}_p(\mathrm{nnz}(A)+d^\omega)$ time, matching the running time of the framework of Woodruff and Yasuda. We use different techniques in the two regimes. For $1\leq p<2$, we combine a bicriteria low-rank split with Lewis-weight sampling and empirical-process bounds independent of the output dimension. For $p>2$, we give a sharper analysis of the Woodruff-Yasuda construction. By retaining the truncation in its sampling probabilities throughout the row-count recurrence, we show that it achieves the improved $\varepsilon^{-2}$ dependence.

We give a randomized data structure for undirected weighted graphs that are partially dynamic, i.e., that undergo either only edge insertions or only edge deletions. The data structure maintains $(1\pm\epsilon)$-approximations to the maxflow value and effective resistance between any queried pair of vertices, with total update time $\widetilde{O}_{\epsilon}(n^2)$ and worst-case query time $\widetilde{O}_{\epsilon}(1)$. Thus, for dense graphs where $m = \Omega(n^2)$, our guarantees are near-optimal. Our algorithms succeed with high probability against an adaptive adversary. Our result follows from a simple stability principle for partially dynamic graphs. We show how to partition an online sequence of $m$ updates into $\widetilde{O}(n/\epsilon)$ epochs such that every graph within an epoch is a $(1\pm O(\epsilon))$-spectral approximation of the graph at the beginning of the epoch. The epochs are determined by the cumulative leverage score of the updated edges: small leverage-score mass implies small spectral change, while the total leverage-score mass over a monotone update sequence is $\widetilde{O}(n)$. Consequently, a spectral sparsifier needs to be recomputed only once per epoch. Applying known static all-pairs maxflow and effective-resistance oracles to these sparsifiers then yields the result.

Given a probability distribution $p = (p_1, \dots, p_n)$ and an integer $1\leq m \leq n$, a contiguous aggregation of $p$ is a probability distribution $q = (q_1, \dots, q_m)$ such that each $q_i$ is a sum of consecutive elements of $p$. Given $p$ and a positive number $R$, we consider the problem of computing a maximum entropy contiguous aggregation $q$ of $p$, under the constraint that its Shannon entropy $H(q)$ is at most $R$. We devise a dynamic programming algorithm that solves the problem exactly, and two time-efficient greedy algorithms that provide close-to-optimal solutions. We discuss a few scenarios where our problem arises.

We study the computation of minimum spanning trees subject to local degree constraints. Recent work (ICALP 2026) established that three natural formalizations of this problem share the exact same parameterized complexity under standard structural graph parameters, including treewidth, pathwidth and clique-width. This applies to the cases where every vertex has a single target degree (Specified Degree MST), or a degree upper bound (Bounded Degree MST), or is equipped with a set of admissible degrees (Set of Degrees MST). In this paper, we investigate these problems under more restrictive parameterizations and reveal that their complexity landscapes fundamentally diverge on bounded-treedepth graphs. Specifically, we prove that the former two problems are fixed-parameter tractable when parameterized by the treedepth of the input graph. In sharp contrast, we show that Set of Degrees MST remains W[1]-hard parameterized by treedepth, even when combined with the feedback vertex number (i.e., deletion distance to treewidth $1$). Finally, we show that this divergence seems to be specific to treedepth: we exclude an analogous W[1]-hardness result for Set of Degrees MST w.r.t. the vertex cover number and also rule out fixed-parameter algorithms for the former two problems w.r.t. deletion distance to constant pathwidth.

It is proved that, for every $\delta \in (0,1)$, the Glauber dynamics for the uniform distribution on proper $q$-colorings is rapidly mixing when $q \geq (1+\delta)\Delta$ and the underlying graph has girth at least $5$ and maximum degree $\Delta = \Omega_\delta(1)$. This result also extends to general multi-spin systems satisfying a $\textit{local spectral contraction}$ condition, including the anti-ferromagnetic Potts model with $q\geq (1+\delta)(1-\beta)\Delta$. These results are achieved by a new spectral local-to-global principle on graphs with girth at least five for general multi-spin systems, and a novel Fourier analysis for Glauber dynamics on a star. The main ideas behind all the proofs were developed through several rounds of interaction with GPT-5.6 Sol Ultra.

Classical paging serves every miss immediately. We study paging with per-replacement maximum delay, where loading a pending page costs one movement plus the age of its oldest outstanding request and clears that page's entire episode. Equivalently, the holding rate is the number of pending pages. For cache size $k$, threshold LRU is strictly $(5k+3)$-competitive, while a randomized algorithm is strictly $5H_k$-competitive against an oblivious adversary; classical constructions give matching $\Omega(k)$ and $\Omega(H_k)$ orders. Offline, we obtain an exact $O(nk)$ dynamic program with one hole, an exact configuration algorithm for any fixed number of holes, and a nonproactive polynomial-time $5$-approximation in general. Physical farthest-next-use can nevertheless fail with only three pages. For page-dependent fetch costs with spread $\rho=w_{\max}/w_{\min}$, the exact fixed-number-of-holes algorithms persist. We obtain a $(3\rho+2)$-approximation and online guarantees with multiplicative factors $O(\rho k)$ deterministically and $O(\rho H_k)$ randomly against an oblivious adversary, plus any additive term inherited from the corresponding classical guarantee. A strict $\Omega(\sqrt\rho)$ randomized lower bound already holds for one cache slot. Thus maximum delay preserves the unit-cost competitive hierarchy but disrupts classical offline structure and makes weighted timing spread-sensitive.

We analyze exact-metric, Metropolis-adjusted Dikin walks by keeping the proposal determinant and reverse quadratic form together. Their leading uncentered terms cancel in the complete logarithmic acceptance ratio, leaving centered fluctuations that can be controlled with second-order tools. For a polytope given by $n$ inequalities and a convex $L$-Lipschitz potential, this yields warm-start mixing in $\widetilde O((d^{2}+dL^{2}R^{2})\log(w/\delta))$ steps for the regularized Lee–Sidford walk. For a spectrahedron with $n\times n$ blocks, the log-det walk mixes in $\widetilde O((\psi^\star nd+dL^{2}R^{2})\log(w/\delta))$ steps, where $\psi^\star$ measures matrix leverage. The two analyses share an acceptance-to-mixing reduction. A proposal-comparison argument transfers the polytope bound to an appropriately padded $O(1/d)$-accurate metric computed from high-precision Lewis weights. For spectrahedra, given $\widehat\psi\ge\psi^\star$, a direct-or-two-seed TensorSRHT construction gives an exact-arithmetic implementation with $\psi^\star$ replaced by $\widehat\psi$ in the mixing bound.

The notion of linear neighborhood complexity is a very general structural assumption on a graph class, covering most classes of sparse graphs such as planar graphs, graphs excluding a fixed (topological) minor, or bounded expansion graphs, as well as many structured classes of dense graphs, such as graphs of bounded clique-width, twin-width, merge-width, or flip-width. In this work, we present $O(n^2)$-time optimal algorithms for $n$-vertex graphs coming from a class of linear neighborhood complexity for the following problems: $\bullet$ All-Pairs Shortest Paths, $\bullet$ the multiplication of the adjacency matrix $M$ of the input graph with any $n \times n$ matrix. More specifically, after a quadratic preprocessing, we can multiply $M$ with any $n$-vector in $O(n)$ time. This solves several questions raised in [Bonnet, Kim, Geniet, Moon; ICALP '26], and improves and generalizes results in several other recent papers [Bonnet, Giocanti, Ossona de Mendez, Thomassé; STACS '23], [Bannach, Marwitz, Tantau; STACS '24], [Anand, van den Brand, McCarty; NeurIPS '26], [Kozma, Opler '26], and [Cardinal, McCarty, Yuditsky '26]. We also extend our results to classes of bounded VC density. In classes of linear neighborhood complexity, we also give a triangle-detection algorithm in randomized linear time $O(n+m)$ in $n$-vertex $m$-edge graphs, a $K_4$-detection algorithm in randomized $O(n \log^5 n + m \log n)$ or deterministic $O(n^2)$ time, and a $K_5$-detection algorithm in randomized $O(n \log^9 n + m \log^5 n)$ time.

The optimal discretization problem asks, given two disjoint sets of points $R$ and $B$ in the plane, for a minimal family of horizontal and vertical lines that separate the two sets, so that no cell delimited by the lines contains points from both sets. The problem arises as a pre-processing in supervised machine learning, and has received significant attention in parameterized algorithmics. Answering the question raised by Bonnet, Giannopoulos, and Lampis [IPEC 2017] and Froese [PhD thesis, 2018], it was shown by Kratsch, Masařík, Muzi, Pilipczuk, and Sorge [SODA 2021] that optimal discretization admits a fixed-parameter algorithm with running time $2^{\mathcal{O}(k^2 \log k)} \cdot n^{\mathcal{O}(1)}$, where $k$ is the solution size and $n = |R| + |B|$. In this paper we give an algorithm for optimal discretization that runs in time $\mathcal{O}(1.9602^n)$. We also study the related point separation problem that asks to separate all input points by axis-parallel lines. For this problem we obtain an algorithm with runtime $\mathcal{O}(1.8906^n)$. Our guarantees follow from structural observations about bichromatic and monochromatic point sets, and hold even if points are allowed to share coordinates. To our knowledge, these are the first improvements over the trivial $2^n$ bound for both problems.

Suppose two parties, Alice and Bob, hold long character strings, $X$ and $Y$, respectively, and they are interested in determining how similar $X$ and $Y$ are. {Moreover, they want to exchange the strings with cost proportional to their degree of dissimilarity.} Such problems arise, for example, in database and file system synchronization operations, as well as in DNA sequence comparisons. Since the strings are long, we are interested in methods that are communication-efficient and have low overhead in terms of the computations that Alice and Bob must perform, when the strings are similar enough. In this paper, we provide a simple low-overhead communication-efficient algorithms for such string reconciliation and edit distance problems, determining the edit distance $k$ between $X$ and $Y$ using only $O(k\log^3 n)$ bits of communication and $O(n\log k)$ time overhead, with high probability.

The problem of pattern matching, that is, finding all occurrences of a given pattern in a string, is one of the fundamental problems in computer science that has applications in many areas. In this paper, we consider fuzzy patterns, defined as sequences of fuzzy properties over the basic alphabet. We first consider fuzzy pattern matching for sequences of elements of the basic alphabet and then extend the problem to partially ordered sets of nodes labeled by elements of the basic alphabet. For sequences, we seek segments that match the pattern, whereas for partially ordered structures, we seek saturated chains of nodes that match the pattern. The key concept underlying the solutions to these problems is the notion of a trajectory, which generalizes the concept of the prefix function used in the Knuth–Morris–Pratt (KMP) algorithm. A trajectory is processed together with the corresponding data structure, allowing the proposed algorithms to be represented as transition systems whose states are trajectories for sequences and trajectories associated with nodes for partially ordered structures. The trajectory-based approach provides a unified framework for fuzzy pattern matching in various data structures.

We study the Lipschitz stability of attention retrieval in hyperbolic spaces. Existing methods lack deterministic guarantees on attention-weight preservation under finite-precision representations. We introduce HCC+, a theoretical framework exploiting three properties of the Poincaré ball: exponential volume growth enabling query-independent boundary truncation; logarithmic covering radius of hyperbolic 1-centers enabling dimension-independent critical-key identification; and a packing bound with constants independent of the embedding dimension. We prove two deterministic guarantees: for exact retrieval, the per-layer attention deviation is bounded by 10% of its ideal value; for soft attention, the total variation distance decays as $O(1/\sqrt{n})$, the rate of finite-sample variance. As a consequence of the guarding mechanism, the framework achieves a storage reduction factor of $6.1\times$ relative to FP16. We provide the first deterministic, query-independent retrieval certificate in non-Euclidean geometry.

Affine modular linear hashing is one of the simplest classical hash families. For a prime $p > u$, the hash function is obtained by choosing $s,t$ uniformly from $\mathbb{Z}_p$ and mapping each key $x \in \{0,\ldots,u-1\}$ to one of $n$ bins by $h(x) = [(sx+t) \bmod p] \bmod n$. Despite its simplicity, the maximum load of linear hashing remains poorly understood. For $n$ keys hashed into $n$ bins, the best known upper bound is $O((n \log n)^{1/3})$, whereas the best known lower bound is only $\Omega(\log n / \log\log n)$. We prove a lower bound of $\exp(\Omega(\log n / \log\log n))$ for universes of size $n^{1+o(1)}$. Surprisingly, there is a key set for which this load holds not just in expectation, but for every random seed. The proof is driven by two simple reductions: one transfers lower bounds from a real version of linear hashing to modular linear hashing, and the other transfers arithmetic Kakeya constructions to real hashing. We further show that, for sufficiently large $p$, the expected maximum loads in the modular and real settings are essentially the same, giving an alternative route to an $n^{1/3+o(1)}$ upper bound. Finally, we show that any uniform subpolynomial upper bound for either setting would imply a polynomial-length arithmetic Kakeya conjecture and hence the Kakeya conjecture for upper Minkowski dimension.

We study the fully dynamic edge orientation problem, focusing on worst-case time bounds. An undirected graph undergoes edge insertions and deletions, and the goal is to maintain an orientation with small maximum outdegree (hereafter, outdegree) and small worst-case update time. The outdegree of any orientation is at least $\alpha-1$, where $\alpha$ is the graph's arboricity, i.e., the minimum number of forests into which its edge set can be partitioned. When $\alpha = O(1)$, it is long known that both the outdegree and the worst-case update time can be bounded by $O(\log n)$. Despite numerous follow-ups, no $o(\log^3 n)$ worst-case update time is known for maintaining constant outdegree, even for very basic graph families—with a notable exception, forests. For forests, a simple folklore algorithm maintains outdegree 2 via random walks: When an insertion creates a vertex of outdegree 3, the algorithm repeatedly chooses a uniformly random outgoing edge until reaching a vertex of outdegree at most 1, and then flips the resulting directed path. As the underlying graph is cycle-free, the path length is easily shown to be $O(\log n)$ in expectation, and also with high probability for polynomially long update sequences. We prove that this simple random walk paradigm extends to outerplanar graphs. Our algorithm maintains constant outdegree with $O(\log n)$ worst-case update time, where the time bound holds in expectation, and also with high probability for polynomially long update sequences. We give a tight analysis: outdegree 4 is achievable with $O(\log n)$-length paths, while outdegree 3 incurs $\mathtt{poly}(n)$-length paths. We also extend the argument to $K_{2,t}$-minor-free graphs, for any $t \ge 2$, with the outdegree bound depending only on $t$ and with the same update time guarantees. The locality of [...]

We show that there exists a polynomial-time algorithm to find a stable matching in network hypergraphic preference systems. The key connection that drives the algorithm was discovered by chatting with ChatGPT-5.6 Sol Max. We verified it independently and present the details in our own words.

In this paper we study the fractional vertex cover problem on trees in two related models: online and incremental. In the online model, the vertices of the tree are known a priori and the edges arrive one at a time. The goal is to maintain a fractional vertex cover of the tree, i.e., an assignment of fractional weights from [0,1] to the vertices such that the weights of endpoints of every edge sum up to at least one. After each edge arrival, we need to modify the fractional vertex cover to cover the new edge as well. However, we can only increase the values assigned to vertices. The problem was studied before (in the vertex arrival model) by Wang and Wong, who motivated it as a generalization of the ski-rental problem, but also (more importantly) by its close connection to the dual online matching problem. They presented a 1.901-competitive algorithm for general graphs in the vertex arrival model. We present an $\frac{11}{6} \approx 1.83$-competitive algorithm for trees in the more general edge arrival model. In addition, we study the fractional vertex cover problem in an incremental model, where we again seek a fractional vertex cover after every update, but all the updates to the tree are known to the algorithm a priori. In this model, we give a 1.5-competitive algorithm and provide a matching lower bound.

Maximizing Nash welfare over indivisible goods is a central problem in resource allocation. For additive valuations, the best-known approximation factor is roughly $e^{-1/e}\approx0.692$, and the problem is APX-hard. We study Nash welfare maximization under lexicographic valuations, where every good is worth more than the total value of all lower-ranked goods. This large-gap structure makes preferences almost ordinal, which might suggest that the problem becomes easy. We show, however, that the picture is more nuanced: although lexicographic valuations enable stronger algorithmic guarantees, they retain significant computational hardness. Our first main result is a $(1/\sqrt{2}-\epsilon)\approx(0.707-\epsilon)$-approximation algorithm for weighted Nash welfare under lexicographic valuations, improving over the inherited guarantee of roughly $e^{-1/e}$. The algorithm rounds the configuration LP for Nash welfare, for which we show a matching integrality gap of $\sqrt{2}$. Our second main contribution is an exact algorithmic polynomial time framework for ordered lexicographic instances and doubling lexicographic instances. We introduce a domination-based branch-and-prune method for which we prove a mutual-exclusion property between sibling subtrees and use a matrix-based leaf-counting argument to bound the pruned recursion tree by a polynomial when the number of agents is constant. Finally, we show that large gaps do not eliminate hardness, as Nash welfare maximization is NP-hard even for ordered lexicographic valuations, and it is NP-hard to obtain a $0.9996$-approximation even for doubling lexicographic valuations. Thus, lexicographic valuations make Nash welfare maximization easier, but not easy: they admit tighter approximation and exact algorithms in important cases, yet still require intricate techniques and preserve some of the hardness of the general additive setting.

We prove the first quantum-classical separation for a sampling problem over a continuous domain. For a class of Gibbs states $p\propto e^{-\beta E}$ on the torus $\mathbb{T}^d$ with smooth ($s$-Gevrey) potential and barrier amplitude $\alpha=e^{\beta\Delta}$, where $\Delta = \max E-\min E$, every classical algorithm querying the value, gradient, or any higher-order derivatives of the log-density requires $\Omega(\alpha)$ queries to sample at constant accuracy in total variation distance, while a quantum algorithm based on quantum singular value thresholding and temperature annealing samples with $\tilde{O}\left(\sqrt{\alpha}\right)$ queries to an oracle for the gradient. The advantage is quadratic in the barrier amplitude, which becomes exponential in the dimension, $e^{\Omega(d)}$, at low temperature. The classical bound is information-theoretic, holding for every classical algorithm with query access to the Gibbs potential and its derivatives at any order.

It is shown in this manuscript that a random graph $G$ drawn from the Erdős–Rényi model $\mathcal{G}(n,p)$ with \[ p=p(n)\leq 1/2, \qquad \lim_{n\to+\infty}(np-\log n-\log\log n)=+\infty, \] is a homomorphic core, i.e., every homomorphism from $G$ to itself is an automorphism. This implies tight ETH-based lower bounds of the subgraph isomorphism problem for almost all $k$-vertex patterns with polynomial average degree.

The classical Minimum Linear Arrangement (MLA) problem has been studied extensively. It is known to be NP-hard and it admits an $O(\sqrt{\log n}\log\log n)$-approximation [Feige and Lee, IPL, 2007]. MLA can be defined as follows as design problem: Given a graph $G$ with vertex set $V(G)$, design a path $H$ on the same vertex set that minimizes the linear arrangement cost $\sum_{uv\in E(G)}\textrm{dist}_H(u,v)$, where $\textrm{dist}_H(u,v)$ indicates the distance of $u$ and $v$ in $H$. We initiate the study of the generalization in which $H$ is allowed to be a caterpillar graph of maximum degree at most $\Delta$. Caterpillars are the simplest generalization of paths, having pathwidth one and interpolating between paths and stars via the degree parameter $\Delta$. We give an algorithm that lifts any $\alpha$-approximation for MLA to an $(\alpha+3-2/(\Delta-1))$-approximation for our problem, thus obtaining an $O(\sqrt{\log n}\log\log n)$-approximation for our more general problem as well. Moreover, we derive a $4$-approximation whenever MLA is polynomial-time solvable, in particular, for trees. Complementing these results, we prove NP-hardness for every constant $\Delta\geq 2$, and, in stark contrast to MLA, show it remains NP-hard on trees when $\Delta$ is part of the input.

In the problem of ground-state energy estimation, one aims to estimate the smallest eigenvalue of a Hamiltonian, often given a guiding state, with some promised overlap $\gamma$ with the ground space. The main approach to this problem is to simulate its evolution, and estimate the smallest (or equivalently, largest) eigenphase of the resulting unitary $U$. We give a quantum algorithm that estimates the largest eigenphase of a unitary $U$ in this guided setting using a factor of $\log\frac{1}{\gamma}$ fewer queries to $U$ than the previous best approach. The result matches an existing lower bound, and answers an open question from Mande and de Wolf. The algorithm is based on transducers, which often allow composition of quantum algorithms without overhead from error reduction.

Suppose we can apply the unitary $U=e^{i H}$ for some Hamiltonian $H$, and are given access to a unitary that prepares a guiding state promised to have overlap at least $\gamma >0$ with the ground space of $H$. Our goal is to estimate the ground-state energy of $H$ within additive error $\delta > 0$ and success probability at least $1-\varepsilon$, $\varepsilon>0$. How many applications of $U$ and its inverse $U^{-1}$ are necessary and sufficient? This quantity corresponds to the total Hamiltonian-simulation time needed. An upper bound $O(\log(1/\varepsilon)\log(1/\gamma)/\gamma\delta)$ was known, and was improved to $O(\log(1/\varepsilon)/\gamma\delta)$ very recently [JW26]. A matching lower bound was known whenever one of the three parameters $\delta,\gamma,\varepsilon$ was held constant [MdW26]. In this paper we prove the joint lower bound $\Omega(\log(1/\varepsilon)/\gamma\delta)$ with the tight $\varepsilon$-dependence provided the dimension of $H$ is at least $\log(1/\varepsilon)/\gamma^2$. Furthermore, we show that this same lower bound (with slightly larger dimension) holds for both the special case in which the ground state is guaranteed to be unique and $H$ has a gap of $\delta$ between its first and second eigenvalue; and for ground-state preparation, where $\delta$ denotes the spectral gap and $\varepsilon$ now is the approximation error. The lower bounds also apply when the Hamiltonian can be accessed via its block-encoding, and when fractional powers of $U$ are allowed, as in continuous-time Hamiltonian simulation. Lastly, improved upper bounds are known when $H$ is nonnegative and presented as a sum of squares; and our results imply the lower bound $\Omega(\log(1/\varepsilon)/\gamma\sqrt{\delta})$ for this case.