Papers

The concept of subspace designs was introduced by Guruswami and Xing (STOC'13), and explicit constructions were given by Guruswami and Kopparty (FOCS'13). These are families of subspaces that have small intersection with any given subspace of a fixed dimension. We introduce robust subspace designs. Informally, these are a quantitative extension in which we demand that not too many subspaces of the family contain directions that lie close to any given subspace of a fixed dimension. We give a probabilistic construction of such a robust subspace design of polynomial size, as well as a non-trivial explicit construction of superpolynomial size. Our main application of this new concept is a quantum space-bounded variant of the Valiant-Vazirani theorem (Theor.Comput.Sci.'86), which shows that restricting $\mathsf{NP}$-complete problems to instances with at most one accepting witness preserves hardness under randomized reductions. For quantum witnesses, the analogous quantity is the dimension of an accepting witness subspace. We use our probabilistic construction of robust subspace designs to isolate a unique witness for space-bounded quantum Merlin-Arthur protocols with perfect completeness and an acceptance gap outside their perfectly accepting subspace. As further applications, we give a randomized reduction of well-conditioned nullity testing to space-bounded quantum Merlin-Arthur protocols with perfect completeness. Using a similar idea, we find that ordinary subspace designs allow us to recover the classical $\mathsf{C_= L}$ containment of Allender, Beals, and Ogihara (STOC'96) for general nullity testing through a simpler proof.

A temporal graph is a sequence of graphs on a common set of $n$ vertices, its snapshots, one for each time step. An agent, knowing the entire sequence in advance, may at each time step wait or move along an edge of the current snapshot, and the temporal graph is explored once the agent has visited every vertex. We consider always-connected temporal graphs, in which every snapshot is connected, and ask how long exploration can be forced to take when the underlying graph, the union of all snapshots, has maximum degree at most~$\Delta$. Two lower bounds were known in this setting, $\Omega(\Delta n)$ and $\Omega(n\log n)$, realised by different constructions. We establish a stronger lower bound, answering a question of Bastide, Groenland, Michel and Rambaud. Specifically, for every $n$ and $\Delta$ with $\Delta_0\le\Delta\le n-1$ we construct an always-connected temporal graph on $n$ vertices with underlying maximum degree at most~$\Delta$ that cannot be explored in fewer than $\gamma\,\Delta n\,(1+\log(n/\Delta))$ time steps from any start vertex, where $\gamma>0$ and $\Delta_0$ are absolute constants. The construction is deterministic, and every snapshot is a spanning tree with exactly one vertex of degree greater than three. Time is divided into phases. In each phase, a rotating-cycle gadget prevents the agent from reaching more than half of the trees attached to it. Between phases, we reassign target vertices among these trees using walks on a constant-degree expander, and a Gray code order of the targets keeps the underlying degree bounded. The reassignment guarantees that, for any walk of the agent, some target remains unvisited throughout all phases.

In this paper, we study CONFLICT-FREE EDGE CUT (CF-CUT), which is a recently introduced conflict-free version of the MIN-CUT problem that asks to find the minimum number of edges to disconnect a connected graph. The CF-CUT takes as input a connected undirected graph G = (V, E), a conflict graph $\widehat{G}$ such that $E(G) = V(\widehat{G})$, and the objective is to decide whether there exists $F \subseteq E(G)$ such that $G - F$ is disconnected and $F$ is an independent set in $\widehat{G}$. Rauch et. al. [IPL-2025] proved that CF-CUT is NP-Complete and also provided some results on the parameterized complexity of CF-CUT. A related variant MIN CONFLICT-FREE EDGE CUT (MIN-CF-CUT) takes a connected graph $G$, a conflict graph $\widehat{G}$ such that $V(\widehat{G}) = E(G)$, and an integer $k$ as input and asks if there is a set $F$ of at most $k$ edges such that $G - F$ is disconnected and $F$ is an independent set in $\widehat{G}$. In this paper, we extend the work of Rauch et al. [IPL-2025] and provide a systematic study on the CF-CUT and MIN-CF-CUT from the perspective of parameterized complexity and polynomial kernelization. We prove that CF-CUT is NP-hard when the dissociation number of the input graph is at most two. We also complement it by proving that CF-CUT is poly-time solvable when the dissociation number of an input graph is at most one. Additionally, for MIN-CF-CUT, we consider both solution size and various structural parameters of the input graph as parameters, and provide fixed-parameter tractability and W[1]-hardness results when the conflict graph is restricted to various graph classes. We also prove that unless NP $\subseteq$ coNP/poly, MIN-CF-CUT admits no polynomial kernel when parameterized by the vertex cover number of the input graph; and also when parameterized by the sum of the solution size and the vertex integrity of the input graph.

We consider a matching problem in which the cost of each edge is a vector with $k$ components. The cost of a matching is the sum of the bottlenecks over all components, and we ask whether there is a perfect matching of cost at most some value $Z$. This type of matching has applications in heavily synchronized job-shop scheduling problems and in reconfiguration problems, where movement is restricted to a single direction per step. In this paper, we analyze the problem from a parameterized complexity perspective and provide various results including FPT-membership for parameters $k$ and $Z$ combined, as well as W[P]-membership and W[SAT]-hardness for each of the two parameters individually. The reduction also implies para-NP-hardness parameterized by either maximum degree or treewidth. We further show hardness of approximation within a super-logarithmic factor for the optimization variant and provide a $k/d$-approximation algorithm for any constant $d \leq k$ as well as an efficient approximation scheme parameterized by $k$. With parameter $Z$, we show that no FPT-time $F(Z)$-approximation algorithm is possible for any computable function $F$, unless W[1] = FPT.

In this work we consider the Maximal Independent Set (MIS) problem and the metric Steiner Forest problem in the sublinear time setting, under the adjacency/distance matrix query model. First, we give an algorithm that estimates the size of an MIS up to a multiplicative factor of $(1+\eps)$ using $\tO(n^{4/3}/\eps^2)$ queries. This improves the best previous algorithm by Mahabadi, Roghani, Tarnawski, and Vakilian (SODA 2026), which had a query complexity of $\tO(n^{3/2}/\eps^2)$. Via a reduction from that work, this would automatically imply the same improvement for the problem of estimating the metric Steiner Forest cost up to an $O(\log n)$ factor. However, as our second contribution, we consider the Steiner Forest problem directly and provide an algorithm with $\tO(n)$ query complexity that is very simple and does not proceed via MIS.

The Independent Chip Model (ICM) converts the chip stacks of the players remaining in a poker tournament into finishing-place probabilities and prize equities. It is the standard model in tournament solvers, but its definition considers all possible finishing orders, and exact computation has been regarded as intractable for large fields. The same mathematics appears in other fields; for instance, the ICM is a Plackett-Luce ranking model with stacks as weights. We present DE-ICM, a deterministic algorithm that computes the placement probabilities of all $n$ players for all paid places in $O(M n^{2})$ time, where the number $M$ of quadrature nodes is fixed for the admissible inputs and the target accuracy. Its numerical scheme allows the target to be set close to the accuracy of double-precision arithmetic. The algorithm evaluates a classical integral representation in which, conditional on a player's exponential clock, the number of players ahead is Poisson-binomial. A double-exponential quadrature on a single grid evaluates every integral; its truncation errors have closed-form bounds, and the step size, which controls the discretization error, follows from the field size. A numerically stable deconvolution then recovers each player's leave-one-out coefficients in $O(n)$ time from one shared product. On exactly solvable instances with up to 4,000 players, the worst observed relative error of an equity is $4.0 \times 10^{-14}$ and the worst absolute error of a placement probability $1.1 \times 10^{-14}$. The full placement matrix for 1,000 players takes 0.2 seconds on one CPU core.

The Sachdev-Ye-Kitaev (SYK) model is a strongly interacting fermionic system. Its dynamics are difficult to analyze with Lieb-Robinson bounds and cluster expansions due to its all-to-all disordered interactions. We prove that, with high probability over the disorder, the SYK model admits a quantum Gibbs sampler with a system-size-independent spectral gap at sufficiently high constant temperatures. This yields polynomial-time preparation of SYK Gibbs states on a quantum computer. While prior non-rigorous computations that suggested SYK thermalization were limited to studying local correlators, we emphasize that our result holds up to arbitrary inverse polynomial trace distance. Our proof introduces the pseudo-Lindbladian approach to fermions and uses an especially simple SYK analysis based on a comparison to classical dynamics.

Every balanced instance of the stable marriage problem with strict complete preferences has a unique finest partition into prime blocks, and that single partition simultaneously factors three different structures: the reachable execution digraph as a Cartesian product, the proposal-prefix antimatroid as a direct sum, and the stable-matching lattice as a direct product. The converse fails, and fails at every size from two on: two explicit families share the identical Boolean-cube execution while one is maximally decomposable with a single stable matching and the other is prime with n. Uniqueness yields an exact census, a recursion counting the prime instances at every size, under which exactly 88,478,208 of the 110,075,314,176 profiles with four agents on each side are decomposable and the decomposable fraction is asymptotically n! / n^(2n). The blocks are characterised as the square components of the mutual-rank filtration, so the partition is computable in polynomial time and the factorisation is a tool rather than only a fact.

The Firefighter Problem models a spreading process (originally a fire, alternatively an infection or rumour, for example) on a graph. A defender saves a single vertex per turn; after each defence, the fire spreads to the unburned and undefended neighbours of all burning vertices. Deciding whether a strategy exists for the defender to protect some targeted number of vertices is computationally hard in graphs in general, but tractable in some restricted cases. Inspired by research into spreadable rabies vaccines for bats, we study a variant of the Firefighter problem in which defence also spreads. Some approximation results are already known for this problem; we provide algorithmic and hardness results, as well as containment results for the infinite $n$-dimensional Cartesian and strong grid graphs.

We present Guanaco, an algorithm for performing conjunctive query evaluation where, for each Boolean conjunctive query, and positive epsilon, the algorithm achieves polynomial time with exponent equal to the submodular width plus epsilon. The algorithm and its running time generalize smoothly to general conjunctive queries. We believe the algorithm and its analysis to be notably simple, indeed, together we believe they form a highly simple argument that conjunctive query evaluation can be performed in essentially submodular width time. In the case of Boolean conjunctive queries, the algorithm is based on interleaving three simple primitives: a subroutine for establishing a form of consistency; a subroutine for establishing global uniformity, which, briefly speaking, partitions relations as needed to control discrepancies between average degree and maximum degree; and, a simple step that joins pairs of existing relations to form new relations.

Time series often contain recurring structural patterns, and efficiently mining such patterns into compact representations is essential for scalable analysis of long sequences. Cartesian tree (CT) equivalence provides a well-established structural abstraction that preserves hierarchical order structure while discarding exact values and fine-grained ordinal variations. By grouping multiple ordinal patterns into a shared structural form, CT equivalence offers a principled way to compress recurring temporal structure. However, mining frequent CT-equivalent patterns at scale remains computationally expensive. A naive pairwise approach repeatedly constructs and counts CT representations over subsequences, requiring $O(n^4)$ time for a sequence of length $n$, which severely limits its applicability to long sequences. We propose a new Cartesian pattern mining algorithm based on a Cartesian suffix tree that compactly organizes CT-equivalent subsequences and reuses shared structural information. Our method reduces exhaustive CT-pattern occurrence collection from $O(n^4)$ to $O(n^2)$ time, and we formally prove the correctness and complexity bounds. We further show that this computational gain translates into effective compact representations. Across diverse time-series datasets, a small set of mined CT patterns preserves meaningful clustering structure, and comparisons with finer-grained order-preserving representations show that CT equivalence reduces redundant ordinal distinctions under limited feature budgets. Our implementation is available at https://github.com/hyundong98/CT-Miner .

In the matroid secretary problem, weighted elements arrive in random order, and an online algorithm must irrevocably accept elements forming a high-weight independent set. Dynamic Thinning is a recent, conceptually simple $3.1462$-competitive algorithm for the matroid secretary problem that maintains a random reference set. Given this reference set, the elements in its maximum-weight independent subset have been accepted independently with a time-dependent probability. We extend this approach by replacing the single reference set with a finite hierarchy of nested reference sets. For every fixed $\eps>0$, the resulting algorithm is $(e+\eps)$-probability-competitive for arbitrary matroids, with $O_\eps(n^2)$ queries in the worst case.

A kernel $k(x,y)$ is LSHable if there exists a locality sensitive hashing scheme $H$ such that $k(x,y)=\Pr_{h\sim H}[h(x)=h(y)]$ for all $x,y$. This notion plays a key role in efficient kernel methods in high dimensions. In this work, we show that the $p$-exponential kernel $k(x,y)=\exp(-\lVert x-y \rVert_p)$ is LSHable in bounded regions for all $1<p\leq2$. Previously, this was known only for $p=1$. Our new "mosaic LSH" scheme is based on a Poisson hyperplane process with hyperplanes sampled as $\ell_1$-biased $p$-stable vectors, for which we develop efficient sampling procedures. As applications, our results yield new and efficient density estimation methods based on LSHability for those $p$-exponential kernels.

Estimating the volume of a high-dimensional convex body is a fundamental problem in theoretical computer science. Given a quantum membership oracle for a convex body $K\subset\mathbb{R}^d$, we show that estimating $\mathrm{vol}(K)$ to relative error $\varepsilon$ takes $\widetilde O(d^{5/2}+d^{3/2}/\varepsilon)$ queries. This improves the previous quantum upper bound $\widetilde O(d^3+d^{9/4}/\varepsilon)$ of Chakrabarti et al. (ACM TQC 2023) and Cornelissen–Hamoudi (SODA 2023), and achieves a larger quantum speedup over the currently best classical upper bound $\widetilde O(d^{7/2}+d^3/\varepsilon^2)$ of Jia et al. (JACM 2026). Our quantum algorithm combines two new ingredients: an input-dependent effective spectral-gap analysis of quantum walks based on uniform classical warm-start mixing bounds, and a single quantum estimator for products of normalizer ratios in simulated annealing based on classical bridge sampling. We also prove an $\Omega(d)$ quantum query lower bound for constant relative error, improving the previous $\Omega(\sqrt d)$ lower bound.

A clique transversal of a graph is a set of vertices intersecting every maximal clique. We prove that deciding whether a graph has an inclusion-wise minimal clique transversal of size at least $k$ is W[1]-hard when parameterized by $k$.

Let $n=2^d$. We prove that the worst-start total-variation mixing time of the full labelled Thorp shuffle satisfies, for every fixed $0<\varepsilon<1$, \[ t_{\mathrm{mix}}(\varepsilon) \le C_* d^{5/3}+O_\varepsilon\!\left(d^{4/3}\log(d+2)\right), \qquad C_*=5.1536951454\ldots . \] Time is measured in ordinary Thorp shuffles. This improves the previously published full-deck upper bound $O(d^3)$ of Morris. The geometric starting point is elementary: label the $n=2^d$ positions by binary strings. One physical Thorp shuffle is a random swap layer on one coordinate matching of the $d$-dimensional cube, followed by a deterministic cyclic rotation of the coordinates. In a rotating frame that deterministic rotation disappears, so $d$ successive physical shuffles become one sweep: one independent random swap layer in each coordinate direction. The proof then combines five ingredients. First, a collision-graph representation and negative dependence give pointwise smoothing for ordered marked cards. Second, an entropy argument is sharpened by moving an endpoint moment up to the critical exponent $\beta_c=\log_2(3/2)$, yielding an entropy-preparation clock $s^2/(2\log(3/2))+O(s\log s)$. Third, exact multiplicities in the ordered $h$-tuple permutation representation amplify marked-card bounds into Fourier operator and energy estimates on low representation levels. Fourth, entropy truncation produces a positive mixture whose regular component retains the marked-card budget. Finally, only two long blocks need this entropy regularization: a shorter third block is used solely for additional low-level smoothing. Optimizing the three block lengths gives the constant above.

We provide potential function proofs of the competitive ratios of DoubleCoverage for the hard $k$-taxi problem on HSTs and general weighted trees of bounded depth. Buchbinder, Coester, and Naor (2023) obtained these ratios using time-reverse dual fitting and noted that they did not know a pure potential proof beyond $k=2$. We observe that the minimum matching distance between the online and offline taxi configurations can be expressed as an integral over the rooted tree of the absolute difference between their taxi counts below each point. This representation lets us analyse DoubleCoverage over short movement intervals and apply the resulting estimates to potentials that remain unchanged when both serving taxis are relocated from pickup to destination.

We study the fundamental problem of computing Personalized PageRank (PPR) on an undirected and unweighted graph $G$. We focus on the $\eps$-error guarantee introduced by Andersen, Chung, and Lang [ACL; FOCS 2006 $\&$ Internet Math 2007]. Their classic local push method computes an approximate PPR vector satisfying the ACL $\eps$-error guarantee in $O((\alpha\eps)^{-1})$ time, where $\alpha$ is the teleportation parameter. In this paper, we eliminate the dependence on $\alpha$ and present an algorithm that achieves the same error guarantee in $O(\eps^{-1-o(1)})$ time. Consequently, we obtain a nearly optimal algorithm for PPR computation under the ACL error guarantee, and a nearly linear algorithm for local graph clustering with complexity independent of $\phi$.

We study the quantum query complexity of maximizing a non-negative submodular function, considering both the unconstrained setting and, for monotone functions, a cardinality constraint $k$ on an $n$-element ground set. In the exact reversible digital value-oracle model, our unconstrained algorithm achieves an expected $(1/2-\varepsilon)$-approximation using only $O_\varepsilon(\log n)$ queries. In contrast, any classical randomized algorithm that attains a fixed expected ratio above $1/4$ requires $\Omega(n/\log n)$ queries (Li, Feldman, Kazemi, and Karbasi, 2022), establishing an exponential separation in query complexity. For cardinality-constrained maximization, we give a bounded-error quantum algorithm that achieves a $(1-1/e-\varepsilon)$-approximation using $\widetilde O_\varepsilon(\min\{\sqrt n,n/k\})$ queries. When $k=o(n)$, our algorithm achieves at least a quadratic speedup up to logarithmic factors over classical randomized algorithms (Mirzasoleiman, Badanidiyuru, Karbasi, Vondrák, and Krause, 2015; Peng and Rubinstein, 2025). Moreover, when $k=cn$ for any fixed rational $c<1-1/e-\varepsilon$, the query complexity reduces to $O_{\varepsilon,c}(\log n)$, yielding an exponential separation from the classical $\Omega(n/\log n)$ lower bound (Li, Feldman, Kazemi, and Karbasi, 2022). We further prove quantum lower bounds of $\exp(\Omega(\varepsilon^2n))$ queries for achieving a ratio beyond $1/2+\varepsilon$ without constraints, and $\exp(\Omega(\varepsilon^2k))$ queries for exceeding $1-1/e+\varepsilon$ when $k/n\le\varepsilon$. These barriers demonstrate that quantum computation offers no exponential speedup at these approximation thresholds.

A collage system is a grammar-based compression model that extends straight-line programs with repetition and substring truncation. Internal collage systems additionally require every nonterminal to be structurally reachable from the start symbol. Migita, Uehata, and I (CPM 2026) showed that any collage system of size $m$ can be converted into an internal one generating the same string with size at most $9m$, and left the improvement of this constant as an open problem. We show that a simple refinement of their top-down conversion reduces the bound to $5m$. The proof combines three elementary ideas: canonicalization of generated truncations that remain aligned with a target endpoint; a repetition decomposition that absorbs every complete copy of the repetition base into a maximal core; and monotonicity of structural reachability, which prevents an input truncation rule from both becoming structurally reachable and later acting as a hidden target at which endpoint alignment is lost. The conversion runs in deterministic $O(m^2)$ worst-case time in the stated unit-cost random-access machine model. Consequently, for every string, the minimum size of an internal collage system is at most five times the minimum size of a general collage system.

Diversity maximization is a fundamental optimization problem with applications in machine learning, data summarization, information retrieval, and recommendation systems. In many such applications, the data are partitioned into groups, and the selected subset must satisfy prescribed group quotas. We study Fair Diversity Maximization: given a set of points in a metric space partitioned into $m$ groups, the goal is to select exactly $k_i$ points from each group $i$ while maximizing the minimum pairwise distance among the selected points. The best previously known approximation guarantee is $m+1$, which grows linearly with the number of groups. We show that this dependence on $m$ is not fundamental. We present a new local-search framework that yields a $4$-approximation for any constant number of groups, with no restrictions on the metric space or on the size of the selected set. To the best of our knowledge, this is the first constant-factor approximation whose guarantee is independent of the number of groups in this general setting. Our framework maintains all group quotas exactly while progressively eliminating violations of the diversity objective. We further develop a specialized algorithm for two groups that achieves a $2$-approximation, improving the previous best factor of $3$. This factor is optimal: unless $\mathrm{P}=\mathrm{NP}$, no polynomial-time algorithm can achieve an approximation factor strictly better than $2$, even for the unconstrained case.

Balanced discrete optimal transport between n sources and n targets of unit mass is exactly the minimum-cost assignment problem-a bipartite perfect matching-and is therefore solvable exactly by industrial matching engines in milliseconds to seconds. We ask when the exact approach beats the standard approximate alternatives, entropic Sinkhorn and its accelerated variant Greenkhorn, and make the sparse-exact side certified by a textbook LP dual-feasibility clip. Three contributions. (i) Measurement: on dense 2-D instances, exact matching (Jonker-Volgenant) is faster and strictly more accurate than either approximate method throughout the moderate-n regime (0.01 s at n=500 to 11.5 s at n=8000); reaching a 1% quality target on the same hardware requires roughly 10-80 min for Greenkhorn (factors 4e2-6e4 over exact; plain Sinkhorn is 20-650x slower still), a rough power-law projection beyond the measured range. Greenkhorn's measured speedup over plain Sinkhorn is only 1.0-1.5x on most converged cells. (ii) A simple kNN-pool gap certificate: given a pool matching and its Blossom dual, a one-pass O(n^2) clip produces a dense-feasible lower bound; combined with the Sinkhorn dual potential (valid at every iterate, not just at convergence), the bound is valid on all 45 measured configurations and tightens monotonically with k. (iii) A multi-robot task-allocation sanity check where the discrete plan is the deliverable: per-round exact assignment costs 0.1-68 ms, while a Sinkhorn-plus-hardening pipeline costs 0.12-15.9 s and accumulates 6-27% extra travel over 15 rounds. The Sinkhorn family's large-n dense regime is acknowledged and left untouched. Code, data, and results under MIT: https://github.com/dimkadimon/OT-Blossom.

$\mathsf{Datalog}^\circ$ has been introduced as an extension to Datalog that increases expressiveness, yet retains simple least fixpoint semantics and admits optimization techniques such as semi-naïve evaluation and demand transformation. $\mathsf{Datalog}^\circ$ accomplishes this by generalizing the or and and operators of Datalog to addition and multiplication over a semiring. Finding a (minimal) fixpoint of a $\mathsf{Datalog}^\circ$ program is equivalent to finding a solution to a system of polynomial equations over the underlying semiring. Solving these systems of polynomial equations is not only a fundamental problem in the theory of $\mathsf{Datalog}^\circ$, but also has many applications in computer science, such as in databases, program analysis, and optimization. This paper resolves a key open problem in the theory of $\mathsf{Datalog}^\circ$: we prove a tight upper bound on the number of steps until convergence of iterative methods for solving these polynomial equation systems over a commutative $p$-stable semiring. In particular, we show that the number of steps until convergence is $O((p+1)n)$ where $n$ is the output size of the $\mathsf{Datalog}^\circ$ program. As the number of steps until convergence is only a proxy for the runtime of $\mathsf{Datalog}^\circ$ evaluation, we also consider the number of semiring operations used and show that, for a natural class of algorithms, $O((p+1)mn)$ is a tight upper bound, where $m$ is the maximum number of semiring operations per iteration.

We study nonnegative, non-monotone $k$-submodular maximization with $k\ge2$ labels under support constraints, and show how the certified approximation coefficient improves as the support region permits more uniform selection. For a compact convex down-closed support region $P\subseteq[0,1]^n$, the diagonal level $\zeta(P)=\max\{t\in[0,1]:t {\bf 1} \in P\}$ ranges from $\zeta=0$, which carries no geometric promise, to $\zeta=1$, which is unrestricted support. Our main structural result is a comparator-uniform linearization of the multilinear extension, built from an objective-independent action and a comparator-independent update field. For $k\ge3$, its validity reduces, independently of the number of elements, to four polynomial inequalities of degree at most three in one or two variables, only one of which depends on $k$. Explicit parameter choices give a nondecreasing certified profile $\underline{\alpha}_k(\zeta)$, in closed form on all of $[0,1]$ when $k=2$. At $\zeta=0$ we certify $0.4456\ldots$ for $k=2$ and $0.4541\ldots$ for every $k\ge3$, improving the recent $\sqrt2-1$ guarantee for one matroid or one knapsack, as well as the $1/3$-type guarantees for a fixed number of budgets; at $\zeta=1$ we certify $1/2$ for $k=2$, $(\sqrt{17}-3)/2$ for $k=3,4$, and $k/(2k-1)$ for $k\ge5$, whose excess over $1/2$ is of order $1/k$ rather than the previous $1/k^2$. Value-retaining rounding transfers these guarantees to matroid and knapsack constraints, and the same field yields $O(\sqrt T)$ approximate regret online under gradient or post-decision value feedback.

We study constant-delay enumeration for conjunctive queries with negation ($\texttt{CQ}^{\neg}$). It is known that $\textit{signed-acyclic}$ queries can be enumerated with constant delay after linear preprocessing time, using linked lists with skip links. We use a hypergraph width measure for $\texttt{CQ}^{\neg}$, the $\textit{fractional signed hyperorder width}$ ($\textsf{sfhow}$), which requires a single variable order to simultaneously handle every subset of the negative atoms, which Capelli et al. introduced to characterize the preprocessing time of direct access. $\textsf{sfhow}$ strictly generalizes signed-acyclicity (recovered when $\textsf{sfhow} = 1$) and fractional hypertree width (recovered on queries without negation). Our main algorithmic result is a variable-elimination algorithm that achieves constant-delay enumeration on input $I$ for any full $\texttt{CQ}^{\neg}$ query with preprocessing time $O(|I|^{\textsf{sfhow}})$. Furthermore, we show that using multiple variable orders can improve this bound, exhibiting an $O(|I|^{3/2})$ algorithm for $k$-cycle queries with at least one negative edge.

We study the minimum Vertex Cover problem under two non-adaptive offline advice models, where predictions about a fixed optimum solution are provided once as part of the input. These forms of prediction were introduced independently by Cohen-Addad, d'Orsi, Gupta, Lee, and Panigrahi (2024) and by Ghoshal, Makarychev, and Makarychev (2025). In Partial Predictions, independently revealed vertices come with correct membership labels. For every sufficiently small fixed $\varepsilon>0$, we give a randomized polynomial-time algorithm that has expected approximation ratio at most $2-(2-o(1))\frac{\log\log(1/\varepsilon)}{\log(1/\varepsilon)}$. In Noisy Predictions, every vertex instead receives a mutually independent noisy membership label of bias $\varepsilon$. For every sufficiently small fixed $\varepsilon>0$, we give a randomized polynomial-time algorithm that has expected approximation ratio at most $2-(1-o(1))\frac{\log\log(1/\varepsilon)}{\log(1/\varepsilon)}$. Our Noisy Predictions bound matches the leading asymptotic improvement below $2$ obtained by Aamand, Chen, Gollapudi, Silwal, and Wu (2025), despite using only one noisy label per vertex rather than a separate independent label for each endpoint of each incident edge. Both our bounds hold as $\varepsilon$ goes to $0$ and are strictly below $2$, the optimal approximation threshold for minimum Vertex Cover without advice under the Unique Games Conjecture, as shown by Khot and Regev (2008).

In network speed scaling, jobs arrive over time at a network of servers whose speeds can be tuned, every job must be processed by the servers along one of its allowed routes, and the goal is to minimize the total flow time plus the total energy. For stochastic arrivals, the competitive ratio of Vaze and Nair depends on the network, through the lengths of the routes it uses. We show that this dependence can be removed: we give an algorithm, which routes the jobs by solving a convex program and runs every server at a fixed speed, whose competitive ratio depends only on the power functions; for $P(s)=s^2$, it is at most the golden ratio $\varphi\approx1.618$. The key idea is a lower bound on the optimal cost which, like the cost of our algorithm, is a sum over the servers of a function of each server's load, so the analysis reduces to a single server. We also show that the routing rule of Vaze and Nair can be a factor $\Omega(L)$ away from optimal, where $L$ is the length of the longest route.

We investigate algorithms for the quantum simulation of the Schrödinger equation on a Riemannian manifold, where the kinetic operator is defined by the Laplace–Beltrami operator corresponding to the metric. Our first algorithms are based on a global spectral method based on the identification of an efficient transform to the eigenbasis of the Laplace–Beltrami operator. We use this method to provide explicit, efficient, quantum simulation algorithms for the Riemannian Schrödinger equation on tori and spheres with their standard metrics, simplices with the Wright–Fisher metric, truncated positive orthants and their invertible affine images with the log-barrier Hessian metric, and $\ell_p$ balls with a metric induced by the Duffy map. Our second algorithm is based on a coherent simulation of local spectral methods on multiple charts, and is in principle applicable to any compact manifold. We first analyze this algorithm in the continuum and derive conditions under which a polynomial spectral cutoff suffices. We also provide a discretization analysis of a polynomial spectral cutoff for tensor-products of constant-dimensional manifolds. Finally, we consider applications of these methods to optimization and physical simulation. For optimization, we provide results including a generalization and convergence analysis of Quantum Hamiltonian Descent for geodesically convex functions that leads to explicit algorithms on the sphere and simplex, and a Riemannian generalization of the Real-Space Adiabatic Algorithm. For physical simulation, we show that our algorithms can simulate certain spatially discretized field theories, including a variant of the nonlinear sigma model.

We study single-sample prophet inequalities for online combinatorial allocation. Our main contribution is a general reduction from combinatorial to single-item prophet inequalities for valuation classes admitting suitable supporting prices. The reduction uses a free-disposal value to separate buyer-side combinatorial constraints from item-side supply constraints, yielding a modular framework that applies in the stronger Game of Googol model. This framework yields a $\frac{1}{6\sqrt{3}}\approx\frac{1}{10.4}$-competitive single-sample prophet inequality and a $(\beta_{k-1}/4)$-competitive $k$-sample prophet inequality for XOS valuations, where $\beta_k$ is the competitive ratio of a $k$-sample single-item prophet inequality, improving upon the work of [DKL+24]. Both results extend directly to divisible resources with capped-XOS valuations. Along the way, we obtain new results for online free disposal and an optimal single-sample prophet inequality for fractional knapsack in the Game of Googol model.

There has been a surge of recent work on correlated equilibrium concepts in Markov games. However, existing results focus on concepts weaker than normal-form correlated equilibria (NFCEs), leaving open the more challenging question of computing such equilibria, which goes back to the seminal work of Papadimitriou and Roughgarden (JACM'08). Here, we establish the first efficient algorithm for NFCEs in finite-horizon Markov games with a fixed number of players $n$. In particular, with $S$ states, horizon $H$, and at most $A$ actions per player, it computes an $\epsilon$-NFCE in time $S(AH/\epsilon)^{O(n)}$. This is the first algorithm polynomial in $1/\epsilon$ and the description of the game for NFCEs in an interesting class of problems beyond the normal-form setting. Moreover, under the usual assumption that recommendations are independent across states, we show PPAD-completeness—that is, computational equivalence to Nash equilibria—either in many-player games or when the precision is exponentially small. The key idea behind our approach is to run backward induction on a sequence of auxiliary stage games, but with the twist that in each step we compute a constant-expectation correlated equilibrium. This is a natural refinement of correlated equilibrium in which the conditional expected payoff from obeying is independent of the recommendation. In fact, our reduction goes both ways, establishing an equivalence between constant-expectation CEs and NFCEs in Markov games. For a fixed number of players, we observe that a constant-expectation CE can be computed approximately by combining linear programming with suitable discretization. In contrast, it is PPAD-hard in i) polymatrix (many-player) games at constant precision, and ii) two-player games at exponentially small precision. The latter result follows from an unexpected connection to rank-2 two-player games.