让下一篇重要论文，不再错过。

We give the first polynomial-time algorithm to estimate the partition function of the antiferromagnetic Ising model on the uniformly random $d$-regular graph to arbitrarily high accuracy for all inverse temperatures up to the Kesten-Stigum threshold. This is the entire regime in which the model is "replica symmetric," and is conjectured to be optimal for polynomial-time counting and sampling. To achieve our algorithmic result, we establish the absence of Fisher zeros for the partition function in a constant-width strip around the interval from $0$ up to the Kesten-Stigum threshold, with high probability over the randomness of the input graph. AI Disclosure: The overarching vision and proof architecture were developed solely by the authors without AI input. GPT 5.5 Pro was used, with extensive guidance from the authors, for only a few minor technical aspects of the proof. The authors independently verified correctness and wrote everything by hand. We take full responsibility for its contents. A more detailed statement can be found in Section 1.2.

We study the problem of finding minimum spanning trees in semi-random corrupted metrics. In this problem proposed by Bateni, Dharangutte, Jayaram, and Wang (COLT'24), we start with a metric space $\mathcal{M} = (X, d)$. The adversary independently corrupts each pairwise distance with probability $p$. The algorithm only observes these possibly corrupted distances and must output a spanning tree whose expected cost (in the true metric) is multiplicatively close to the minimum spanning tree cost. Prior work showed that losing a factor of $Ω(\log n)$ is necessary, even if the algorithm may adaptively select $O(n/\log n)$ points and probe the true distances between them. However, no upper bounds were known. We study three regimes based on whether the adversary can only increase, decrease, or arbitrarily change the corrupted distances. We show the following: (1) In the increase-only setting, we give a polynomial-time $O(1)$-approximation algorithm for any constant $0 \le p < 1$. (2) In the decrease-only setting, we give an $O(\log n)$-approximation algorithm for any constant $0 \le p < 1$. Additionally, we show an $O(\log^4 n)$-approximation algorithm that runs in quasi-polynomial time $n^{O(\log n)}$, for any constant $p \leq 0.1$. Moreover, we show that a polynomial-time algorithm computing an $n^{1/3-\varepsilon}$-approximation is unlikely to exist. (3) In the general setting, where the adversary can both increase and decrease distances, we show an $O(\log n)$-approximation algorithm for every $0 \leq p < \frac{1}{4}$. Further, we show that no $o(n)$-approximation is possible for $\frac{1}{2} \leq p < 1$. Our algorithms answer the main open question in Bateni et al.'s work; moreover, the $O(\log n)$-approximations in the latter two settings are information-theoretically optimal.

We consider the maximum cardinality matching problem in general graphs, and give an $n^{2+o(1)}$-time algorithm for the problem. Our work builds on recent work on combinatorial algorithms for maximum flow and extends the techniques to skew-symmetric flow problems which capture general matchings as a special case.

Sensor time series routinely reach similarity search with missing values, and the retrieved neighbours then depend on how the gaps were filled. We ask which retrieval conclusions hold for every admissible completion of an incomplete query. For a family of window-minimum distances that contains the $L_1$ distance as its separable end we give a pairwise certificate that one candidate is strictly closer than another, a guaranteed candidate set that contains every possible nearest neighbour, and a certificate that a nearest-neighbour label cannot change. The guarantees hold uniformly over the specified box and apply to the true query whenever it lies inside. At the separable end a closed form is known, but beyond it the worst-case gap need not be attained at a corner of the box. We prove a finite reduction to critical values, give an exact interval dynamic programme for the worst-case ranking gap, and a quadratic-time fast layer of valid bounds and feasible witnesses, supported by an exact criterion for when the feature-box relaxation is tight. On three traffic-sensor collections with real gap patterns, linear interpolation changes the nearest neighbour in 15 to 33 percent of the queries. At the full band the feature-box relaxation misses up to 14 percent of the exactly stable boundary pairs, whereas fixed-completion and Monte Carlo checks issue false certificates. The fast layer resolves about 90 percent of all tested boundary pairs without dynamic programming, and on paired network-scale queries the exact method is 82 to 110 times faster than the programme alone. Label certificates transfer to three public benchmarks, and the band width trades forecasting accuracy and certification coverage for lower imputation sensitivity, with dataset-dependent crossovers.

Minimum Label $k$-Cut asks for a minimum-size set of colors such that deleting all edges of these colors leaves at least $k$ connected components. We study how the structure of the underlying graph and its color classes affects the complexity of this problem. Regarding fixed-parameter tractability with respect to $k$ on simple planar graphs, we prove that the problem is W[1]-hard, even when the underlying graph is a cactus graph and each color appears at most twice. Furthermore, for cactus graphs in which every color class is connected, we establish a complexity boundary in terms of the maximum monochromatic degree: when this degree is at most two, an optimal solution can be found in $O(Nk^2)$ time, where $N$ is the input size, with no restriction on color frequency or the number of blocks spanned by a color; allowing the maximum monochromatic degree to increase to three already makes the decision version NP-complete. Finally, we prove that the problem remains W[1]-hard parameterized by $k$ on fan graphs, even when every color class is an induced path. These results show that bounding color frequency does not suffice to eliminate parameterized hardness. They also establish a monochromatic-degree boundary for connected colors on cactus graphs and identify complexity barriers to extending this tractable case to outerplanar graphs.

The online Steiner Forest problem is a fundamental network design problem in which terminal pairs arrive one at a time and an algorithm must irrevocably purchase edges connecting each pair. The natural greedy algorithm connects each terminal pair by a shortest path after making all previously purchased edges free. Awerbuch, Azar, and Bartal (SODA 1996) proved that this algorithm is $O(log^2 k)$-competitive and conjectured an $O(log k)$ bound, where $k$ is the number of terminal pairs. Bamas, Drygala, and Maggiori (SODA 2022) obtained an $O(\log \NumPairs\log\log \NumPairs)$ bound for greedy on a restricted class of inputs, leaving the general case open. In this paper, we resolve the competitive ratio of the greedy algorithm for online Steiner Forest, proving that it is tightly $Θ(log(k) log log(k))$. Specifically, we establish an $O(log(k) log log(k))$ upper bound for arbitrary instances and construct a matching $Ω(log(k) log log(k))$ lower bound. Our results refute the $O(log k)$ conjecture of Awerbuch, Azar, and Bartal and establish a tight $Θ(log log k)$ separation between the greedy algorithm and the best possible online algorithm for Steiner Forest.

Retailers can learn customer preferences by experimenting with different assortments. However, the cost of each experiment and the operational burden of repeatedly changing product offerings can make extensive experimentation impractical. We study how much a retailer can improve its final assortment decision using customer observations from one (or a very small number of) fixed experimental assortments. We consider the multinomial logit model, where product revenues are known but customer preferences are unknown. In this setting, we jointly design the experimental assortment and the rule for selecting a final assortment from observed customer choices to minimize worst-case regret: the expected revenue loss of the final assortment relative to an optimal assortment under the true preferences, maximized over possible customer preferences. We first characterize exactly the minimax regret and an optimal assortment when no experimentation is possible. We then show, surprisingly, that offering a single experimental assortment to just one customer can already substantially reduce regret, depending on how the products' revenues compare. We derive the exact minimax regret and an optimal experimental policy in this setting. When a single fixed experimental assortment is offered to many customers, we give a geometric characterization of the limiting minimax regret and construct an asymptotically optimal policy. Depending on product revenues, this single experiment can yield no improvement over no experimentation or nearly complete regret reduction. Finally, we extend our analysis to multiple experimental assortments chosen in advance, characterizing the limiting minimax regret, constructing asymptotically optimal policies, and identifying which collections of assortments allow worst-case regret to vanish as the number of customer observations grows.

We revisit scheduling with obligatory testing, where each job consists of two operations: a test with an initially known time and an operation whose processing time is revealed upon completion of the job's test. We develop an iterative competitive-ratio-preserving instance-reduction method that enables us to compute exact competitive ratios directly on highly structured instances for various index rules. Our main result is a clean analysis showing that the competitive ratio of the algorithm $1$-SORT is exactly the Golden Ratio $\varphi\approx 1.618$, improving the recent best-known upper bound of $1.861$ for arbitrary testing times. We further leverage this method to obtain near-tight bounds for a simple randomized algorithm on instances with uniform testing times.

We propose a novel quantum algorithm for solving linear optimization problems by quantum-mechanical simulation of the central path. While interior point methods follow the central path with an iterative algorithm that works with successive linearizations of the perturbed KKT conditions, we perform a simulation working directly with the nonlinear complementarity equations. Using a two-stage quantum eigenpath traversal schedule, this approach yields an algorithm for solving linear optimization problems involving $m$ constraints and $n$ variables to $\varepsilon$-optimality using $\widetilde{O}\!\left((m+n)^{3/2}\,\varepsilon^{-1}+(m+n)^{1/2}\,\varepsilon^{-3/2}\right)$ queries to an oracle that evaluates a potential function. In the standard gate model (i.e., without access to quantum RAM), our algorithm can obtain $\varepsilon$-precise solutions to LO problems using at most $$\widetilde{O}\!\left(\left[\frac{(m+n)^{3/2}}{\varepsilon} + \frac{(m+n)^{1/2}}{\varepsilon^{3/2}}\right] \cdot \mathrm{nnz}(A)\right)$$ elementary gates, where $\mathrm{nnz}(A)$ is the total number of non-zero elements found in the constraint matrix.

Continuum Glauber dynamics is a spatial birth-death process whose stationary distribution is a Gibbs distribution. We establish a spectral gap for Continuum Glauber dynamics applied to Gibbs point processes with repulsive pair potentials, a well-known special case of which is the hard sphere model. For arbitrary-range repulsive pair potentials, we show that a continuous version of Spectral Independence suffices to establish a spectral gap. This extends the regime of activity for which Continuum Glauber dynamics is known to mix, yielding a simple efficient sampling algorithm for arbitrary-range pair potentials that matches the known efficient sampling regime for finite-range pair potentials currently based on specialized algorithms. As a consequence, we also improve the threshold up to which packings of fixed size/density can be efficiently sampled from a bounded domain, the first improvement since Kannan, Mahoney and Montenegro (2003). To prove these results, we develop continuous analogs of Spectral Independence and negative fields localization. We show that a stronger variant of zero-freeness implies Spectral Independence, which in turn allows us to run the localization scheme to boost the spectral gap of Continuum Glauber dynamics from smaller activity to larger activity. While this follows the high-level blueprint of Chen and Eldan (2022) for the discrete setting, we have to address several novel difficulties due to the continuous setting. Notably, we avoid discretization in the algorithm and the analysis and work directly in the continuous setting.

We introduce a new tool, Express, for converting a non-causal attention approximation into a causal approximation with matching approximation guarantees. When combined with the state-of-the-art Thinformer approximation, Express improves upon the best known causal attention guarantees, delivering $\log^{3/2}(n)/s$ approximation error with only $O(s)$ memory and $O(s^2 \log^2(n))$ compression overhead for a sequence of length $n$. We pair these developments with an efficient I/O-aware Triton implementation, demonstrate substantial speedups over FlashAttention 2, and use Express to overcome four resource bottlenecks in the language modeling pipeline: long-context prefill, KV cache compression, long-form memory-constrained decoding, and long-form compute-constrained decoding.

For many optimization problems on random graphs, most prominently the largest independent set problem, natural algorithms reach only half the optimum, and such factor-two gaps are known to be sharp within broad algorithmic classes. We prove such a gap for finding large induced matchings in Erdős-Rényi random graphs. In the worst case, requiring a matching to be induced turns a polynomial-time solvable problem into an NP-hard one; our results reveal an average-case counterpart. In $G(n,p)$ with $C/n\le p\le 1-κ$, the largest induced matching has size $(1+o(1))L$ with high probability, where $L=\log_{1/(1-p)}(np)$. A simple online greedy algorithm finds an induced matching of size $(1/2-ε)L$, while no online algorithm, even with adaptive inspection order and unbounded computation, finds one of size $(1/2+ε)L$ except with probability $\exp(-Ω(L\log(np)))$. In the sparse regime $p=d/n$, low-degree polynomial algorithms exhibit the same sharp threshold. Our guarantees rest on a maximality principle: with high probability, every maximal induced matching is half-optimal. Greedy outputs a maximal induced matching and our low-degree algorithm a near-maximal one. We bypass the delicate coupling arguments classically used to analyze greedy since Karp. For low-degree hardness, we adapt Wein's overlap gap argument, with forbidden structures that track overlaps between endpoint sets rather than edge sets, since a new edge may join two previously used endpoints and thus encounter no fresh randomness. For online hardness, we give a new and significantly simpler argument that avoids the correlated instances of prior OGP-based proofs: at a suitable stopping time, all pairs involving uninspected vertices remain unrevealed, so a direct first-moment bound suffices.

In the restricted shortest paths problem, we are given an $n$-vertex $m$-edge directed graph $G$ whose edges are assigned two non-negative weights (or resources) called lengths and costs, a source vertex $s$, and a cost threshold $C$. The goal is to find, for each target vertex $t$, the length of the shortest $(s,t)$-path whose total cost is at most $C$. Ashvinkumar, Bernstein, and Karczmarz (SODA 2025) gave the first randomized bicriteria $(1+\varepsilon,1+\varepsilon)$-approximate algorithm on directed graphs running in sub-$mn$ time, answering a problem posed by Bernstein (SODA 2012). We substantially improve this on two fronts, supplying a deterministic bicriteria $(1+\varepsilon,1+\varepsilon)$-approximate algorithm running in near-linear time. Our approach straightforwardly generalizes to more than two resources. For $k$ resources, a deterministic $(1+\varepsilon_1,\ldots,1+\varepsilon_k)$-approximate algorithm running in near-linear time is possible, with $\varepsilon_1=\ldots=\varepsilon_k=\varepsilon$.

We study the weighted $k$-path vertex cover problem for any fixed integer $k \ge 2$, in which a vertex non-negatively weighted simple graph $G = (V, E)$ is given and one wants to find a minimum-weight vertex subset $C \subseteq V$ such that every simple path on $k$ vertices in $G$ contains at least one vertex of $C$. The problem generalizes the classic weighted vertex cover problem, that is, $k = 2$, and it admits a $k$-approximation algorithm of Hochbaum~[SIAM J. Comput., 1982] and a $(k - \frac 12)$-approximation algorithm of Brüstle et al.~[IWOCA 2021]. In this paper, we carefully design local weight functions in the Layering and Reverse Extension local ratio framework of Bar-Yehuda et al.~[ACM Comput. Surv., 2004], leading to an approximation algorithm of ratio $\frac {3k-1}4$ for odd $k \ge 3$ and of ratio $\frac {3k}4$ for even $k \ge 4$. When $k = 3$ and $k = 4$, these ratios are $2$ and $3$, respectively, matching the state-of-the-art approximation ratios for the corresponding special cases.