ALT We prove the all-exp property for the Traveling Salesman Problem with the 2-opt neighborhood, and for (3,2,2)- and (2,3,2)-Maximum Constraint Assignment with the flip neighborhood. The all-exp property states that infinitely many instances have an initial solution from which every sequence of improving moves to a local optimum, and hence every run of local search, has exponential length. For the Traveling Salesman Problem, the all-exp property was previously known for every k-opt neighborhood with k ≥ 3, so our result for 2-opt completes the picture for all k-kopt neighborhoods. In (p,q,r)-Maximum Constraint Assignment, each constraint involves at most p variables, each variable occurs in at most q constraints, and each domain has size r. Decreasing any of the parameters of (3,2,2) or (2,3,2) yields a problem in which every sequence of improving flips has polynomial length, so both Maximum Constraint Assignment results are tight. All three results settle long-standing open problems. Ou

ALT Data summarization techniques, such as coresets, are essential for scaling clustering algorithms to massive datasets. We study uniform sampling, which is an extremely simple and ultra-efficient method to construct coresets, in the context of weak, stable, and strong coresets for the k-means problem in Hilbert space. We begin with the fundamental case k=1, and establish matching upper and lower bounds on the sample complexity of stable and weak coresets, finding that the two notions require different sample complexities. As a consequence, we obtain tight bounds for stable 1-median coresets in ℓ₁ and ℓ₂ (and thus also Kendall–tau and Jaccard metrics), removing the dimension dependence of Carmel and Krauthgamer (ICLR 2026). Our analysis is substantially simpler than prior work and reduces to bounding the deviation of the sample mean. For the general k-means problem on ρ-separated instances, a well-studied structural assumption on the input, we show that uniform sampling yields a stable co

ALT We introduce peak-aware one-way trading (PA-OWT), an online problem in which a player must purchase a fixed quantity of an asset by a deadline, minimizing its total cost plus a charge proportional to the largest single purchase at any time. The model is motivated by electricity procurement, where the cost of energy rewards buying at low prices, but peak demand charges discourage concentrating purchases in short periods. Balancing these incentives is challenging when future prices are unknown: a large purchase can establish a peak that later decisions cannot undo. We develop a peak-aware online algorithm that balances price and peak control with deadline feasibility and achieves the order-optimal competitive ratio. We further extend our results to a learning-augmented algorithm that uses predictions of optimal purchasing decisions to achieve an order-optimal consistency-robustness tradeoff, exploiting accurate advice while retaining worst-case guarantees in the face of arbitrary predict

ALT We consider the class of open addressing hash tables that uses a fixed offset sequence r₀,r₁,…,rₙ₋₁ to find an open slot in the table. As special cases this subsumes linear probing (rᵢ=i) and quadratic probing (rᵢ=i²). We give a simple analysis that the expected insertion time when the load factor is 1-∊ is O(∊⁻²). This bound is tight for linear probing (but not for quadratic probing), so the analysis cannot be improved at this level of generality.

ALT The delayed deterministic finite automaton (D²FA) is a powerful tool for compressing the prohibitively large deterministic finite automata (DFAs) used in modern signature-based deep packet inspection systems. A D²FA augments a DFA with unlabeled default transitions to remove transitions that are similar. This technique works well in practice and has inspired numerous subsequent studies. However, existing approaches do not guarantee a D²FA of optimal size as they restrict the default transitions to form a forest. In this paper, we consider the problem of finding an optimal D²FA. We make the key observation that cycles of default transitions can be valid and produce a smaller D²FA, provided that for every character in the alphabet, at least one state in the cycle has an outgoing transition labeled by that character. This observation raises fundamental questions concerning the computational complexity of finding an optimal D²FA and the approximation guarantee of the standard (forest-restr

ALT A proper vertex coloring assigns colors to vertices so that adjacent vertices have distinct colors. A Kempe change in a proper vertex coloring exchanges two colors on one component of the subgraph induced by the vertices of those colors. For a fixed palette of k colors, textscKempe Reachability asks whether two given proper vertex colorings of the same graph are connected by a sequence of Kempe changes. This problem is known to be PSPACE-complete for every fixed k≥ 3. A graph is claw-free if no vertex has three pairwise nonadjacent neighbors. The line graph of a graph has one vertex for each edge, with two vertices adjacent exactly when the corresponding edges share an end. Line graphs are claw-free, and their Kempe changes correspond to edge-Kempe changes in the original graph, which exchange two colors on one component of the subgraph formed by the edges of those colors. A claw-free graph with a proper k-coloring has maximum degree at most 2k-2, whereas for k≥ 4 all proper k-coloring

ALT A k-path vertex cover (k-PVC) of a graph is a set of vertices meeting every path on k vertices. In Token Sliding, one token occupies each vertex of a cover; a move slides one token along an edge to an unoccupied vertex, preserving the cover property. We study whether two covers of a tree are joined by such moves. The case k=2 admits a linear-time decision algorithm. For every integer k≥2, we give an O(kn²)-time decision algorithm on trees of order n, with k part of the input. Two covers are reachable precisely when their rigid tokens, which never move, occupy the same set and every component remaining after their deletion has the same token count. The proof transforms configurations of equal size without rigid tokens into a canonical configuration built on properly rooted subtrees, whose paths on k vertices pass through their roots. A rooted-subtree recurrence identifies movable candidate tokens. We also give a direct recursive construction between the supplied endpoints, returning at

ALT We study the classic k-median problem and refine the iterative rounding framework recently introduced by Byrka, Guo, Hu, Li, Wan, and Wang [FOCS'26], which yields a (2+ε)-approximation. Our algorithm opens k+O(1) facilities and achieves an expected connection cost of at most 1.9999 times the value of the standard LP relaxation. Although the numerical improvement over 2 is small, it establishes a conceptual result: the integrality gap of the standard relaxation is strictly below 2 when a constant number of extra facilities is allowed. This partially answers an open question of Li and Svensson [STOC'13]. Combined with their reduction, our result yields a 1.9999-approximation algorithm for k-median. Our approach builds on several technical ingredients. First, we develop a preprocessing routine that makes the LP solution poly(ε)-integral, at the cost of a (1+ε) factor in the connection cost and O(1) extra facilities. Unlike the preprocessing of Byrka et al. [FOCS'26], our conversion is unc

ALT The Equal Subset Sum problem asks, given n positive integers bounded by M, to find two distinct non-empty subsets with the same sum. We study this problem in the regime M = 2ᵒ⁽ⁿ⁾, where the pigeonhole principle guarantees that a solution exists, and the main challenge is to find one efficiently. A straightforward dynamic-programming algorithm runs in time poly(n).M. After a sequence of improvements, the current best algorithm, due to Jin, Williams, and Zhang (ESA 2025), achieves running time poly(n).M¹/³. In this work, we give the first poly(n).Mᵒ⁽¹⁾-time algorithm for Equal Subset Sum in the regime M = 2ᵒ⁽ⁿ⁾. We complement this algorithmic result with a tight conditional lower bound showing that any substantial improvement would require a major breakthrough in approximation algorithms for lattice problems, building on recent techniques of Vafa and Vaikuntanathan (STOC 2025). Finally, we show that our techniques yield an asymptotically faster algorithm for the regime when M = Θ(2ⁿ): a

ALT Let \F be a finite field. Given N distinct points α₁,…,α_N ∈ \F^m and associated values β₁,…,β_N ∈ \F, we study the problem of computing an m-variate polynomial f(\vy) ∈ \F[\vy] of individual degree less than d, with d^m ≥ N, that satisfies f(α_i)=β_i for all i∈[N], whenever such an f exists. This is known as the multivariate polynomial interpolation problem. In this work, given N=d^m, we present a deterministic algorithm for multivariate polynomial interpolation over an arbitrary finite field \F that runs in near-quadratic time N²+o(1)poly(m,d,log|\F|), assuming d is ω(1). This bound holds whenever the associated multivariate Vandermonde matrix is nonsingular, has constant co-rank, or admits some special structure. In the general case, we give a randomized Las Vegas algorithm attaining the same near-quadratic running time, provided the finite field is sufficiently large and d grows similar to the previous case. Unlike other multivariate interpolation strategies, we do not assume black