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

ALT We study the mixing time of two local Markov chains for uniformly sampling proper edge colorings of trees. For a tree with n vertices and maximum degree Δ, we establish optimal mixing-time bounds for the single-edge Glauber dynamics when q≥Δ+2 and for the neighboring-edge heat-bath dynamics when q≥Δ+1. Both chains mix in OΔ(n(log n+log(1/ε))) steps, uniformly over the number of colors q. These bounds are tight in their dependence on n for fixed Δ. Our proof establishes log-Sobolev inequalities through local color release, reusable recoloring buffers, and component entropy mixing. The two dynamics require different local constructions: a two-spare-color buffer for single-edge updates and a second-layer controller for neighboring-edge updates. The resulting entropy analysis applies to general bounded-degree trees, including regimes where worst-case spatial mixing fails.

ALT An online tracker retains one state and may reset only to the current observation, although its complete observed history is available. Offline knows the future and obeys the same reset rule. Under a hard discrepancy radius, for every horizon T≥5 and every feasible randomized causal policy, a fixed path on a known five-point metric requires at least (T+1)/2 expected refreshes, against exactly two offline refreshes with a strict radius margin. Missing the useful prefix center forces a refresh at every suffix observation. The path visits four values, with resolution fixed as T grows. Write Jλ for cumulative discrepancy plus a charge λ>0 per refresh, and V_T for the sum of successive observation distances. A candidate prefix adapts Bienkowski and Schmid's randomized lower bound to this objective under current resets, even with a known horizon and V_T<3λ. This construction needs finer grids; a fixed finite metric with positive minimum separation instead admits a competitive policy for pena

ALT We study the paging with time windows problem, a natural generalization of classical paging, in which each request must be served within a specified time interval. This model allows the algorithm to delay service and batch service requests over time, but also introduces significant challenges due to overlapping requests. While the problem is well understood in the unweighted setting, where optimal O(log k)-competitive algorithms are known, the weighted case remains more elusive. The best previously known algorithm, due to Gupta, Kumar, and Panigrahi, achieves an O(log k log n)-competitive ratio, leaving a gap between weighted and unweighted settings. We obtain an O(log k)-competitive algorithm for weighted paging with time windows, matching the optimal competitive ratio. The core of our approach is a structural decomposition that separates the evolution of the cache from the scheduling of page loads. We show that, given a sequence of cache configurations, computing a near-optimal seque

ALT Temporal graphs are graphs whose edge relation changes in discrete time steps. The Temporal Exploration Problem (TEXP) asks for a temporal walk that visits all vertices. We show that every connected temporal graph with n vertices can be explored in O(qnlog³(n)) time steps, where q is the number of vertex color classes induced by the 2-dimensional Weisfeiler-Leman (2-WL) coloring. For small q, our bound substantially improves on the known worst-case bound of Θ(n²) in general connected temporal graphs. A corresponding walk can be constructed in deterministic polynomial time. The upper bound is complemented by lower bounds of Ω(nlog(n)) when q is constant and Ω(qn) when 1 ≤ q ≤ n. We also apply our techniques to the Temporal Rendezvous Problem (TRP), in which two agents that do not communicate have to meet on a vertex of the temporal graph. Our exploration algorithm yields that, with only polynomial-time computational power, the agents can meet in O(nlog³(n)) time steps, leaving a gap of

ALT The non-preemptive dial-a-ride problem (DaRP) asks for a minimum-cost route of a capacitated vehicle that serves a collection of pickup-and-delivery requests, without allowing objects to be transferred or temporarily unloaded. We study this problem and the closely related minimum-ratio Steiner k-Forest (SkF) in general symmetric metrics on n locations. We develop a randomized polynomial-time algorithm with expected approximation ratio O(n¹/4log¹1n) for the minimum-ratio SkF. Our algorithm combines a deterministic approximation on hierarchically well-separated trees with randomized metric embeddings. Using this algorithm, we obtain expected O(n¹/4log¹2n)-approximations for both the non-preemptive DaRP and the SkF. These guarantees improve on the previous n⁰.3+ε-approximation ratios for the SkF and the non-preemptive DaRP, where ε>0 is any fixed constant. Complementing these algorithmic results, we establish the first superconstant inapproximability bounds for the non-preemptive DaRP. Un

ALT Power grids use a constantly changing mix of fossil fuels and clean renewables like solar and wind. Because of this, the carbon footprint fluctuates drastically based on time and geographic location. Time-varying carbon intensity creates opportunities to reduce carbon emissions by shifting workloads to cleaner periods, but deadlines, task dependencies, and limited edge capacity constrain this flexibility. We study carbon-aware scheduling of jobs with release times, deadlines, and arbitrary precedence constraints under piecewise-constant carbon costs. Jobs execute preemptively on a single edge server or are outsourced at job-dependent costs to an unlimited-capacity cloud, modeled as providing instantaneous execution. The objective is to minimize the sum of edge carbon costs and cloud outsourcing costs while satisfying all timing and precedence constraints. For edge-only execution, we give an exact polynomial-time algorithm. Allowing cloud outsourcing makes the problem NP-hard, but we ob

ALT The k-center problem is one of the most fundamental clustering problems and has applications in a wide range of areas, including facility location, network design, data mining, and machine learning. Given a set of objects together with a distance measure, the goal is to select k centers such that the maximum distance from any object to its closest center is minimized. The k-center problem has been studied extensively from a theoretical perspective, both in metric spaces and on graphs. While there have been experimental studies for k-center clustering in metric spaces, a comprehensive experimental comparison of graph-based k-center algorithms on large-scale graphs is still lacking. In this paper, we bridge this gap by evaluating a set of algorithms ranging from classical approaches such as Gonzalez's greedy algorithm [TCS 1985], an algorithm based on the framework established by Hochbaum and Shmoys [Math. Oper. Res. 1985], to more recent methods, including algorithms by Thorup [SICOMP 2

ALT Counting small patterns is a core primitive in large-scale graph analytics, with numerous applications in data management and network analysis. As modern graphs are massive and continuously evolving, maintaining such counts efficiently under updates is essential. Although specialized dynamic algorithms for patterns with up to four vertices exist, they remain largely theoretical; practitioners instead rely on general-purpose or approximate methods. We present an in-depth study of two dynamic approaches by Eppstein et al. and Hanauer et al. for maintaining counts of triangles and all connected four-vertex subgraph patterns, both based on variants of heavy-light vertex partitioning. We introduce theoretical and practical optimizations and develop hybrid variants that combine complementary strengths of both approaches. Experiments on 56 large real-world and synthetic dynamic graphs show that our optimized algorithms outperform the original methods and strong static and dynamic baselines on