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

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

ALT An approximate distance oracle (ADO) is a compact data structure that answers distance queries within a multiplicative stretch k. The classical oracle of Thorup and Zwick [STOC'01, JACM'05] has stretch k=2ℓ-1 and uses O(ℓ n¹⁺¹/ℓ) space, for every integer ℓ≥1, and this trade-off is optimal for dense graphs under the Erdős girth conjecture. However, for sparse graphs (m=O(n)), their lower bound yields nothing beyond Ω(m) space. A pivot-based 1-hop ADO gives every vertex Õ(1) pivots and returns as an estimate the length of a walk of the form u→ pᵤ → pᵥ → v, where pᵤ is a pivot of u, pᵥ is a pivot of v, and pᵤ→ pᵥ is a stored distance. We observe that all the state-of-the-art ADOs with Õ(1) query time are in fact pivot-based 1-hop ADOs. Our main result is that, under the Erdős girth conjecture for graphs with constant degree, there are sparse graphs on which every pivot-based 1-hop ADO with stretch k requires Ω̃(m¹⁺²/⁽ᵏ⁺¹)) space, for every real k≥1. This matches the best known upper bound

ALT In STOC 1972, Jürg Nievergelt and Edward M. Reingold introduced the BB[α]-tree, also known as a weight-balanced tree, which for any α ∈ (0,¹⁄₂] is a binary search tree where every node with induced subtree size s has children with induced subtree sizes at least α(s+1)-1. A new rotation-based binary search tree, the PopTree, is introduced. The PopTree stores n elements and is a BB[tfrac120]-tree after the execution of each search and a BB[tfrac1729]-tree at all times, notably while rebalancing. For any constant α, its performance on search sequences of length Ω(n log n) is proved to be within a constant factor (which can depend on α) of the optimal offline rotation-based binary search tree that maintains the BB[α] balance invariant.

ALT The Fréchet distance (also known as the dog-leash distance) is a well-studied and prevalent distance measure for polygonal curves. In this paper we present parallel algorithms for computing the discrete frechet{ dis}tance between two curves, in both the PRAM and MPC models. For two polygonal curves of length n, we show that the distance between them can be computed in the PRAM model in O(log n/ σ) rounds and Õ(n²⁺σ) total work for any 0<σ<1. In the sublinear MPC model with nρ space per machine, it can be computed in O(log n / ρ²) rounds and Õ(n²⁺ρ) total space. For c-packed curves, we match our bounds to the sequential algorithms, showing that a (1+\eps)-approximation of the distance can be computed in the PRAM model in O(logn+1/σ³) rounds and Õ(n¹⁺σ) total work, and in the sublinear MPC model in O(1 /ρ³) rounds and Õ(n¹⁺ρ) total space, for any constant \eps>0. As part of our algorithms for c-packed curves, we also present parallel algorithms for computing curve simplifications, which

ALT We study Largest Cardinality Subset Sum (LCSS): given n nonnegative integers and a target t, compute a subset summing to t with maximum cardinality. We also consider its all-targets version (ATLCSS), which asks for this maximum for every sum s∈{1,…,t}. The LCSS problem sits squarely between the subset sum and the knapsack problem. A straightforward dynamic-programming algorithm runs in time O(nt) and solves ATLCSS, and using Jin and Bringmann (both in STOC 2024, independent work) 0-1 knapsack results we achieve a Õ(n + aₘₐₓ²)-time deterministic algorithm for LCSS. Our main result is a randomized Õ(n+√n t)-time algorithm for LCSS. Then, we complement this result with a randomized Õ(n+t³/²)-time algorithm for ATLCSS. Together, our two algorithms give a Õ(n+tmin{√n,√t})-time algorithm for LCSS. As an application of ATLCSS, we solve the search version of Projection Subset Sum, introduced by Dutta and Rajasree (SYNSAC 2023), in Õ(n+t³/²) time for every fixed ℓₚ norm with p≥ 1. For ℓ₁,

ALT We study labeling schemes for p × q Γ-free binary matrices, which do not contain a submatrix of the form Γ = 1 1 1 0. We give a labeling scheme that uses O(log² N)-bit labels for rows and columns, where N=p+q, such that any matrix entry can be determined from the corresponding two labels in constant time in the word RAM model. The worst-case label size matches the counting lower bound up to constant factors. We also show that the entrywise least Γ-free supermatrix of an arbitrary p× q binary matrix, together with its labels, can be constructed in O(pq) time. Our construction represents a Γ-free matrix by a sequence of rooted forests, where each column corresponds to a node and the 1-entries of each row form a root path in its corresponding forest. For n-vertex strongly chordal and chordal bipartite graphs, our result gives asymptotically optimal O(log² n)-bit adjacency labeling schemes with constant-time queries. These results answer both representation questions posed by Spinrad in 19

ALT We study the problem of characterizing the optimal consistency–robustness tradeoff for single-machine non-clairvoyant scheduling with predicted processing times. For a fixed number of jobs n, Wei and Zhang (2020) established a lower bound on the achievable consistency–robustness tradeoff which is known to be tight only in a few restrictive cases, leaving the optimal tradeoff for general n unresolved. A different worst-case-n regime was introduced by Benomar et al. (2025), who applied the former work's construction to show that for λ ∈ (0, 1), any (1+λ)-consistent algorithm must be 1 + Ω(√(1/λ))-robust. On the other hand, the best known upper bound gives (1+λ)-consistency and 1+O(1/λ)-robustness, leaving a polynomial gap. In this work, we develop a new lower bound using a multi-stage adversarial construction and a new algorithm that together give a near-optimal characterization of the consistency–robustness tradeoff: for every n, our upper bound is within a multiplicative factor 3/2 of

ALT In interval graphs and many generalizations thereof, problems such as Matching, Independent Set, and Dominating Set can be solved with simple greedy algorithms in linear (i.e., O(m+n)) time. However, it is sometimes possible to improve on this runtime if a suitable representation as intersection graph is given; for example it is possible to find a minimum dominating set in an interval graph in O(n) time. We study RDV graphs, a generalization of interval graphs, namely, the intersection graphs of directed paths in a rooted tree. It was known that the above three problems can be solved in O(nlog n) time if a suitably small representation is given. In this paper, we reduce the runtime to O(n). Furthermore, while the previous results used non-trivial operations such as range-minimum queries and ray-shooting queries, we need nothing more complicated than lists and (for Matching) a Union-Find data structure.

ALT We study deterministic gossiping in unknown strongly connected directed radio networks with n vertices, m arcs, and distinct identifiers in {1,…,N}, where N=n^O(1). Messages are unrestricted; collision detection is unavailable. Our protocol combines amplified reception certificates, adaptive pruning, and permutation selectors. Each pruning iteration costs O(nlog²nloglog n) rounds. A two-step quadratic contraction limits each receiver and each class of identical closed in-neighborhoods to O(loglog n) qualifying selections. Under an explicit degree-concentration condition, our protocol takes O(mlog²n(loglog n)²) rounds. For directed regular graphs with common in- and out-degree Δ, our strategy achieves Õ(min{n√Δ,n²/Δ}), extending the identifier range of Wu and Chrobak and improving their high-degree bound. At completion entry, let a≥1 count original sources not yet credited as globally disseminated. Permutation selectors reduce our balanced completion budget by a factor of log²/3(a+1) r