In the covering version of the pinwheel scheduling problem, a daily task must be assigned to agents under the constraint that agent i can perform the task at most once in any a_i-day interval. In this paper, we determine the optimal constant α^* = 1.264… such that every instance with ∑_i 1/a_i ≥ α^* is schedulable. This resolves an open problem posed by Kawamura and Soejima (2020). Our proof combines Kawamura’s (2026) techniques for the packing version with new mathematical insights to reduce the analysis to a finite set of instances, which are then verified through an exhaustive computer-aided search that draws on ideas from Gąsieniec, Smith, and Wild (2022). The same result was obtained independently by Mishra (2026).

A real-valued sequence f = {f(n)}_{n ∈ ℕ} is said to be second-order holonomic if it satisfies a linear recurrence f (n + 2) = P (n) f (n + 1) + Q (n) f (n) for all sufficiently large n, where P, Q ∈ ℝ(x) are rational functions. We study the ultimate sign of such a sequence, i.e., the repeated pattern that the signs of f (n) follow for sufficiently large n. For each P, Q we determine all ultimate signs that f can have, and show how they partition the space of initial values of f. This completes the prior work by Neumann, Ouaknine and Worrell, who have settled some restricted cases. As a corollary, it follows that when P, Q have rational coefficients, f either has an ultimate sign of length 1, 2, 3, 4, 6, 8 or 12, or never falls into a repeated sign pattern. We also give a partial algorithm that finds the ultimate sign of f (or tells that there is none) in almost all cases.

Suppose that a set of mobile agents, each with a predefined maximum speed, want to patrol a fence together so as to minimize the longest time interval during which a point on the fence is left unvisited. In 2011, Czyzowicz, Gąsieniec, Kosowski and Kranakis studied this problem for the settings where the fence is an interval (a line segment) and a circle, and conjectured that the following simple strategies are always optimal: for Interval Patrolling, the simple strategy partitions the fence into subintervals, one for each agent, and lets each agent move back and forth in the assigned subinterval with its maximum speed; for Circle Patrolling, the simple strategy is to choose a number r, place the r fastest agents equidistantly around the circle, and move them at the speed of the rth agent. Surprisingly, these conjectures were then proved false: schedules were found (for some settings of maximum speeds) that slightly outperform the simple strategies. In this paper, we are interested in the ratio between the performances of optimal schedules and simple strategies. For the two problems, we construct schedules that are 4/3 times (for Interval Patrolling) and 21/20 times (for Circle Patrolling) as good, respectively, as the simple strategies. We also propose a new variant, in which we want to patrol a single point under the constraint that each agent can only visit the point some predefined time after its previous visit. We obtain some similar ratio bounds and NP-hardness results related to this problem.

We apply average-case complexity theory to physical problems modeled by continuous-time dynamical systems. The computational complexity when simulating such systems for a bounded time-frame mainly stems from trajectories coming close to complex singularities of the system. We show that if for most initial values the trajectories do not come close to singularities the simulation can be done in polynomial time on average. For Hamiltonian systems we relate this to the volume of "almost singularities" in phase space and give some general criteria to show that a Hamiltonian system can be simulated efficiently on average. As an application we show that the planar circular-restricted three-body problem is average-case polynomial-time computable.

The last years have seen an increasing interest in classifying (existence claims in) classical mathematical theorems according to their strength. We pursue this goal from the refined perspective of computational complexity. Specifically, we establish that rigorously solving the Dirichlet Problem for Poisson's Equation is in a precise sense ‘complete’ for the complexity class ${\#\mathcal{P}}$ and thus as hard or easy as parametric Riemann integration (Friedman 1984; Ko 1991.Complexity Theory of Real Functions).

It is proved that the total length of any set of countably many rectifiable curves, whose union meets all straight lines that intersect the unit square U, is at least 2.00002. This is the first improvement on the lower bound of 2 by Jones in 1964. A similar bound is proved for all convex sets U other than a triangle.

We promote the theory of computational complexity on metric spaces: as natural common generalization of (i) the classical discrete setting of integers, binary strings, graphs etc. as well as of (ii) the bit-complexity theory on real numbers and functions according to Friedman, Ko (1982ff), Cook, Braverman et al.; as (iii) resource-bounded refinement of the theories of computability on, and representations of, continuous universes by Pour-El&Richards (1989) and Weihrauch (1993ff); and as (iv) computational perspective on quantitative concepts from classical Analysis: Our main results relate (i.e. upper and lower bound) Kolmogorov's entropy of a compact metric space X polynomially to the uniform relativized complexity of approximating various families of continuous functions on X. The upper bounds are attained by carefully crafted oracles and bit-cost analyses of algorithms perusing them. They all employ the same representation (i.e. encoding, as infinite binary sequences, of the elements) of such spaces, which thus may be of own interest. The lower bounds adapt adversary arguments from unit-cost Information-Based Complexity to the bit model. They extend to, and indicate perhaps surprising limitations even of, encodings via binary string functions (rather than sequences) as introduced by Kawamura&Cook (SToC'2010, §3.4). These insights offer some guidance towards suitable notions of complexity for higher types.

The synthesis of (discrete) Complexity Theory with Recursive Analysis provides a quantitative algorithmic foundation to calculations over real numbers, sequences, and functions by approximation up to prescribable absolute error 1/2n (roughly corresponding to n binary digits after the radix point). In this sense Friedman and Ko have shown the seemingly simple operators of maximization and integration 'complete' for the standard complexity classes NP and #P — even when restricted to smooth (=C∞) arguments. Analytic polynomial-time computable functions on the other hand are known to get mapped to polynomial-time computable functions: non-uniformly, that is, disregarding dependences other than on the output precision n. The present work investigates the uniform parameterized complexity of natural operators Λ on subclasses of smooth functions: evaluation, pointwise addition and multiplication, (iterated) differentiation, integration, and maximization. We identify natural integer parameters k=k(f) which, when given as enrichment to approximations to the function argument f, permit to computably produce approximations to Λ(f); and we explore the asymptotic worst-case running time sufficient and necessary for such computations in terms of the output precision n and said k. It turns out that Maurice Gevrey's 1918 classical hierarchy climbing from analytic to (just below) smooth functions provides for a quantitative gauge of the uniform computational complexity of maximization and integration that, non-uniformly, exhibits the phase transition from tractable (i.e. polynomial-time) to intractable (in the sense of NP-'hardness'). Our proof methods involve Hard Analysis, Approximation Theory, and an adaptation of Information-Based Complexity to the bit model.

Given a polygonal region containing a target point (which we assume is the origin), it is not hard to see that there are two points on the perimeter that are antipodal, i.e., whose midpoint is the origin. We prove three generalizations of this fact. (1) For any polygon (or any bounded closed region with connected boundary) containing the origin, it is possible to place a given set of weights on the boundary so that their barycenter (center of mass) coincides with the origin, provided that the largest weight does not exceed the sum of the other weights. (2) On the boundary of any 3-dimensional bounded polyhedron containing the origin, there exist three points that form an equilateral triangle centered at the origin. (3) On the 1-skeleton of any 3-dimensional bounded convex polyhedron containing the origin, there exist three points whose center of mass coincides with the origin.

The bisector of two nonempty sets P and Q in a metric space is the set of all points with equal distance to P and to Q. A distance k-sector of P and Q, where k ≥ 2 is an integer, is a (k-1)-tuple (C1, C2, ..., Ck-1) such that Ci is the bisector of Ci-1 and Ci+1 for every i= 1, 2, ..., k-1, where C0 = P and Ck = Q. This notion, for the case where P and Q are points in Euclidean plane, was introduced by Asano, Matousek, and Tokuyama, motivated by a question of Murata in VLSI design. They established the existence and uniqueness of the distance trisector in this special case. We prove the existence of a distance k-sector for all k and for every two disjoint, nonempty, closed sets P and Q in Euclidean spaces of any (finite) dimension, or more generally, in proper geodesic spaces (uniqueness remains open). The core of the proof is a new notion of k-gradation for P and Q, whose existence (even in an arbitrary metric space) is proved using the Knaster-Tarski fixed point theorem, by a method introduced by Reem and Reich for a slightly different purpose.

Zone diagram is a variation on the classical concept of a Voronoi diagram. Given n sites in a metric space that compete for territory, the zone diagram is an equilibrium state in the competition. Formally it is defined as a fixed point of a certain "dominance" map.

We propose a new framework for discussing computational complexity of problems involving uncountably many objects, such as real numbers, sets and functions, that can be represented only through approximation. The key idea is to use a certain class of string functions, which we call regular functions, as names representing these objects. These are more expressive than infinite sequences, which served as names in prior work that formulated complexity in more restricted settings. An important advantage of using regular functions is that we can define their size in the way inspired by higher-type complexity theory. This enables us to talk about computation on regular functions whose time or space is bounded polynomially in the input size, giving rise to more general analogues of the classes P, NP, and PSPACE. We also define NP- and PSPACE-completeness under suitable many-one reductions.

In answer to Ko's question raised in 1983, we show that an initial value problem given by a polynomial-time computable, Lipschitz continuous function can have a polynomial-space complete solution. The key insight is simple: the Lipschitz condition means that the feedback in the differential equation is weak. We define a class of polynomial-space computation tableaux with equally restricted feedback, and show that they are still polynomial-space complete. The same technique also settles Ko's two later questions on Volterra integral equations.

We present a redevelopment of the theory of real-valued recursive functions that was introduced by C. Moore in 1996 by analogy with the standard formulation of the integer-valued recursive functions. While his work opened a new line of research on analog computation, the original paper contained some technical inaccuracies. We discuss possible attempts to remove the ambiguity in the behavior of the operators on partial functions, with a focus on his “primitive recursive” functions generated by the differential recursion operator that solves initial value problems. Under a reasonable reformulation, the functions in this class are shown to be analytic and computable in a strong sense in computable analysis. Despite this well-behavedness, the class turns out to be too big to have the originally purported relation to differentially algebraic functions, and hence to C. E. Shannon's model of analog computation.