arXiv subjects

Akitoshi Kawamura

Publications and source records attributed to Akitoshi Kawamura.

At least 19 recordsLinked to original sources

arXiv2026

Proof of the Density Threshold Conjecture for Pinwheel Scheduling

In the pinwheel scheduling problem, each task $i$ is associated with a positive integer $a_i$ called its period, and we want to (perpetually) schedule one task per day so that each task $i$ is performed at least once every $a_i$ days. An obvious necessary condition for schedulability is that the density, defined as the sum of execution rates $1/a_i$, does not exceed $1$. We prove that all instances with density not exceeding $5/6$ are schedulable, as was conjectured by Chan and Chin in 1993. Like some of the known partial progress towards the conjecture, our proof involves computer search for schedules for a large but finite set of instances. A key idea in our reduction to these finite cases is to generalize the problem to fractional (non-integer) periods in an appropriate way. As byproducts of our ideas, we obtain a simple proof that every instance with two distinct periods and density at most $1$ is schedulable, as well as a fast algorithm for the bamboo garden trimming problem with approximation ratio $4/3$.

cs.DM↗

arXiv2025

A Computer-Assisted Proof of the Optimal Density Bound for Pinwheel Covering

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 $\alpha^* = 1.264\ldots$ such that every instance with $\sum_{i} 1 / a_i \ge \alpha^*$ 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 G\k{a}sieniec, Smith, and Wild (2022). The same result was obtained independently by Mishra (2026).

cs.DM↗

arXiv2025

The Ultimate Signs of Second-Order Holonomic Sequences

A real-valued sequence $f = \{ f(n) \}_{n \in \mathbb{N}}$ 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 $P, Q \in \mathbb{R}(x)$ 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 the 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.

cs.DM↗

arXiv2021

Weight Balancing on Boundaries

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, that is, whose midpoint is the origin. We prove three generalizations of this fact. (1) For any polygon (or any compact planar set) 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 compact set containing the origin, there exist three points that form an equilateral triangle centered at the origin. (3) For any $d$-dimensional bounded convex polyhedron containing the origin, there exists a pair of antipodal points consisting of a point on a $\lfloor d/2 \rfloor$-face and a point on a $\lceil d/2\rceil$-face.

cs.CG↗

arXiv2021

Simple strategies versus optimal schedules in multi-agent patrolling

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.

cs.DS↗

arXiv2018

Representation Theory of Compact Metric Spaces and Computational Complexity of Continuous Data

Choosing an encoding over binary strings for input/output to/by a Turing Machine is usually straightforward and/or inessential for discrete data (like graphs), but delicate -- heavily affecting computability and even more computational complexity -- already regarding real numbers, not to mention more advanced (e.g. Sobolev) spaces. For a general theory of computational complexity over continuous data we introduce and justify QUANTITATIVE admissibility as requirement for sensible encodings of arbitrary compact metric spaces, a refinement of qualitative 'admissibility' due to [Kreitz&Weihrauch'85]: An admissible representation of a T0 space $X$ is a (i) continuous partial surjective mapping from the Cantor space of infinite binary sequences which is (ii) maximal w.r.t. continuous reduction. By the Kreitz-Weihrauch (aka "Main") Theorem of computability over continuous data, for fixed spaces $X,Y$ equipped with admissible representations, a function $f:X\to Y$ is continuous iff it admits continuous a code-translating mapping on Cantor space, a so-called REALIZER. We define a QUANTITATIVELY admissible representation of a compact metric space $X$ to have (i) asymptotically optimal modulus of continuity, namely close to the entropy of $X$, and (ii) be maximal w.r.t. reduction having optimal modulus of continuity in a similar sense. Careful constructions show the category of such representations to be Cartesian closed, and non-empty: every compact $X$ admits a linearly-admissible representation. Moreover such representations give rise to a tight quantitative correspondence between the modulus of continuity of a function $f:X\to Y$ on the one hand and on the other hand that of its realizer: the MAIN THEOREM of computational complexity. This suggests (how) to take into account the entropies of the spaces under consideration when measuring algorithmic cost over continuous data.

cs.LO↗

arXiv2018

Invitation to Real Complexity Theory: Algorithmic Foundations to Reliable Numerics with Bit-Costs

While concepts and tools from Theoretical Computer Science are regularly applied to, and significantly support, software development for discrete problems, Numerical Engineering largely employs recipes and methods whose correctness and efficiency is demonstrated empirically. We advertise REAL COMPLEXITY THEORY: a resource-oriented foundation to rigorous computations over continuous universes such as real numbers, vectors, sequences, continuous functions, and Euclidean subsets: in the bit-model by approximation up to given absolute error. It offers sound semantics (e.g. of comparisons/tests), closure under composition, realistic runtime predictions, and proofs of algorithmic optimality by relating to known classes like NP, #P, PSPACE.

cs.CC↗

arXiv2017

Polynomial running times for polynomial-time oracle machines

This paper introduces a more restrictive notion of feasibility of functionals on Baire space than the established one from second-order complexity theory. Thereby making it possible to consider functions on the natural numbers as running times of oracle Turing machines and avoiding second-order polynomials, which are notoriously difficult to handle. Furthermore, all machines that witness this stronger kind of feasibility can be clocked and the different traditions of treating partial operators from computable analysis and second-order complexity theory are equated in a precise sense. The new notion is named "strong polynomial-time computability", and proven to be a strictly stronger requirement than polynomial-time computability. It is proven that within the framework for complexity of operators from analysis introduced by Kawamura and Cook the classes of strongly polynomial-time computable operators and polynomial-time computable operators coincide.

cs.CC↗

arXiv2015

Function spaces for second-order polynomial time

In the context of second-order polynomial-time computability, we prove that there is no general function space construction. We proceed to identify restrictions on the domain or the codomain that do provide a function space with polynomial-time function evaluation containing all polynomial-time computable functions of that type. As side results we show that a polynomial-time counterpart to admissibility of a representation is not a suitable criterion for natural representations, and that the Weihrauch degrees embed into the polynomial-time Weihrauch degrees.

cs.CC↗

arXiv2014

Fence patrolling by mobile agents with distinct speeds

Suppose we want to patrol a fence (line segment) using k mobile agents with given speeds v_1, ..., v_k so that every point on the fence is visited by an agent at least once in every unit time period. Czyzowicz et al. conjectured that the maximum length of the fence that can be patrolled is (v_1 + ... + v_k)/2, which is achieved by the simple strategy where each agent i moves back and forth in a segment of length v_i/2. We disprove this conjecture by a counterexample involving k = 6 agents. We also show that the conjecture is true for k = 2, 3.

cs.CG↗

arXiv2014

A lower bound on opaque sets

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 established by Jones in 1964. A similar bound is proved for all convex sets U other than a triangle.

cs.CG↗

arXiv2014

Computational Complexity of Smooth Differential Equations

The computational complexity of the solutions $h$ to the ordinary differential equation $h(0)=0$, $h'(t) = g(t, h(t))$ under various assumptions on the function $g$ has been investigated. Kawamura showed in 2010 that the solution $h$ can be PSPACE-hard even if $g$ is assumed to be Lipschitz continuous and polynomial-time computable. We place further requirements on the smoothness of $g$ and obtain the following results: the solution $h$ can still be PSPACE-hard if $g$ is assumed to be of class $C^1$; for each $k\ge2$, the solution $h$ can be hard for the counting hierarchy even if $g$ is of class $C^k$.

cs.CC↗

arXiv2014

Morpion Solitaire 5D: a new upper bound of 121 on the maximum score

Morpion Solitaire is a pencil-and-paper game for a single player. A move in this game consists of putting a cross at a lattice point and then drawing a line segment that passes through exactly five consecutive crosses. The objective is to make as many moves as possible, starting from a standard initial configuration of crosses. For one of the variants of this game, called 5D, we prove an upper bound of 121 on the number of moves. This is done by introducing line-based analysis, and improves the known upper bound of 138 obtained by potential-based analysis.

cs.DM↗

arXiv2013

Complexity Theory for Operators in Analysis

We propose an extension of the framework for discussing the 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 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 advantage of using string 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 string 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. Because our framework separates machine computation and semantics, it can be applied to problems on sets of interest in analysis once we specify a suitable representation (encoding). As prototype applications, we consider the complexity of functions (operators) on real numbers, real sets, and real functions. For example, the task of numerical algorithms for solving a certain class of differential equations is naturally viewed as an operator taking real functions to real functions. As there was no complexity theory for operators, previous results only stated how complex the solution can be. We now reformulate them and show that the operator itself is polynomial-space complete.

cs.CC↗

arXiv2012

Parameterized Uniform Complexity in Numerics: from Smooth to Analytic, from NP-hard to Polytime

The synthesis of classical Computational Complexity Theory with Recursive Analysis provides a quantitative foundation to reliable numerics. Here the operators of maximization, integration, and solving ordinary differential equations are known to map (even high-order differentiable) polynomial-time computable functions to instances which are `hard' for classical complexity classes NP, #P, and CH; but, restricted to analytic functions, map polynomial-time computable ones to polynomial-time computable ones -- non-uniformly! We investigate the uniform parameterized complexity of the above operators in the setting of Weihrauch's TTE and its second-order extension due to Kawamura&Cook (2010). That is, we explore which (both continuous and discrete, first and second order) information and parameters on some given f is sufficient to obtain similar data on Max(f) and int(f); and within what running time, in terms of these parameters and the guaranteed output precision 2^(-n). It turns out that Gevrey's hierarchy of functions climbing from analytic to smooth corresponds to the computational complexity of maximization growing from polytime to NP-hard. Proof techniques involve mainly the Theory of (discrete) Computation, Hard Analysis, and Information-Based Complexity.

math.NA↗

arXiv2010

Generalized Semimagic Squares for Digital Halftoning

Completing Aronov et al.'s study on zero-discrepancy matrices for digital halftoning, we determine all (m, n, k, l) for which it is possible to put mn consecutive integers on an m-by-n board (with wrap-around) so that each k-by-l region holds the same sum. For one of the cases where this is impossible, we give a heuristic method to find a matrix with small discrepancy.

cs.CG↗

arXiv2010

Lipschitz Continuous Ordinary Differential Equations are Polynomial-Space Complete

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 weak feedback, and show that they are still polynomial-space complete. The same technique also settles Ko's two later questions on Volterra integral equations.

cs.CC↗

arXiv2009

Distance k-Sectors Exist

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 is an integer, is a (k-1)-tuple (C_1, C_2, ..., C_{k-1}) such that C_i is the bisector of C_{i-1} and C_{i+1} for every i = 1, 2, ..., k-1, where C_0 = P and C_k = 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.

cs.CG↗