Akitoshi Kawamura

Akitoshi Kawamura

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.

Akitoshi Kawamura, Norbert Th. Müller, Carsten Rösnick et al.

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.

Akitoshi Kawamura, Martin Ziegler

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.

Akitoshi Kawamura, Yusuke Kobayashi

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.