Peter Bro Miltersen

On the Complexity of Numerical Analysis

We study two quite different approaches to understanding the complexity of fundamental problems in numerical analysis: (a) the Blum-Shub-Smale model of computation over the reals; and (b) a problem we call the “generic task of numerical computation,” which captures an aspect of doing numerical computation in floating point, similar to the “long exponent model” that has been studied in the numerical computing community. We show that both of these approaches hinge on the question of understanding the complexity of the following problem, which we call PosSLP: Given a division-free straight-line program producing an integer

SOFSEM 2009: Theory and Practice of Computer Science

N

Lower bounds for union-split-find related problems on random access machines

, decide whether

Strategy Iteration Is Strongly Polynomial for 2-Player Turn-Based Stochastic Games with a Constant Discount Factor

N>0

On the complexity of numerical analysis

. In the Blum-Shub-Smale model, polynomial-time computation over the reals (on discrete inputs) is polynomial-time equivalent to PosSLP when there are only algebraic constants. We conjecture that using transcendental constants provides no additional power, beyond nonuniform reductions to PosSLP, and we present some preliminary results supporting this conjecture. The generic task of numerical computation is also polynomial-time equivalent to PosSLP. We prove that PosSLP lies in the counting hierarchy. Combining this with work of Tiwari, we obtain that the Euclidean traveling salesman problem lies in the counting hierarchy—the previous best upper bound for this important problem (in terms of classical complexity classes) being PSPACE. In the course of developing the context for our results on arithmetic circuits, we present some new observations on the complexity of the arithmetic circuit identity testing (ACIT) problem. In particular, we show that if

Fusion trees can be implemented with AC 0 instructions only

n!

The Complexity of Solving Stochastic Games on Graphs

is not ultimately easy, then ACIT has subexponential complexity.

Send mixed signals: earn more, work less

SOFSEM 2009: Theory and Practice of Computer Science: 35th Conference on Current Trends in Theory and Practice of Computer Science : Spindlerův Mlýn, Czech Republic, January 2009, Proceedings of the Conference

Computing sequential equilibria for two-player games

We prove Ω(√log log n) lower bounds on the random access machine complexity of several dynamic, partially dynamic and static data structure problems, including the union-split-find problem, dynamic prefix problems and one-dimensional range query problems. The proof techniques include a general technique using perfect hashing for reducing static data structure problems (with a restriction of the size of the structure) into partially dynamic data structure problems (with no such restriction), thus providing a way to transfer lower bounds. We use a generalization of a method due to Ajtai for proving the lower bounds on the static problems, but describe the proof in terms of communication complexity, revealing a striking similarity to the proof used by Karchmer and Wigderson for proving lower bounds on the monotone circuit depth of connectivity.

Linear hash functions

Ye [2011] showed recently that the simplex method with Dantzig’s pivoting rule, as well as Howard’s <i>policy iteration</i> algorithm, solve discounted Markov decision processes (MDPs), with a constant discount factor, in strongly polynomial time. More precisely, Ye showed that both algorithms terminate after at most <i>O</i>(<i>mn</i>1−<i>γ</i> log <i>n</i>1−<i>γ</i>) iterations, where <i>n</i> is the number of states, <i>m</i> is the total number of actions in the MDP, and 0 < <i>γ</i> < 1 is the discount factor. We improve Ye’s analysis in two respects. First, we improve the bound given by Ye and show that Howard’s policy iteration algorithm actually terminates after at most <i>O</i>(<i>m</i>1−<i>γ</i> log <i>n</i>1−<i>γ</i>) iterations. Second, and more importantly, we show that the same bound applies to the number of iterations performed by the <i>strategy iteration</i> (or <i>strategy improvement</i>) algorithm, a generalization of Howard’s policy iteration algorithm used for solving 2-player turn-based <i>stochastic games</i> with discounted zero-sum rewards. This provides the first strongly polynomial algorithm for solving these games, solving a long standing open problem. Combined with other recent results, this provides a complete characterization of the complexity the standard strategy iteration algorithm for 2-player turn-based stochastic games; it is strongly polynomial for a fixed discount factor, and exponential otherwise.