Rasmus Ibsen-Jensen

The complexity of solving reachability games using value and strategy iteration

Two standard algorithms for approximately solving two-player zerosum concurrent reachability games are value iteration and strategy iteration. We prove upper and lower bounds of 2mΘ(N) on the worst case number of iterations needed for both of these algorithms to provide non-trivial approximations to the value of a game with N non-terminal positions and m actions for each player in each position.

Computational complexity of ecological and evolutionary spatial dynamics

Significance An important question in evolution is: how does population structure affect the outcome of the evolutionary process? The theory of evolution in structured population has provided an impressive range of results, but an understanding of the computational complexity of even simple questions was missing. We prove that some fundamental problems in ecology and evolution can be precisely characterized by well-established computational complexity classes. This implies that the problems cannot be answered by simple equations. For example, there cannot be simple formulas for the fixation probability of a mutant given frequency-dependent selection in a structured population. We also show that, for example, calculating the molecular clock of neutral evolution in structured populations admit efficient algorithmic solutions. There are deep, yet largely unexplored, connections between computer science and biology. Both disciplines examine how information proliferates in time and space. Central results in computer science describe the complexity of algorithms that solve certain classes of problems. An algorithm is deemed efficient if it can solve a problem in polynomial time, which means the running time of the algorithm is a polynomial function of the length of the input. There are classes of harder problems for which the fastest possible algorithm requires exponential time. Another criterion is the space requirement of the algorithm. There is a crucial distinction between algorithms that can find a solution, verify a solution, or list several distinct solutions in given time and space. The complexity hierarchy that is generated in this way is the foundation of theoretical computer science. Precise complexity results can be notoriously difficult. The famous question whether polynomial time equals nondeterministic polynomial time (i.e., P = NP) is one of the hardest open problems in computer science and all of mathematics. Here, we consider simple processes of ecological and evolutionary spatial dynamics. The basic question is: What is the probability that a new invader (or a new mutant) will take over a resident population? We derive precise complexity results for a variety of scenarios. We therefore show that some fundamental questions in this area cannot be answered by simple equations (assuming that P is not equal to NP).

Faster Algorithms for Algebraic Path Properties in Recursive State Machines with Constant Treewidth

Interprocedural analysis is at the heart of numerous applications in programming languages, such as alias analysis, constant propagation, etc. Recursive state machines (RSMs) are standard models for interprocedural analysis. We consider a general framework with RSMs where the transitions are labeled from a semiring, and path properties are algebraic with semiring operations. RSMs with algebraic path properties can model interprocedural dataflow analysis problems, the shortest path problem, the most probable path problem, etc. The traditional algorithms for interprocedural analysis focus on path properties where the starting point is fixed as the entry point of a specific method. In this work, we consider possible multiple queries as required in many applications such as in alias analysis. The study of multiple queries allows us to bring in a very important algorithmic distinction between the resource usage of the one-time preprocessing vs for each individual query. The second aspect that we consider is that the control flow graphs for most programs have constant treewidth. Our main contributions are simple and implementable algorithms that support multiple queries for algebraic path properties for RSMs that have constant treewidth. Our theoretical results show that our algorithms have small additional one-time preprocessing, but can answer subsequent queries significantly faster as compared to the current best-known solutions for several important problems, such as interprocedural reachability and shortest path. We provide a prototype implementation for interprocedural reachability and intraprocedural shortest path that gives a significant speed-up on several benchmarks.

Solving simple stochastic games with few coin toss positions

Gimbert and Horn gave an algorithm for solving simple stochastic games with running time O(r! n) where n is the number of positions of the simple stochastic game and r is the number of its coin toss positions. Chatterjee et al. pointed out that a variant of strategy iteration can be implemented to solve this problem in time 4rnO(1). In this paper, we show that an algorithm combining value iteration with retrograde analysis achieves a time bound of O(r 2r (r logr+n)), thus improving both time bounds. We also improve the analysis of Chatterjee et al. and show that their algorithm in fact has complexity 2rnO(1).

The Complexity of Ergodic Mean-payoff Games

We study two-player (zero-sum) concurrent mean-payoff games played on a finite-state graph. We focus on the important sub-class of ergodic games where all states are visited infinitely often with probability 1. The algorithmic study of ergodic games was initiated in a seminal work of Hoffman and Karp in 1966, but all basic complexity questions have remained unresolved. Our main results for ergodic games are as follows: We establish (1) an optimal exponential bound on the patience of stationary strategies (where patience of a distribution is the inverse of the smallest positive probability and represents a complexity measure of a stationary strategy); (2) the approximation problem lies in FNP; (3) the approximation problem is at least as hard as the decision problem for simple stochastic games (for which NP ∩ coNP is the long-standing best known bound). We present a variant of the strategy-iteration algorithm by Hoffman and Karp; show that both our algorithm and the classical value-iteration algorithm can approximate the value in exponential time; and identify a subclass where the value-iteration algorithm is a FPTAS. We also show that the exact value can be expressed in the existential theory of the reals, and establish square-root sum hardness for a related class of games.

A faster algorithm for solving one-clock priced timed games

One-clock priced timed games is a class of two-player, zero-sum, continuous-time games that was defined and thoroughly studied in previous works. We show that one-clock priced timed games can be solved in time m 12nnO(1), where n is the number of states and m is the number of actions. The best previously known time bound for solving one-clock priced timed games was

Algorithms for algebraic path properties in concurrent systems of constant treewidth components

2^{O(n^2+m)}

Faster Algorithms for Quantitative Verification in Constant Treewidth Graphs

, due to Rutkowski. For our improvement, we introduce and study a new algorithm for solving one-clock priced timed games, based on the sweep-line technique from computational geometry and the strategy iteration paradigm from the algorithmic theory of Markov decision processes. As a corollary, we also improve the analysis of previous algorithms due to Bouyer, Cassez, Fleury, and Larsen; and Alur, Bernadsky, and Madhusudan.

Edit Distance for Pushdown Automata

We study algorithmic questions for concurrent systems where the transitions are labeled from a complete, closed semiring, and path properties are algebraic with semiring operations. The algebraic path properties can model dataflow analysis problems, the shortest path problem, and many other natural problems that arise in program analysis. We consider that each component of the concurrent system is a graph with constant treewidth, a property satisfied by the controlflow graphs of most programs. We allow for multiple possible queries, which arise naturally in demand driven dataflow analysis. The study of multiple queries allows us to consider the tradeoff between the resource usage of the one-time preprocessing and for each individual query. The traditional approach constructs the product graph of all components and applies the best-known graph algorithm on the product. In this approach, even the answer to a single query requires the transitive closure (i.e., the results of all possible queries), which provides no room for tradeoff between preprocessing and query time. Our main contributions are algorithms that significantly improve the worst-case running time of the traditional approach, and provide various tradeoffs depending on the number of queries. For example, in a concurrent system of two components, the traditional approach requires hexic time in the worst case for answering one query as well as computing the transitive closure, whereas we show that with one-time preprocessing in almost cubic time, each subsequent query can be answered in at most linear time, and even the transitive closure can be computed in almost quartic time. Furthermore, we establish conditional optimality results showing that the worst-case running time of our algorithms cannot be improved without achieving major breakthroughs in graph algorithms (i.e., improving the worst-case bound for the shortest path problem in general graphs). Preliminary experimental results show that our algorithms perform favorably on several benchmarks.

Edit distance for timed automata

We consider the core algorithmic problems related to verification of systems with respect to three classical quantitative properties, namely, the mean-payoff property, the ratio property, and the minimum initial credit for energy property. The algorithmic problem given a graph and a quantitative property asks to compute the optimal value (the infimum value over all traces) from every node of the graph. We consider graphs with constant treewidth, and it is well-known that the control-flow graphs of most programs have constant treewidth. Let n denote the number of nodes of a graph, m the number of edges (for constant treewidth graphs \(m=O(n)\)) and W the largest absolute value of the weights. Our main theoretical results are as follows. First, for constant treewidth graphs we present an algorithm that approximates the mean-payoff value within a multiplicative factor of \(\epsilon \) in time \(O(n \cdot \log (n/\epsilon ))\) and linear space, as compared to the classical algorithms that require quadratic time. Second, for the ratio property we present an algorithm that for constant treewidth graphs works in time \(O(n \cdot \log (|a\cdot b|))=O(n\cdot \log (n\cdot W))\), when the output is \(\frac{a}{b}\), as compared to the previously best known algorithm with running time \(O(n^2 \cdot \log (n\cdot W))\). Third, for the minimum initial credit problem we show that (i) for general graphs the problem can be solved in \(O(n^2\cdot m)\) time and the associated decision problem can be solved in \(O(n\cdot m)\) time, improving the previous known \(O(n^3\cdot m\cdot \log (n\cdot W))\) and \(O(n^2 \cdot m)\) bounds, respectively; and (ii) for constant treewidth graphs we present an algorithm that requires \(O(n\cdot \log n)\) time, improving the previous known \(O(n^4 \cdot \log (n \cdot W))\) bound. We have implemented some of our algorithms and show that they present a significant speedup on standard benchmarks. Open image in new window