arXiv subjects

Rasmus Ibsen-Jensen

Publications and source records attributed to Rasmus Ibsen-Jensen.

30 records · Page 2Linked to original sources

arXiv2015

Faster Algorithms for Quantitative Verification in Constant Treewidth Graphs

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 $ε$ in time $O(n \cdot \log (n/ε))$ 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.

cs.DS↗

arXiv2014

The complexity of interior point methods for solving discounted turn-based stochastic games

We study the problem of solving discounted, two player, turn based, stochastic games (2TBSGs). Jurdzinski and Savani showed that 2TBSGs with deterministic transitions can be reduced to solving $P$-matrix linear complementarity problems (LCPs). We show that the same reduction works for general 2TBSGs. This implies that a number of interior point methods for solving $P$-matrix LCPs can be used to solve 2TBSGs. We consider two such algorithms. First, we consider the unified interior point method of Kojima, Megiddo, Noma, and Yoshise, which runs in time $O((1+κ)n^{3.5}L)$, where $κ$ is a parameter that depends on the $n \times n$ matrix $M$ defining the LCP, and $L$ is the number of bits in the representation of $M$. Second, we consider the interior point potential reduction algorithm of Kojima, Megiddo, and Ye, which runs in time $O(\frac{-δ}θn^4\log ε^{-1})$, where $δ$ and $θ$ are parameters that depend on $M$, and $ε$ describes the quality of the solution. For 2TBSGs with $n$ states and discount factor $γ$ we prove that in the worst case $κ= Θ(n/(1-γ)^2)$, $-δ= Θ(\sqrt{n}/(1-γ))$, and $1/θ= Θ(n/(1-γ)^2)$. The lower bounds for $κ$, $-δ$, and $1/θ$ are obtained using the same family of deterministic games.

cs.GT↗

arXiv2014

Faster Algorithms for Algebraic Path Properties in RSMs 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 \emph{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 \emph{one-time} preprocessing vs for \emph{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.

cs.PL↗

arXiv2014

The Value 1 Problem Under Finite-memory Strategies for Concurrent Mean-payoff Games

We consider concurrent mean-payoff games, a very well-studied class of two-player (player 1 vs player 2) zero-sum games on finite-state graphs where every transition is assigned a reward between 0 and 1, and the payoff function is the long-run average of the rewards. The value is the maximal expected payoff that player 1 can guarantee against all strategies of player 2. We consider the computation of the set of states with value 1 under finite-memory strategies for player 1, and our main results for the problem are as follows: (1) we present a polynomial-time algorithm; (2) we show that whenever there is a finite-memory strategy, there is a stationary strategy that does not need memory at all; and (3) we present an optimal bound (which is double exponential) 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).

cs.GT↗

arXiv2014

Qualitative Analysis of Concurrent Mean-payoff Games

We consider concurrent games played by two-players on a finite-state graph, where in every round the players simultaneously choose a move, and the current state along with the joint moves determine the successor state. We study a fundamental objective, namely, mean-payoff objective, where a reward is associated to each transition, and the goal of player 1 is to maximize the long-run average of the rewards, and the objective of player 2 is strictly the opposite. The path constraint for player 1 could be qualitative, i.e., the mean-payoff is the maximal reward, or arbitrarily close to it; or quantitative, i.e., a given threshold between the minimal and maximal reward. We consider the computation of the almost-sure (resp. positive) winning sets, where player 1 can ensure that the path constraint is satisfied with probability 1 (resp. positive probability). Our main results for qualitative path constraints are as follows: (1) we establish qualitative determinacy results that show that for every state either player 1 has a strategy to ensure almost-sure (resp. positive) winning against all player-2 strategies, or player 2 has a spoiling strategy to falsify almost-sure (resp. positive) winning against all player-1 strategies; (2) we present optimal strategy complexity results that precisely characterize the classes of strategies required for almost-sure and positive winning for both players; and (3) we present quadratic time algorithms to compute the almost-sure and the positive winning sets, matching the best known bound of algorithms for much simpler problems (such as reachability objectives). For quantitative constraints we show that a polynomial time solution for the almost-sure or the positive winning set would imply a solution to a long-standing open problem (the value problem for turn-based deterministic mean-payoff games) that is not known to be solvable in polynomial time.

cs.GT↗

arXiv2014

Generalized Risk-Aversion in Stochastic Multi-Armed Bandits

We consider the problem of minimizing the regret in stochastic multi-armed bandit, when the measure of goodness of an arm is not the mean return, but some general function of the mean and the variance.We characterize the conditions under which learning is possible and present examples for which no natural algorithm can achieve sublinear regret.

cs.LG↗

arXiv2014

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 lie in FNP; (3) the approximation problem is at least as hard as the decision problem for simple stochastic games (for which NP intersection 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.

cs.GT↗

arXiv2013

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 12^n n^(O(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 2^(O(n^2+m)), 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.

cs.GT↗

arXiv2012

Strategy complexity of finite-horizon Markov decision processes and simple stochastic games

Markov decision processes (MDPs) and simple stochastic games (SSGs) provide a rich mathematical framework to study many important problems related to probabilistic systems. MDPs and SSGs with finite-horizon objectives, where the goal is to maximize the probability to reach a target state in a given finite time, is a classical and well-studied problem. In this work we consider the strategy complexity of finite-horizon MDPs and SSGs. We show that for all $ε>0$, the natural class of counter-based strategies require at most $\log \log (\frac{1}ε) + n+1$ memory states, and memory of size $Ω(\log \log (\frac{1}ε) + n)$ is required. Thus our bounds are asymptotically optimal. We then study the periodic property of optimal strategies, and show a sub-exponential lower bound on the period for optimal strategies.

cs.GT↗

arXiv2012

Patience of Matrix Games

For matrix games we study how small nonzero probability must be used in optimal strategies. We show that for nxn win-lose-draw games (i.e. (-1,0,1) matrix games) nonzero probabilities smaller than n^{-O(n)} are never needed. We also construct an explicit nxn win-lose game such that the unique optimal strategy uses a nonzero probability as small as n^{-Omega(n)}. This is done by constructing an explicit (-1,1) nonsingular nxn matrix, for which the inverse has only nonnegative entries and where some of the entries are of value n^{Omega(n)}.

cs.DM↗

arXiv2012

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 4^r r^{O(1)} n^{O(1)}. In this paper, we show that an algorithm combining value iteration with retrograde analysis achieves a time bound of O(r 2^r (r log r + n)), thus improving both time bounds. While the algorithm is simple, the analysis leading to this time bound is involved, using techniques of extremal combinatorics to identify worst case instances for the algorithm.

cs.GT↗

arXiv2012

The complexity of solving reachability games using value and strategy iteration

Two standard algorithms for approximately solving two-player zero-sum concurrent reachability games are value iteration and strategy iteration. We prove upper and lower bounds of 2^(m^(Theta(N))) on the worst case number of iterations needed by both of these algorithms for providing non-trivial approximations to the value of a game with N non-terminal positions and m actions for each player in each position. In particular, both algorithms have doubly-exponential complexity. Even when the game given as input has only one non-terminal position, we prove an exponential lower bound on the worst case number of iterations needed to provide non-trivial approximations.

cs.GT↗