Results 1  10
of
18
Violator Spaces: Structure and Algorithms
, 2007
"... Sharir and Welzl introduced an abstract framework for optimization problems, called LPtype problems or also generalized linear programming problems, which proved useful in algorithm design. We define a new, and as we believe, simpler and more natural framework: violator spaces, which constitute a p ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
(Show Context)
Sharir and Welzl introduced an abstract framework for optimization problems, called LPtype problems or also generalized linear programming problems, which proved useful in algorithm design. We define a new, and as we believe, simpler and more natural framework: violator spaces, which constitute a proper generalization of LPtype problems. We show that Clarkson’s randomized algorithms for lowdimensional linear programming work in the context of violator spaces. For example, in this way we obtain the fastest known algorithm for the Pmatrix generalized linear complementarity problem with a constant number of blocks. We also give two new characterizations of LPtype problems: they are equivalent to acyclic violator spaces, as well as to concrete LPtype problems (informally, the constraints in a concrete LPtype problem are subsets of a linearly ordered ground set, and the value of a set of constraints is the minimum of its intersection).
Fast algorithms for monotonic discounted linear programs with two variables per inequality
 700 70 0 701.062 70.362 1.036 500 100 200 500.0873 99.849 200.039 670 0 200 670.978 0.0388 199.963 700 0 200 700.663 0.020 199.989 Calibration Data Reconstructed Data R T R T 0.083 676.421 0.083 606.801
, 2006
"... We suggest new strongly polynomial algorithms for solving linear programs min ( � xiS) with constraints S of the monotonic discounted form xi ≥ λxj + β with 0 < λ < 1. The algorithm for the case when the discounting factor λ is equal for all constraints is O(mn 2), whereas the algorithm for ..."
Abstract

Cited by 7 (2 self)
 Add to MetaCart
(Show Context)
We suggest new strongly polynomial algorithms for solving linear programs min ( � xiS) with constraints S of the monotonic discounted form xi ≥ λxj + β with 0 < λ < 1. The algorithm for the case when the discounting factor λ is equal for all constraints is O(mn 2), whereas the algorithm for the case when λ may vary between the constraints is O(mn 2 log m), where n is the number of variables and m is the number of constraints. As applications, we obtain the best currently available algorithm for twoplayer discounted payoff games and a new faster strongly subexponential algorithm for the ergodic partition problem for mean payoff games. 1
An Improved Algorithm for Discounted Payoff Games
"... Abstract. We show that an optimal counterstrategy against a fixed positional strategy in a generalized discounted payoff game, where edges have individual discounts, can be computed in O(mn 2 log m) strongly polynomial time, where n and m are the number of vertices and edges in the game graph. This ..."
Abstract

Cited by 6 (0 self)
 Add to MetaCart
(Show Context)
Abstract. We show that an optimal counterstrategy against a fixed positional strategy in a generalized discounted payoff game, where edges have individual discounts, can be computed in O(mn 2 log m) strongly polynomial time, where n and m are the number of vertices and edges in the game graph. This results in the best known strongly subexponential time bound for solving twoplayer generalized discounted payoff games. 1
Discounted deterministic Markov decision processes and discounted allpairs shortest paths
 ACM Transcations on Algorithms
"... We present two new algorithms for finding optimal strategies for discounted, infinitehorizon, Deterministic Markov Decision Processes (DMDP). The first one is an adaptation of an algorithm of Young, Tarjan and Orlin for finding minimum mean weight cycles. It runs in O(mn + n 2 log n) time, where n ..."
Abstract

Cited by 6 (1 self)
 Add to MetaCart
(Show Context)
We present two new algorithms for finding optimal strategies for discounted, infinitehorizon, Deterministic Markov Decision Processes (DMDP). The first one is an adaptation of an algorithm of Young, Tarjan and Orlin for finding minimum mean weight cycles. It runs in O(mn + n 2 log n) time, where n is the number of vertices (or states) and m is the number of edges (or actions). The second one is an adaptation of a classical algorithm of Karp for finding minimum mean weight cycles. It runs in O(mn) time. The first algorithm has a slightly slower worstcase complexity, but is faster than the first algorithm in many situations. Both algorithms improve on a recent O(mn 2)time algorithm of Andersson and Vorobyov. We also present a randomized Õ(m1/2 n 2)time algorithm for finding Discounted AllPairs Shortest Paths (DAPSP), improving several previous algorithms. 1
A subexponential lower bound for the Random Facet algorithm for Parity Games
"... Parity Games form an intriguing family of infinite duration games whose solution is equivalent to the solution of important problems in automatic verification and automata theory. They also form a very natural subclass of Deterministic Mean Payoff Games, which in turn is a very natural subclass of t ..."
Abstract

Cited by 6 (5 self)
 Add to MetaCart
Parity Games form an intriguing family of infinite duration games whose solution is equivalent to the solution of important problems in automatic verification and automata theory. They also form a very natural subclass of Deterministic Mean Payoff Games, which in turn is a very natural subclass of turnbased Stochastic Mean Payoff Games. It is a major open problem whether these game families can be solved in polynomial time. The currently fastest algorithms for the solution of all these games are adaptations of the randomized generalizationof linear programming. We refer to the algorithm ofMatouˇsek, Sharir and Welzl as the Random Facet algorithm. The expected running time of these algorithmsis subexponential in the size of the game, i.e., 2
THE COMPLEXITY OF NASH EQUILIBRIA IN STOCHASTIC MULTIPLAYER GAMES
 VOL. 7 (3:20) 2011, PP. 1–45
, 2011
"... ..."
(Show Context)
Meanpayoff games and propositional proofs
, 2010
"... We associate a CNFformula to every instance of the meanpayoff game problem in such a way that if the value of the game is nonnegative the formula is satisfiable, and if the value of the game is negative the formula has a polynomialsize refutation in Σ2Frege (a.k.a. DNFresolution). This reduces ..."
Abstract

Cited by 4 (1 self)
 Add to MetaCart
We associate a CNFformula to every instance of the meanpayoff game problem in such a way that if the value of the game is nonnegative the formula is satisfiable, and if the value of the game is negative the formula has a polynomialsize refutation in Σ2Frege (a.k.a. DNFresolution). This reduces the problem of solving meanpayoff games to the weak automatizability of Σ2Frege, and to the interpolation problem for Σ2,2Frege. Since the interpolation problem for Σ1Frege (i.e. resolution) is solvable in polynomial time, our result is close to optimal up to the computational complexity of solving meanpayoff games. The proof of the main result requires building lowdepth formulas that compute the bits of the sum of a constant number of integers in binary notation, and lowcomplexity proofs of the required arithmetic properties. 1
Automatizability and Simple Stochastic Games
 In Proc. of 38th International Colloquium on Automata, Languages and Programming (ICALP), Luca Aceto, Monika Henzinger, Jiri Sgall (Eds
, 2011
"... The complexity of simple stochastic games (SSGs) has been open since they were dened by Condon in 1992. Despite intensive eort, the complexity of this problem is still unresolved. In this paper, building on the results of [4], we establish a connection between the complexity of SSGs and the complexi ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
The complexity of simple stochastic games (SSGs) has been open since they were dened by Condon in 1992. Despite intensive eort, the complexity of this problem is still unresolved. In this paper, building on the results of [4], we establish a connection between the complexity of SSGs and the complexity of an important problem in proof complexity{the proof search problem for low depth Frege systems. We prove that if depth3 Frege systems are weakly automatizable, then SSGs are solvable in polynomialtime. Moreover we identify a natural combinatorial principle, which is a version of the wellknown Graph Ordering Principle (GOP), that we call the integervalued GOP (IGOP). This principle states that for any graph G with nonnegative integer weights associated with each node, there exists a locally maximal vertex (a vertex whose weight is at least as large as its neighbors). We prove that if depth2 Frege plus IGOP is weakly automatizable, then SSG is in P. Supported by NSERC. 1