A subexponential lower bound for the Random Facet algorithm for Parity Games

Transcription

1 A subexponential lower bound for the Random Facet algorithm for Parity Games Oliver Friedmann 1 Thomas Dueholm Hansen 2 Uri Zwick 3 1 Department of Computer Science, University of Munich, Germany. 2 Center for Algorithmic Game Theory, Department of Computer Science, Aarhus University, Denmark. 3 School of Computer Science, Tel Aviv University, Israel. January 23, 2011 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 1/19

2 The RandomFacet algorithm RandomFacet: Randomized pivoting rule for the simplex algorithm for linear programming (LP-type problems). Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 2/19

3 The RandomFacet algorithm RandomFacet: Randomized pivoting rule for the simplex algorithm for linear programming (LP-type problems). Introduced by Matoušek, Sharir and Welzl (1992). A different variant was introduced by Kalai (1992). Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 2/19

4 The RandomFacet algorithm RandomFacet: Randomized pivoting rule for the simplex algorithm for linear programming (LP-type problems). Introduced by Matoušek, Sharir and Welzl (1992). A different variant was introduced by Kalai (1992). Terminates in a subexponential number of steps: 2 O( n log m), (n variables, m constraints). Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 2/19

5 The RandomFacet algorithm RandomFacet: Randomized pivoting rule for the simplex algorithm for linear programming (LP-type problems). Introduced by Matoušek, Sharir and Welzl (1992). A different variant was introduced by Kalai (1992). Terminates in a subexponential number of steps: 2 O( n log m), (n variables, m constraints). This is the best known upper bound in n and m for any algorithm for the problem. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 2/19

6 The RandomFacet algorithm RandomFacet: Randomized pivoting rule for the simplex algorithm for linear programming (LP-type problems). Introduced by Matoušek, Sharir and Welzl (1992). A different variant was introduced by Kalai (1992). Terminates in a subexponential number of steps: 2 O( n log m), (n variables, m constraints). This is the best known upper bound in n and m for any algorithm for the problem. Matoušek (1994): 2 Ω( n) lower bound in abstract setting. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 2/19

7 The RandomFacet algorithm RandomFacet: Randomized pivoting rule for the simplex algorithm for linear programming (LP-type problems). Introduced by Matoušek, Sharir and Welzl (1992). A different variant was introduced by Kalai (1992). Terminates in a subexponential number of steps: 2 O( n log m), (n variables, m constraints). This is the best known upper bound in n and m for any algorithm for the problem. Matoušek (1994): 2 Ω( n) lower bound in abstract setting. Until recently a candidate for solving linear programs in strongly polynomial time: Disproved by the continuation of the work presented here. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 2/19

8 Games and LP-type problems Stochastic games were introduced by Shapley (1953). Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 3/19

9 Games and LP-type problems Stochastic games were introduced by Shapley (1953). Turn-based stochastic games are played on directed graphs with vertices controlled by maximizer, minimizer and nature. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 3/19

10 Games and LP-type problems Stochastic games were introduced by Shapley (1953). Turn-based stochastic games are played on directed graphs with vertices controlled by maximizer, minimizer and nature. Major open problem, first stated by Condon (1992): The problem of solving turn-based stochastic games is in NP conp, but no polynomial time algorithm is known. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 3/19

11 Games and LP-type problems Stochastic games were introduced by Shapley (1953). Turn-based stochastic games are played on directed graphs with vertices controlled by maximizer, minimizer and nature. Major open problem, first stated by Condon (1992): The problem of solving turn-based stochastic games is in NP conp, but no polynomial time algorithm is known. Halman (2007): Turn-based stochastic games are of LP-type. Vertices correspond to variables and edges correspond to constraints. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 3/19

12 Games and LP-type problems Stochastic games were introduced by Shapley (1953). Turn-based stochastic games are played on directed graphs with vertices controlled by maximizer, minimizer and nature. Major open problem, first stated by Condon (1992): The problem of solving turn-based stochastic games is in NP conp, but no polynomial time algorithm is known. Halman (2007): Turn-based stochastic games are of LP-type. Vertices correspond to variables and edges correspond to constraints. The RandomFacet algorithm is the fastest known algorithm for solving turn-based stochastic games. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 3/19

13 Notable special cases 1 player: Markov decision processes (MDPs): maximizer and nature. Introduced by Bellman (1957). Manne (1960): MDPs can be solved by linear programming. No known strongly polynomial time algorithm for MDPs. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 4/19

14 Notable special cases 1 player: Markov decision processes (MDPs): maximizer and nature. Introduced by Bellman (1957). Manne (1960): MDPs can be solved by linear programming. No known strongly polynomial time algorithm for MDPs. Deterministic MDPs: maximizer. Solves the minimum mean cost cycle problem. Karp (1978): O(nm) time algorithm, n vertices and m edges. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 4/19

15 Notable special cases 1 player: Markov decision processes (MDPs): maximizer and nature. Introduced by Bellman (1957). Manne (1960): MDPs can be solved by linear programming. No known strongly polynomial time algorithm for MDPs. Deterministic MDPs: maximizer. Solves the minimum mean cost cycle problem. Karp (1978): O(nm) time algorithm, n vertices and m edges. No vertices controlled by nature: Mean payoff games: maximizer and minimizer. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 4/19

16 Notable special cases 1 player: Markov decision processes (MDPs): maximizer and nature. Introduced by Bellman (1957). Manne (1960): MDPs can be solved by linear programming. No known strongly polynomial time algorithm for MDPs. Deterministic MDPs: maximizer. Solves the minimum mean cost cycle problem. Karp (1978): O(nm) time algorithm, n vertices and m edges. No vertices controlled by nature: Mean payoff games: maximizer and minimizer. Parity games: maximizer and minimizer, special structure. Equivalent to the problem of µ-calculus model checking. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 4/19

17 Overview Abstract LP-type problems Concrete Turn-based stochastic games 2 1 /2 players Linear programming Mean payoff games 2 players Markov decision problems 1 1 /2 players Parity games 2 players Deterministic MDPs 1 player Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 5/19

18 Overview Abstract LP-type problems Concrete Turn-based stochastic games 2 1 /2 players Linear programming Mean payoff games 2 players Markov decision problems 1 1 /2 players Parity games 2 players NP conp Deterministic MDPs 1 player P Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 5/19

19 Main result Theorem The RandomFacet algorithm may require 2 Ω( n/ log n) expected steps to solve n-state parity games, mean payoff games and turn-based stochastic games. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 6/19

20 Parity games EVEN (circle) wins if the largest priority seen infinitely often is even, ODD (square) wins otherwise: Observed priorities: 8 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 7/19

21 Parity games EVEN (circle) wins if the largest priority seen infinitely often is even, ODD (square) wins otherwise: Observed priorities: 8 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 7/19

22 Parity games EVEN (circle) wins if the largest priority seen infinitely often is even, ODD (square) wins otherwise: Observed priorities: 8, 7 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 7/19

23 Parity games EVEN (circle) wins if the largest priority seen infinitely often is even, ODD (square) wins otherwise: Observed priorities: 8, 7, 5 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 7/19

24 Parity games EVEN (circle) wins if the largest priority seen infinitely often is even, ODD (square) wins otherwise: Observed priorities: 8, 7, 5, 4 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 7/19

25 Parity games EVEN (circle) wins if the largest priority seen infinitely often is even, ODD (square) wins otherwise: Observed priorities: 8, 7, 5, 4, 5 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 7/19

26 Parity games EVEN (circle) wins if the largest priority seen infinitely often is even, ODD (square) wins otherwise: Observed priorities: 8, 7, 5, 4, 5 A (positional) strategy, σ or τ, is an outgoing edge from each vertex controlled by the corresponding player. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 7/19

27 Parity games EVEN (circle) wins if the largest priority seen infinitely often is even, ODD (square) wins otherwise: Observed priorities: 8, 7, 5, 4, 5, 4 A (positional) strategy, σ or τ, is an outgoing edge from each vertex controlled by the corresponding player. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 7/19

28 Parity games EVEN (circle) wins if the largest priority seen infinitely often is even, ODD (square) wins otherwise: Observed priorities: 8, 7, 5, 4, 5, 4, 5 A (positional) strategy, σ or τ, is an outgoing edge from each vertex controlled by the corresponding player. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 7/19

29 Parity games EVEN (circle) wins if the largest priority seen infinitely often is even, ODD (square) wins otherwise: Observed priorities: 8, 7, 5, 4, 5, 4, 5, 4,... A (positional) strategy, σ or τ, is an outgoing edge from each vertex controlled by the corresponding player. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 7/19

30 Mean payoff games Priorities are replaced by rewards on edges. Players take the roles of maximizer and minimizer. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 8/19

31 Mean payoff games Priorities are replaced by rewards on edges. Players take the roles of maximizer and minimizer. The value val σ,τ (v) of a vertex v is the average reward of the cycle reached from v when moving according to σ and τ. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 8/19

32 Mean payoff games Priorities are replaced by rewards on edges. Players take the roles of maximizer and minimizer. The value val σ,τ (v) of a vertex v is the average reward of the cycle reached from v when moving according to σ and τ. The path leading to the cycle is of secondary importance. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 8/19

33 Mean payoff games Priorities are replaced by rewards on edges. Players take the roles of maximizer and minimizer. The value val σ,τ (v) of a vertex v is the average reward of the cycle reached from v when moving according to σ and τ. The path leading to the cycle is of secondary importance. Reduction from parity games: ( n) 5 5 ( n) 5 ( n) 5 ( n) 5 A cycle has positive value iff its largest priority is even. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 8/19

34 A simpler view In our lower bound examples we always reach a cycle with value zero (essentially). Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 9/19

35 A simpler view In our lower bound examples we always reach a cycle with value zero (essentially). The path leading to the cycle is then the primary focus. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 9/19

36 A simpler view In our lower bound examples we always reach a cycle with value zero (essentially). The path leading to the cycle is then the primary focus. For simplicity I will use val σ,τ (v) to denote the total sum of rewards on the path to the cycle. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 9/19

37 Optimal strategies and improving switches We generally assume that the minimizer plays an optimal counter-strategy: val σ (v) = min τ val σ,τ (v) Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 10/19

38 Optimal strategies and improving switches We generally assume that the minimizer plays an optimal counter-strategy: val σ (v) = min τ val σ,τ (v) σ is optimal from v if for all σ, val σ (v) val σ (v). Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 10/19

39 Optimal strategies and improving switches We generally assume that the minimizer plays an optimal counter-strategy: val σ (v) = min τ val σ,τ (v) σ is optimal from v if for all σ, val σ (v) val σ (v). σ is optimal if it is optimal from all v. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 10/19

40 Optimal strategies and improving switches We generally assume that the minimizer plays an optimal counter-strategy: val σ (v) = min τ val σ,τ (v) σ is optimal from v if for all σ, val σ (v) val σ (v). σ is optimal if it is optimal from all v. Shapley (1957): Optimal positional strategies are guaranteed to exist. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 10/19

41 Optimal strategies and improving switches We generally assume that the minimizer plays an optimal counter-strategy: val σ (v) = min τ val σ,τ (v) σ is optimal from v if for all σ, val σ (v) val σ (v). σ is optimal if it is optimal from all v. Shapley (1957): Optimal positional strategies are guaranteed to exist. An edge (u, v) is an improving switch w.r.t. σ if the value of u is improved by switching to (u, v): val σ[(u,v)] (u) val σ (u) Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 10/19

42 Optimal strategies and improving switches We generally assume that the minimizer plays an optimal counter-strategy: val σ (v) = min τ val σ,τ (v) σ is optimal from v if for all σ, val σ (v) val σ (v). σ is optimal if it is optimal from all v. Shapley (1957): Optimal positional strategies are guaranteed to exist. An edge (u, v) is an improving switch w.r.t. σ if the value of u is improved by switching to (u, v): val σ[(u,v)] (u) val σ (u) σ is optimal iff there are no improving switches. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 10/19

43 The RandomFacet algorithm Function RandomFacet(G, σ) if E 0 = σ then return σ else Choose e E 0 \ σ uniformly at random σ RandomFacet(G \ {e}, σ) if e is improving switch w.r.t. σ then σ σ [e] return RandomFacet(G, σ ) else return σ Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 11/19

44 Example: Binary choices e σ e σ e 1 σ σ e 1 σ Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 12/19

45 Example: Binary choices e σ e σ e 1 σ σ e 1 σ σ Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 12/19

46 Example: Binary choices e σ e σ e 1 σ σ e 1 σ σ σ Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 12/19

47 Example: Binary choices e σ e σ e 1 σ σ σ e 1 σ σ σ Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 12/19

48 Example: Binary choices e 2 σ e 2 σ e 1 σ σ σ e 1 σ σ σ val(g \ {e 1 }) val(g \ {e 2 })... val(g \ {e n }) Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 12/19

49 Example: Binary choices The number of steps is upper bounded by: f (0) = 1 f (n) = f (n 1) + 1 n 1 f (i) for n > 0 n i=0 The recurrence is bounded by: f (n) = 2 Θ( n) Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 13/19

50 The construction T : 1 0 c 3 : 8 B 3 : 2 b 3 : 2 D 3 : 3 A 3 : 2 a 3 : 2 0 c 2 : 6 B 2 : 2 b 2 : 2 D 2 : 3 A 2 : 2 a 2 : 2 0 c 1 : 4 B 1 : 2 b 1 : 2 D 1 : 3 A 1 : 2 a 1 : 2 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 14/19

51 The construction T : 1 0 c 3 : 8 B 3 : 2 b 3 : 2 D 3 : 3 A 3 : 2 a 3 : 2 0 c 2 : 6 B 2 : 2 b 2 : 2 D 2 : 3 A 2 : 2 a 2 : 2 0 c 1 : 4 B 1 : 2 b 1 : 2 D 1 : 3 A 1 : 2 a 1 : 2 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 14/19

52 The construction T : 1 0 c 3 : 8 B 3 : 2 b 3 : 2 D 3 : 3 A 3 : 2 a 3 : 2 1 c 2 : 6 B 2 : 2 b 2 : 2 D 2 : 3 A 2 : 2 a 2 : 2 1 c 1 : 4 B 1 : 2 b 1 : 2 D 1 : 3 A 1 : 2 a 1 : 2 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 14/19

53 The construction T : 1 1 c 3 : 8 B 3 : 2 b 3 : 2 D 3 : 3 A 3 : 2 a 3 : 2 1 c 2 : 6 B 2 : 2 b 2 : 2 D 2 : 3 A 2 : 2 a 2 : 2 1 c 1 : 4 B 1 : 2 b 1 : 2 D 1 : 3 A 1 : 2 a 1 : 2 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 14/19

54 The construction T : 1 1 c 3 : 8 B 3 : 2 b 3 : 2 D 3 : 3 A 3 : 2 a 3 : 2 1 c 2 : 6 B 2 : 2 b 2 : 2 D 2 : 3 A 2 : 2 a 2 : 2 1 c 1 : 4 B 1 : 2 b 1 : 2 D 1 : 3 A 1 : 2 a 1 : 2 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 14/19

55 The construction T : 1 1 c 3 : 8 B 3 : 2 b 3 : 2 D 3 : 3 A 3 : 2 a 3 : 2 0 c 2 : 6 B 2 : 2 b 2 : 2 D 2 : 3 A 2 : 2 a 2 : 2 0 c 1 : 4 B 1 : 2 b 1 : 2 D 1 : 3 A 1 : 2 a 1 : 2 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 14/19

56 The construction T : 1 1 c 3 : 8 B 3 : 2 b 3 : 2 D 3 : 3 A 3 : 2 a 3 : 2 0 c 2 : 6 B 2 : 2 b 2 : 2 D 2 : 3 A 2 : 2 a 2 : 2 0 c 1 : 4 B 1 : 2 b 1 : 2 D 1 : 3 A 1 : 2 a 1 : 2 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 14/19

57 Randomized bitcounter Start with n bits with value 0: Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 15/19

58 Randomized bitcounter Start with n bits with value 0: Pick a random bit i and fix it: Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 15/19

59 Randomized bitcounter Start with n bits with value 0: Pick a random bit i and fix it: Count recursively with the remaining n 1 bits: Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 15/19

60 Randomized bitcounter Start with n bits with value 0: Pick a random bit i and fix it: Count recursively with the remaining n 1 bits: Increment the i th bit: Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 15/19

61 Randomized bitcounter Start with n bits with value 0: Pick a random bit i and fix it: Count recursively with the remaining n 1 bits: Increment the i th bit: Reset the i 1 lower bits: Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 15/19

62 Randomized bitcounter Start with n bits with value 0: Pick a random bit i and fix it: Count recursively with the remaining n 1 bits: Increment the i th bit: Reset the i 1 lower bits: Count recursively with the i 1 lower bits: Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 15/19

63 Randomized bitcounter Start with n bits with value 0: Pick a random bit i and fix it: Count recursively with the remaining n 1 bits: Increment the i th bit: Reset the i 1 lower bits: Count recursively with the i 1 lower bits: Expected number of steps: f (0) = 1 f (n) = f (n 1) + 1 n 1 f (i) for n > 0 n i=0 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 15/19

64 Randomized bitcounter Start with n bits with value 0: Pick a random bit i and fix it: Count recursively with the remaining n 1 bits: Increment the i th bit: Reset the i 1 lower bits: Count recursively with the i 1 lower bits: Expected number of steps: f (0) = 1 f (n) = f (n 1) + 1 n 1 f (i) for n > 0 n i=0 Same recurrence as for upper bound: f (n) = 2 Θ( n) Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 15/19

65 The modified RandomFacet algorithm Function RandomFacet (G, σ, ϕ) if E 0 = σ then return σ else e argmin e E 0 \σ ϕ(e ) σ RandomFacet (G \ {e}, σ, ϕ) if e is improving switch w.r.t. σ then σ σ [e] return RandomFacet (G, σ, ϕ) else return σ Same expected number of steps when the permutation ϕ is picked uniformly at random. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 16/19

66 Ensuring worst-case behavior Good permutation: 1 i : ϕ(b i, B i ) < ϕ(a i, A i ) T : 1 c3 : 8 B3 : 2 b3 : 2 D3 : 3 A3 : 2 a3 : 2 c2 : 6 B2 : 2 b2 : 2 D2 : 3 A2 : 2 a2 : 2 c1 : 4 B1 : 2 b1 : 2 D1 : 3 A1 : 2 a1 : 2 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 17/19

67 Ensuring worst-case behavior Good permutation: 1 i : ϕ(b i, B i ) < ϕ(a i, A i ) c3 : 8 B3 : 2 l T : 1 b3 : 2 Increase probability of picking a good permutation at random by duplication: Pr[(1) not satisfied] n (l!)2 (2l)! n 2 l l D3 : 3 A3 : 2 a3 : 2 l c2 : 6 B2 : 2 b2 : 2 l D2 : 3 A2 : 2 a2 : 2 l c1 : 4 B1 : 2 b1 : 2 l D1 : 3 A1 : 2 a1 : 2 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 17/19

68 Ensuring worst-case behavior Good permutation: 1 i : ϕ(b i, B i ) < ϕ(a i, A i ) c3 : 8 B3 : 2 l T : 1 b3 : 2 Increase probability of picking a good permutation at random by duplication: Pr[(1) not satisfied] n (l!)2 (2l)! n 2 l Here we lose a logarithmic factor in the exponent. l D3 : 3 A3 : 2 a3 : 2 l c2 : 6 B2 : 2 b2 : 2 l D2 : 3 A2 : 2 a2 : 2 l c1 : 4 B1 : 2 b1 : 2 l D1 : 3 A1 : 2 a1 : 2 Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 17/19

69 Concluding remarks We constructed parity games where the expected running time of the RandomFacet algorithm is 2 Ω( n), almost matching the upper bound by Matoušek, Sharir and Welzl (1992). Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 18/19

70 Concluding remarks We constructed parity games where the expected running time of the RandomFacet algorithm is 2 Ω( n), almost matching the upper bound by Matoušek, Sharir and Welzl (1992). By replacing the minimizer with vertices controlled by nature, we have later managed to prove the same lower bound for MDPs and linear programming. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 18/19

71 Concluding remarks We constructed parity games where the expected running time of the RandomFacet algorithm is 2 Ω( n), almost matching the upper bound by Matoušek, Sharir and Welzl (1992). By replacing the minimizer with vertices controlled by nature, we have later managed to prove the same lower bound for MDPs and linear programming. Using similar techniques we have also managed to show that RandomEdge (repeatedly switch a random improving edge), when applied to the same settings, may require 2 Ω(n1/4 ) expected steps. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 18/19

72 Concluding remarks We constructed parity games where the expected running time of the RandomFacet algorithm is 2 Ω( n), almost matching the upper bound by Matoušek, Sharir and Welzl (1992). By replacing the minimizer with vertices controlled by nature, we have later managed to prove the same lower bound for MDPs and linear programming. Using similar techniques we have also managed to show that RandomEdge (repeatedly switch a random improving edge), when applied to the same settings, may require 2 Ω(n1/4 ) expected steps. Major open problem: Polynomial time algorithm for Parity Games. Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 18/19

73 The end Thank you for listening! Friedmann, Hansen, Zwick Lower bound for Random Facet algorithm Page 19/19