1. Uri Zwick Tel Aviv University
Deterministic subexponential algorithm for Parity Games
Uri Zwick Tel Aviv University
China Theory Week 2007
2007理论计算机科学明日 之星中国论坛
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2. Simple Stochastic Games
Mean Payoff Games
Parity Games

3. A simple Simple Stochastic GameR R

4. Simple Stochastic game (SSGs) Reachability version [Condon (1992)]
MAX
RAND
min
MAX-sink
min-sink
Two Players: MAX and min
Objective: MAX/min the probability of getting to the MAX-sink

5. Simple Stochastic game (SSGs) [Condon (1992)]
Terminating binary games
The outdegrees of all non-sinks are 2
All probabilities are ½.
The game terminates with prob. 1

6. Simple Stochastic games (SSGs)
Basic properties :
Every vertex in the game has a value v
Both players have positional optimal strategies
Positional strategy for MAX: choice of an outgoing edge from each MAX vertex
Decision version: Is value v

7. “Solving” terminating binary SSGs
The values vi of the vertices of a game are the unique solution of the following equations:
The values are rational numbers requiring only a linear number of bits
Corollary: Decision version in NP  co-NP

8. Simple Stochastic game (SSGs) Payoff version [Shapley (1953)]
MAX
RAND
min
Limiting average version
Discounted version

9. Markov Decision Processes (MDPs)
MAX
RAND
min
Theorem: [Derman (1970)]
Values and optimal strategies of a MDP can be found by solving an LP

10. NP  co-NP – Another proof
Deciding whether the value of a game is at least (at most) v is in NP  co-NP
To show that value  v , guess an optimal strategy  for MAX
Find an optimal counter-strategy  for min by solving the resulting MDP.
Is the problem in P ?

11. Mean Payoff Games (MPGs) [Ehrenfeucht, Mycielski (1979)]
MAX
RAND
min
Non-terminating version
Discounted version
COCOON 1995
Reachability SSGs
MPGs
(PZ’96)
Pseudo-polynomial algorithm
(PZ’96)

12. Mean Payoff Games (MPGs) [Ehrenfeucht, Mycielski (1979)]
Again, both players have optimal positional strategies.
Value – average of the cycle

13. Selecting the second largest element with only four storage locations [PZ’96]

14. Non-emptyness of -tree automata modal -calculus model checking
Parity Games (PGs)
ODD 8
Priorities
EVEN 3
EVEN wins if largest priority seen infinitely often is even
Equivalent to many interesting problems in automata and verification:
Non-emptyness of -tree automata
modal -calculus model checking

15. Parity Games (PGs) A simple example2 3 2 1 4 1

16. Mean Payoff Games (MPGs)
Parity Games (PGs)
Mean Payoff Games (MPGs)
[Stirling (1993)] [Puri (1995)]
ODD 8
EVEN 3
Replace priority k by payoff (n)k
Move payoffs to outgoing edges

17. A randomized subexponential algorithm for simple stochastic games

18. Simple Stochastic games (SSGs) Switches…
A switch is a change of strategy at a single vertex
A switch is profitable for MAX if it increases the value of the game (sum of values of all vertices)
A strategy is optimal iff no switch is profitable

19. A randomized subexponential algorithm for binary SSGs [Ludwig (1995)] [Kalai (1992)] [Matousek-Sharir-Welzl (1992)]
Start with an arbitrary strategy  for MAX
Choose a random vertex iVMAX
Find the optimal strategy ’ for MAX in the game in which the only outgoing edge of i is (i,(i))
If switching ’ at i is not profitable, then ’ is optimal
Otherwise, let  (’)i and repeat

20. Would never be switched !
A randomized subexponential algorithm for binary SSGs [Ludwig (1995)] [Kalai (1992)] [Matousek-Sharir-Welzl (1992)]
MAX vertices
All correct !
Would never be switched !
There is a hidden order of MAX vertices under which the optimal strategy returned by the first recursive call correctly fixes the strategy of MAX at vertices 1,2,…,i

21. Positions 1,..,i where switched and would never be switched again.
The hidden order
Let vi be the value of the optimal strategy for MAX that agrees with σ on i
Order the vertices such that
Positions 1,..,i where switched and would never be switched again.

22. A deterministic subexponential algorithm for parity games

23. Exponential algorithm for PGs [McNaughton (1993)] [Zielonka (1998)]
Vertices of highest priority (even)
First recursive call
Vertices from which EVEN can force the game to enter A
Lemma: (i)
(ii)

24. Exponential algorithm for PGs [McNaughton (1993)] [Zielonka (1998)]
Second recursive call
In the worst case, both recursive calls are on games of size n1

25. Idea: Look for small dominions!
Deterministic subexponential alg for PGs Jurdzinski, Paterson, Z (2006)
Idea: Look for small dominions!
Second recursive call
Dominions of size s can be found in O(ns) time
Dominion
Dominion: A (small) set from which one of the players can win without the play ever leaving this set

26. Open problems Polynomial algorithms?
Faster subexponential algorithms for parity games?
Deterministic subexponential algorithms for MPGs and SSGs?
Faster pseudo-polynomial algorithms for MPGs?