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 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAA

Simple Stochastic Games Mean Payoff Games Parity Games

A simple Simple Stochastic Game R R

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

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

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

“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

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

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

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 ?

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)

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

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

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

Parity Games (PGs) A simple example 2 3 2 1 4 1

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

A randomized subexponential algorithm for simple stochastic games

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

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

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

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.

A deterministic subexponential algorithm for parity games

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)

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

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

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