A DETERMINISTIC SUBEXPONENTIAL ALGORITHM FOR
SOLVING PARITY GAMES
∗
MARCIN JURDZI
´
NSKI
†
, MIKE PATERSON
‡
, AND URI ZWICK
§
Abstract. The existence of polynomial time algorithms for the solution of parity games is
a major open problem . The fastest known algorithms for the problem are randomized algorithms
that run in subexponential time. These algor ithm s are all ultimately based on the randomized
subexponential simplex algorithms of Kalai and of Matouˇsek, Sharir and Welzl. Randomness seems to
play an essential role in these algorithms. We use a completely different, and elementary, approach to
obtain a deterministic subexpon ential algorithm for the solution of parity games. The new algorit hm,
like the existing randomized subexponential algori thm s, uses only polynomial space, and it is almost
as fast as the randomized subexponential algorithms mentioned above.
Key words. analysis of algorithms and problem comple xity, specification and verification,
2-person games, games involving graphs, discrete-time games
AMS subject classifications. 68Q25, 68Q60, 91A05, 91A43, 91A50.
1. Introduction. A parity game [11, 15] is played on a directed graph (V, E) by
two players, Even and Odd, who move a token from vertex to vertex along the edges of
the graph so that an infinite path is formed. A partition (V
0
, V
1
) is given of the set V
of vertices: player Even moves if the token is at a vertex of V
0
and player Odd moves
if the token is at a vertex of V
1
. Finally, a priority function p : V → { 1, 2, . . . , d } is
given. The players compete for the parity of the highest priority occurring infinitely
often: player Even wins if lim sup
i→∞
p(v
i
) is even while player Odd wins if it is odd,
where v
0
, v
1
, v
2
, . . . is the infinite path formed by the players.
The algorithmic problem of solving parity games is, given a parity game G =
(V
0
, V
1
, E, p) and an initial vertex v
0
∈ V , to determine whether player Even has a
winning strategy in the game if the token is initially placed on vertex v
0
. Algorithms
for solving parity games [30, 20, 29, 15, 1] usually compute the winning sets win
0
and
win
1
, i.e., the sets of vertices from which playe rs Even and Odd, respectively, have a
winning strategy. By the Determinacy Theorem for parity games [11, 15] the winning
sets win
0
and win
1
form a partition of the set of vertices V . None of these algorithms
is known to run in polynomial time and the existence of a polynomial time algorithm
for the solution of parity games is a long-standing open problem [12, 15].
The original motivation for the study of parity games comes from the area of
formal verification of systems by temporal logic model checking [5, 15]. The problem
of solving parity games is polynomial time equivalent to the non-emptiness problem of
ω-automata on infinite trees with Rabin-chain acceptance conditions [12], and to the
the model checking problem of the modal µ-calculus (modal fixpoint logic) which is a
specification formalism of choice in formal specification and validation [10, 15]. The
model checking problem is a fundamental algorithmic problem in automated hardware
and software verification [10, 5].
∗
This paper is an updated and extended version of the SODA’06 paper [21]. The work was
partially supported by the London Mathematical Society.
†
Department of Computer Science, University of Warwick, Coventry CV4 7 AL, United Kingdom
(mju@dcs.warwick.ac.uk).
‡
Department of Computer Science, University of Warwick, Coventry CV4 7 AL, United Kingdom
(msp@dcs.warwick.ac.uk).
§
School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel (zwick@cs.tau.ac.il).
1
2 M. JURDZI
´
NSKI, M. PATERSON, AND U. ZWICK
Another important motivation to study the problem of solving parity gam es is its
intriguing complexity theoretic status: the problem is known to be in NP ∩ co-NP [12]
and even in UP ∩ co-UP [19] but, as mentioned, despite considerable efforts of the
community [12, 20, 29, 15, 1, 27] no polynomial time algorithm has been found so
far. Moreover, parity games are polynomial time reducible to mean payoff games [31],
simple stochastic games [6], and discounted games [6, 31]. A stochastic generalization
of parity games was als o studied [9, 3]. The problems of solving all those games are in
UP ∩ co-UP as well [19, 3]. Condon has shown that simple stochastic games are log-
space complete in the class of log-space randomized alternating Turing machines [6].
The task of solving parity, mean payoff, discounted, and simple stochastic games
can be also viewed as a search problem: given a game graph, compute optimal
strategies for both players. The value functions used in strategy improvement al-
gorithms [7, 24, 29, 1] witness membership of all those optimal strategies search prob-
lems in PLS (i.e., the class of polynomial local search problems) [17]. On the other
hand, the problem of computing optimal strategies in simple stochastic games can
be reduced in polynomial time to s olving a P-matrix linear complementarity prob-
lem [14] and to finding a Brouwer fixpoint [18] and hence it is also in PPAD [28]. It
follows that there are polynomial time reductions from the problems of computing
optimal strategies in parity, mean payoff, discounted, and simple stochastic games to
the problem of finding Nash equlibria in bimatrix games [8, 4].
Let n = |V | and m = |E| be the numbers of vertices and edges of a parity
game graph and let d be the largest priority assigned to vertices by the priority
function p : V → { 1, 2, . . . , d }. For parity games with a small number of priorities,
more specifically if d = O(n
1/2
), the progress-measure lifting algorithm [20] gives
currently the best time complexity of O(dm · (2n/d)
d/2
). If d = Ω(n
1/2+ε
) then the
randomized algorithm of Bj¨orklund et al. [1] has a better (expected) running time
bound of n
O(
√
n/ log n)
.
The main contribution of this paper is a deterministic algorithm for solving parity
games which achieves roughly the same complexity as the randomized algorithm of
Bj¨orklund et al. [1]: the complexity of our algorithm is n
O(
√
n/ log n)
if the out-degree
of all vertices is bounded, and it is n
O(
√
n)
otherwise. The new algorithm uses only
polynomial space.
The randomized algorithm of Bj¨orklund et al. [1] is based on the randomized
algorithm of Ludwig [24] for simple stochastic games, which in turn is inspired by the
subexponential randomized simplex algorithms for linear programming and LP-type
problems by Kalai and by Matouˇsek et al. [22, 25]. For games with out-degree two
these algorithms are instantiations of the Random-Facet algorithm for finding the
unique sink in an acyclic unique sink orientation (AUSO) of a hypercube [13]. The
nodes of a hypercube correspond to positional strategies for one of the players and
the orientation of an edge connecting two positional strategies that differ at exactly
one vertex is determined by which of the two strategies has a better value when her
opponent plays a best-response strategy [7, 24, 29, 1].
In contrast, our deterministic algorithm for parity games is obtained by a mod-
ification of a more elementary algorithm of McNaughton and Zielonka for parity
games [30, 15]. The methods we use are thus very different from those of Ludwig
and Bj¨orklund et al. Our method is applicable, so it seems, only to parity games,
while the randomized algorithms for finding the unique sink in an AUSO [13] and
for solving an LP-type problem [25] can be applied to a number of problems includ-
ing computing the values of parity, mean payoff, discounted, and simple stochastic
A DETERMINISTIC SUBEXPONENTIAL ALGORITHM FOR PARITY GAMES 3
games [1, 2, 16].
2. Preliminaries. A parity game G = (V
0
, V
1
, E, p) is composed of two disjoint
sets of vertices V
0
and V
1
, a directed edge set E ⊆ V × V , where V = V
0
∪ V
1
, and a
priority function p : V
0
∪V
1
→ {1, 2, . . . , d}, for some integer d, defined on its vertices.
Every vertex u ∈ V has at least one outgoing edge (u, v) ∈ E. The game is played by
two players: Even, also referred to as Player 0, and Odd, also referred to as Player 1.
The game starts at some vertex v
0
∈ V . The players construct an infinite path
as follows. Let u be the last vertex added so far to the path. If u ∈ V
0
, then Player 0
chooses an edge (u, v) ∈ E. Otherwise, if u ∈ V
1
, then Player 1 chooses an e dge
(u, v) ∈ E. In either case, vertex v is added to the path, and a new edge is then
chosen by either Player 0 or Player 1. As each vertex has at least one outgoing edge,
the path constructed can always be continued.
Let v
0
, v
1
, v
2
, . . . be the infinite path constructed by the two players, and let
p(v
0
), p(v
1
), p(v
2
), . . . be the sequence of the priorities of the vertices on the path.
Player 0 wins the game if the largest priority seen infinitely many times is even, and
Player 1 wins otherwise .
3
2 3 2
4
1
G
a b c
d e f
Fig. 2.1. A parity game G. Double-headed arrows show a positional strategy for each player.
A strategy for Player i in a game G specifies, for every finite path v
0
, v
1
, . . . , v
k
in G that ends in a vertex v
k
∈ V
i
, an edge (v
k
, v
k+1
) ∈ E. A strategy is said to be
a positional strategy if the edge (v
k
, v
k+1
) ∈ E chosen depends only on v
k
, the last
vertex visited. A strategy for player i is said to be a winning strategy if using this
strategy ensures a win for Player i, no matter which strategy is used by the other
player. The Determinacy Theorem for parity games [11, 15] says that for any parity
game G and any start vertex v
0
, either Player 0 has a winning strategy or Player 1
has a winning strategy. (This claim is not immediate as the games considered are
infinite.) Furthermore, if a player has a winning strategy from a vertex in a parity
game then she also has a winning positional strategy from this vertex.
In the parity game G illustrated in Figure 2.1, the initial vertex is labelled a,
Even’s vertices are represented as squares (even number of sides) and Odd’s as tri-
angles. As an example, if each player were to choose the double-headed arrow out
of each of their vertices then the infinite path would be a, d, b, e, d, b, e, . . . , and the
largest priority seen infinitely often would be 4 at vertex e. So Even would win this
play.
The winning set for Player i, denoted by win
i
(G), is the set of vertices of the
game from which Player i has a winning strategy. By the Determinacy Theorem for
parity games [11, 15] we have that win
0
(G) ∪ win
1
(G) = V .
A set B ⊆ V is said to be i-closed, where i ∈ {0, 1}, if for every u ∈ B ∩V
i
, there
4 M. JURDZI
´
NSKI, M. PATERSON, AND U. ZWICK
is some (u, v) ∈ E, such that v ∈ B, and if for every (u, v) ∈ E with u ∈ B ∩ V
¬i
, we
have v ∈ B. (We use ¬i for the complement of i in {0, 1}.) In other words, a set B
is i-closed if Player i can always choose to stay in B while Player ¬i cannot escape
from it, i.e., B is a “trap” for Player ¬i.
Lemma 2.1. For every i ∈ {0, 1}, the set win
i
(G) is i-closed.
Proof. Let j = ¬i, and note that V \ win
i
(G) = win
j
(G) by the Determinacy
Theorem. Let u be an arbitrary vertex in win
i
(G). I f u ∈ V
i
then Player i’s strategy
uses some (u, v) ∈ E such that v ∈ win
i
(G). If u ∈ V
j
then v ∈ win
i
(G) for all
(u, v) ∈ E, since otherwise there would be a move for Player j from u to a vertex of
v ∈ win
j
(G), contradicting the assumption of u ∈ win
i
(G). So, win
i
(G) is i-closed.
Let A ⊆ V be an arbitrary set. The i-reachability set of A, denoted reach
i
(A),
contains all vertices in A and all vertices from which Player i has a strategy to enter
the set A at least once; we call it an i-reachability strategy to set A. (See Figure 2.2
for a simple example.)
3
2 3 2
4
1
G
A
reach
0
(A)
Fig. 2.2. The 0-reachability set of A and a positional 0-reachability strategy to set A.
Lemma 2.2. For any set A ⊆ V and i ∈ {0, 1}, the set reach
i
(A) can be computed
in O(m) time, where m = |E| is the n umber of edges in the game.
Proof. The vertices of A are in reach
i
(A) so we initialize B ← A. We then
iteratively add to B every vertex of V
i
that has at least one edge going into B, and
every vertex of V
¬i
all of whose edges go into B. We stop when no new vertices can
be added to B. It is easy to see that this process can be performed in O(m) time,
and that when it ends we have B = reach
i
(A), as required.
Lemma 2.3. For any set A ⊆ V and i ∈ {0, 1}, the set V \ reach
i
(A) is (¬i)-
closed.
Proof. Let u ∈ V \ reach
i
(A). If u ∈ V
¬i
then there must be an edge (u, v) ∈ E
from vertex u into the set V \reach
i
(A), i.e., such that v 6∈ reach
i
(A), since otherwise
the iterative procedure from the proof of Lemma 2.2 would add vertex u to reach
i
(A).
Similarly, if u ∈ V
i
then all edges from vertex u must go into the set V \ reach
i
(A).
Therefore, the set V \ reach
i
(A) is (¬i)-closed.
If B ⊆ V is such that for every vertex u in V \ B there is an edge (u, v) with v
in V \B, then the game G \B is the game obtained from G by removing the vertices
of B and all the edges that touch them.
Lemma 2.4. Let G be a parity game and let i ∈ {0, 1} and j = ¬i. If U ⊆ win
i
(G)
and U = reach
i
(U), then win
i
(G) = win
i
(G \ U) ∪ U and win
j
(G) = win
j
(G \ U).
Proof. It suffices to exhibit strategies for Players j and i that are winning for
them in the game G, from the sets win
j
(G \U ) and win
i
(G \U ) ∪U respectively. We
claim that a winning strategy for Player j from the set win
j
(G \ U) in the subgame
A DETERMINISTIC SUBEXPONENTIAL ALGORITHM FOR PARITY GAMES 5
G \ U is also winning for her from the same set in the original game G. It suffices to
establish that Player i cannot escape from the set win
j
(G \ U) in the game G. This
follows from Lemma 2.3 (the set V \ U is j-closed in G, and hence Player i cannot
escape to U ) and from Lemma 2.1 (the set win
j
(G \U) is j-closed in G \U , and hence
Player i cannot es cape to win
i
(G \ U)).
Now we exhibit a winning strategy for Player i in the game G from the set win
i
(G\
U) ∪U. By the assumption that U ⊆ win
i
(G), there is a strategy σ for Player i in the
game G which is winning for her from all vertices in U . Let τ be a winning strategy
for Player i from the set win
i
(G\U) in the subgame G \U. We construct a strategy π
for Player i in the game G by composing strategies τ and σ in the following way: if
the play so far is contained in the set win
i
(G\U) then follow strategy τ, otherwise use
the i-reachability strategy to the set U and “restart” the play following the strategy σ
thenceforth. The strategy π is well-defined because, by Lemma 2.1, Player j can
escape from win
i
(G \ U ) only into the set U. It is a winning strategy for Player i
because if strategy π ever switches from following τ into following σ then an infinite
suffix of the play is winning for Player i, and in parity games removing an arbitrary
finite prefix of a play does not change the winner.
Lemma 2.5. Let G be a parity game. Let d = d(G) be the highest priority and
let A = A
d
(G) be the set of vertices of highest priority. Let i = d mod 2 and j = ¬i.
Let G
0
= G \ reach
i
(A). Then, we have win
j
(G
0
) ⊆ win
j
(G). Also, if win
j
(G
0
) = ∅
then win
i
(G) = V (G), i.e., Player i wins from every vertex of G.
3
2 2
1
W
1
'
G' =G \ reach
0
(A)
W
0
'
Fig. 2.3. The game G
0
= G \ reach
0
(A) and winnin g sets W
0
i
= win
i
(G
0
) for i = 0, 1.
Proof. We claim that a winning strategy for Playe r j from vertices in the set
win
j
(G
0
) in the subgame G
0
is also winning for her in the whole game G. Indeed,
by Lemma 2.1 Player i cannot escape from win
j
(G
0
) to win
i
(G
0
), and by Lemma 2.3
Player i cannot escape from win
j
(G
0
) to reach
i
(A). (As an example, consider Fig-
ures 2.2 and 2.3, with i = 0 and j = 1.)
Supp ose now that win
j
(G
0
) = ∅. Let τ be a winning strategy for Player i from
win
i
(G
0
) (which, by determinacy, is equal to V \ reach
i
(A)) in the subgame G
0
. We
construct a strategy π for Player i in the following way: if a play so far is contained in
the set win
i
(G
0
) then follow strategy τ ; otherwise the current vertex is in reach
i
(A) so
follow the i-reachability strategy to the set A; moreover, each time the play re-enters
the set win
i
(G
0
) “restart” the play and follow strategy τ henceforth, etc. If a play
following the strategy π eventually never leaves the set win
i
(G
0
) then it has an infinite
suffix played according to strategy τ and hence it is winning for Player i. Otherwise it
leaves the set win
i
(G
0
) infinitely often, so it visits a vertex of the maximal priority d
infinitely often, and hence it is winning for Player i because i = d mod 2.
