Nathanaël Fijalkow

CNRS, LaBRI, Bordeaux, and The Alan Turing Institute of data science, London

Parity games: the quasipolynomial era

Universal trees

Question: what is the smallest tree containing all trees of height $2$ with $5$ leaves?


Definition: A tree is $(n,h)$-universal if it contains all trees of height $h$ with $n$ leaves

Remark: the number $\binom{h + \log(n)}{h}$ is
  • $O(n^{\log(h)})$ in general
  • $n^{O(1)}$ for $h = O(\log(n))$

Upper bounds

We construct a $(n,h)$-universal tree. Inductively
Fact: there exists a balanced node

Lower bounds

$$g(n,h) = \sum_{d = 1}^n g(\lfloor n / d \rfloor,h-1)$$ Let $T$ a $(n,h)$-universal tree and $\delta \in [1,n]$.
Claim: the number of nodes at depth $h-1$ of degree $\ge \delta$ is at least $g(\lfloor n / \delta \rfloor,h-1)$.
Claim: $T_\delta$ is $(\lfloor n / \delta \rfloor,h-1)$-universal

Parity games

Parity: the maximal priority appearing infinitely often is even

Solving parity games

INPUT: A parity game and initial vertex $v_0$
QUESTION: Does Eve have a winning strategy from $v_0$?

Parameters: $n$ (number of vertices), $m$ (number of edges) and $d$ (number of priorities)

Best algorithm $$O \left( m \cdot \binom{d/2 + \log(n)}{d/2} \right) = O(n^{\log(d)})$$

Why might you care?

Parity games play a crucial role in:
But also complexity: in $\textrm{NP} \cap \textrm{coNP}$, not known to be in $\textrm{P}$!

Also: included in mean payoff, discounted payoff, and simple stochastic games
Positional strategy $$\sigma : V \to E$$ Theorem: If Eve has a winning strategy in a parity game, she also has a positional winning strategy. The same holds for Adam.

Definition: A graph satisfies parity if all paths in the graph satisfy parity
Remark: If $\sigma$ is a positional winning strategy, then $G_{\sigma}$ satisfies parity

Value iteration

Büchi: parity with priorities $1$ and $2$

Lemma: A graph satisfies Büchi if and only if there exists a progress measure $\mu : V \to \mathbb{N}$: $$(v,1,v') \in E \implies \mu(v) < \mu(v')$$
A tree is the graphical representation of nested orders $\triangleleft_p$, for $p \in [1,d]$. Example:

Lemma: A graph satisfies parity if and only if there exists a tree $T$ and $\mu : V \to T$: $$(v,p,v') \in E \implies \mu(v) \triangleleft_p \mu(v')$$
$G$ parity game. A progress measure is $\mu : V \to T \cup \{ \bot \}$: $$ \begin{array}{c} \forall v \in V_{\text{Eve}},\ \exists (v,p,v') \in E \wedge \mu(v) \triangleleft_p \mu(v') \\ \forall v \in V_{\text{Adam}},\ \forall (v,p,v') \in E,\ \mu(v) \triangleleft_p \mu(v') \end{array} $$
Theorem: There exists a tree $T$ and a progress measure $\mu : V \to T \cup \{ \bot \}$ such that $\mu(v) \neq \bot$ if and only if Eve wins from $v$

Corollary: Let $\mathcal{T}$ a $(n,d/2)$-universal tree. There exists a progress measure $\mu : V \to \mathcal{T} \cup \{ \bot \}$ such that $\mu(v) \neq \bot$ if and only if Eve wins from $v$
Key idea: The value iteration algorithm constructs the largest progress measure

Three similar stories

good for small games, value iteration, and fixed point are families of algorithms for parity games, parametrised by the choice of a universal tree!

Beyond parity

Definition: A graph satisfies W if all paths in the graph satisfy $W$
Definition: A (graph) homomorphism is $\phi : V \to V'$ st $$(v,w,v') \in E \Longrightarrow (\phi(v),w,\phi(v')) \in E$$
Definition: A graph $U$ is $(n,W)$-universal if
  • it satisfies $W$
  • all graphs of size $n$ satisfying $W$ can be homomorphically mapped into $U$

Value iteration algorithm

Let $G$ game with objective $W$ and $U$ a $(n,W)$-universal graph

We construct a value iteration algorithm of time complexity $m |U|$ and space complexity $n \log |U|$.

Happening now

with universal graphs