Energy Parity Games

Our next goal is to solve bounded parity games with weights, thus providing the

oracle required by : Algorithm 3.1. To this end we employ a similar approach to that : Sec. 3.2, Page 42

for solving parity games with weights, i.e., we reduce the problem to that of iteratively solving a number of conceptually simpler games, namely energy parity games in this case [CD12].

An energy parity game G = (A,Ω, Weight) consists of an arena A with vertex Def.energy parity

game

set V and edge set E, a coloring Ω : V → N of V, and a weighting Weight: E → Z

ofA. This definition is not compatible with the framework presented in: Section 2, as : Page 11

we have not (yet) defined the winner of the plays. This is because the winner depends on an initial credit, which is existentially quantified in the definition of winning the

gameG. Formally, the set of winning plays with initial credit cI ∈N is defined as Def.initial credit EnergyParitycI(Ω, Weight) =Parity(Ω) ∩

v0v1v2· · · ∈Vω | ∀j∈N. cI+Weight(v0· · ·vj) ≥0

.

Now, we say that Player 0 wins the energy parity game G = (A,Ω, Weight) from Def.winning an

energy parity game

some vertex v ∈ V if there exists some initial credit cI ∈ N such that she wins GcI =

(A, EnergyParitycI(Ω, Weight)) from v (in the sense of the definitions in Section 2). If this is not the case, i.e., if Player 1 wins every GcI from v, then we say that Player 1 wins G from v. In energy parity games, the initial credit is uniform for all plays. This contrasts the bound on the cost-of-response in the parity condition with weights, which may depend on the play.

Unravelling these definitions shows that Player 0 winsG from v if there is an initial credit cIand a strategy σ, such that every play that starts in v and is consistent with σ satisfies the parity condition and the accumulated weight over the play prefixes (the

energy level) never drops below −cI. We call such a strategy σ a winning strategy Def.winning strategy

for Player 0

for Player 0 in G from v. Dually, Player 1 winsG from v if, for every initial credit cI, there is a strategy τcI, such that every play that starts in v and is consistent with τcI violates the parity condition or its energy level drops below −cI at least once. Hence, as the notation suggests, Player 1 may be required to use different strategies τcI to win depending on the initial credit cI. However, Chatterjee and Doyen [CD12] showed that this is, in fact, not necessary for him: There is a uniform strategy τ for Player 1 that is winning for him from v for every initial credit cI.

Proposition 3.11([CD12]). LetG be an energy parity game. If Player 1 winsG from v, then

he has a positional strategy that is winning from v inG_cI for every cI.

We call such a strategy τ as in Proposition 3.11 a winning strategy for Player 1 Def.winning strategy for Player 1

from v. A play beginning in v that is consistent with τ either violates the parity condition, or the energy levels of its prefixes diverge towards−∞, i.e., Player 1 is able

to unbound the energy from below in the latter case.

Furthermore, Chatterjee and Doyen showed an analogous result characterizing the relevant space of strategies for Player 0: They obtained an upper bound on the initial

credit necessary for Player 0 to win an energy parity game, as well as an upper bound on the size of a corresponding finite-state winning strategy.

Proposition 3.12 ([CD12]). LetG be an energy parity game with n vertices, d colors, and

largest absolute weight W. Moreover, let v be a vertex ofG.

If Player 0 winsGfrom v, then she winsG(n−1)W from v with a finite-state strategy with at most ndW states.

This proposition yields that finite-state strategies of bounded size suffice for Player 0 to win, i.e., that they suffice for her to bound the energy level from below. A straight- forward pumping argument yields the additional property that such strategies do not admit long expensive descents. This property later on allows us to reason about the structure of plays consistent with winning strategies for Player 0.

Lemma 3.13. LetG = (A,Ω, Weight)be an energy parity game with n vertices and largest

absolute weight W. Further, let σ be a finite-state strategy of size s, and let ρ be a play that starts in some vertex from which σ is winning, and that is consistent with σ.

Every infix π of ρ satisfies Weight(π) > −Wns.

Proof. Let σ be implemented byM = (M, init, upd)and let ρ= v0v1v2· · ·. Towards a contradiction assume that there is an infix π= vj· · ·vj0 of ρ with Weight(π) ≤ −Wns. Since W is the maximal absolute weight occurring in G, the infix π attains at least ns+1 different nonpositive energy levels. Hence, we obtain ns+1 prefixes of π with increasing length and with strictly decreasing nonpositive energy levels.

Thus, there are positions j0, j1 with j≤ j0 < j1 ≤ j0 with vj0 = vj1, upd

+

(v0· · ·vj0) = upd+(v0· · ·vj1), and Weight(vj0· · ·vj1) <0. Hence, the play v0· · ·vj0−1(vj0· · ·vj1−1)

ω

obtained by repeating the loop between vj0 and vj1 ad infinitum begins in a vertex from which σ is winning and is consistent with σ. This play, however, violates the energy parity condition, which in turn contradicts σ being winning from v0.

As a final result, Chatterjee and Doyen gave an upper bound on the complexity of solving energy parity games. This bound was recently improved to pseudo-quasi- polynomial time by Daviaud, Jurdzi ´nski, and Lazi´c [DJL18].

Proposition 3.14([CD12, DJL18]). The following problem is in NP∩coNP:

“Given an energy parity game G with n vertices, d colors, and largest absolute weight W, and a vertex v inG, does Player 0 winG from v?”

Moreover, the problem can be solved in timeO(dnlog(d/ log n)(W+1/n)).

No non-trivial lower bound on the complexity of solving energy parity games was found since the initial presentation of the problem by Chatterjee and Doyen [CD12].

Thus, the problem belongs to a family of problems that are known to be in NP∩

coNP, but for which no polynomial-time algorithm exists. It shares this property

with, e.g., mean-payoff parity games [CHJ05]. Furthermore, solving energy parity games is polynomial-time equivalent to solving mean-payoff parity games [CD12].