carlocomin.files.wordpress.com · complexity in inﬁnite games on graphs and temporal constraint...

Complexity in Infinite Games on Graphsand Temporal Constraint Networks

Carlo CominAdvisor: Prof. Romeo Rizzi (University of Verona, Italy)

Co-Advisor: D.d.R. Stéphane Vialette (LIGM UPEM, CNRS, France)

Referee: Prof. Luke Hunsberger (Vassar College, NY, US)Referee: Prof. Angelo Montanari (University of Udine, Italy)

University of Trento jointly University of Veronain co-tutélle with

Université Paris-Est in Marne-la-Vallée

Complexity in Infinite Games on Graphs and Temporal Constraint Networks

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Chapter 1Introduction and Context


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Introduction and Context1 Temporal Constraint Networks (Temporal Planning and Scheduling) [Dechter, 1991]

2 Infinite Games on Finite Graphs (Games for Formal Verification) [Grädel, 2002]


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Introduction and Context1 This dissertation provides further evidence that game theoretic arguments

help to study algorithmic problems in the area of automated temporalplanning and formal verification of finite state non-terminating systems.

I Automated temporal planning is a branch of Artificial Intelligence (AI) thatconcerns the realization of temporal strategies or temporal actionsequences, typically for execution by intelligent agents, autonomous robotsand unmanned vehicles.

I Non-terminating computing systems abound in environments as varied ashousehold appliances, medical equipment, industrial control systems, flightcontrol systems in airplanes, etc. Failures caused by design faults may bevery costly and they should be avoided as much as possible. Behaviour ofsuch systems is typically very complex which makes their design andvalidation a challenge. Formal methods try to address this challenge bydeveloping formal models of such systems, and methods to specify andreason about their properties.


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Introduction and ContextWe began our research by studying algorithmic problems in automatedtemporal planning, particularly, conditional temporal planning.At some point, we have identified interesting connections between thealgorithmics of these problems and that of a specific family of infinite2-player pebble games played on finite graphs.

I i.e. Energy (EGs) and Mean-Payoff Games (MPGs).I MPGs are intimately related to Parity Games and the semantics of the

modal µ-calculus, a well-known logic for formal verification.


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In summary...Algorithmic Game Theory (especially, infinite two-player pebble gamesplayed on finite graphs) is the red thread that makes it possible to look atthe various contributions of this dissertation in a sufficiently coherent way.

I A game theoretic formulation helps to abstract away from syntactic andsemantic peculiarities of modelling formalisms and makes the conditionaltemporal constraints problems in question more easily amenable toalgorithmic and complexity analysis.

I On the other side, this have led us to deepen the study of algorithmic andcomplexity issues in infinite games on finite graphs per se. Ultimately, weobtained faster algorithms and improved complexity bounds for some ofthese games, i.e., Update Games, Explicit McNaughton-Müller Games,and finally, for Mean Payoff Games.


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Summary of Results (I)

Temporal Constraint Networks (Temporal Planning and Scheduling) [Dechter, 1991]

1 Hyper Temporal Networks (A Tractable Generalization of Simple Temporal Networksand its relation to Mean Payoff Games)

I 21st International Symposium on Temporal Representation and Reasoning(TIME2014), Verona, Italy, 2014

I Constraints, an International Journal, Springer-US, 20162 Singly-Exponential Time DC-Checking of Conditional Simple Temporal

Networks via Mean Payoff GamesI 22st International Symposium on Temporal Representation and Reasoning

(TIME2015), Kassel, Germany, 2015I Information and Computation, Elsevier, Special Issue of TIME2015

3 Instantaneous Reaction-Time in DC-Checking of Conditional SimpleTemporal Networks

I 23st International Symposium on Temporal Representation and Reasoning(TIME2016), Copenhagen, Denmark, 2016


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Summary of Results (II)

Infinite Games on Finite Graphs (Games for Formal Verification) [Grädel, 2002]

Parity Games, Mean Payoff Games, Energy Games, Update Games, Müller-McNaughton, ...

1 Linear-Time Algorithm for Update Games viaStrongly-Trap-Connected-ComponentsI Submitted.

2 An Improved Upper Bound on the Time Complexity of the Value Problemand Optimal Strategy Synthesis in Mean Payoff Games

I Strategic Reasoning 2015 (SR2015), Oxford, U.K.I Algorithmica, Springer-US, 2016

3 Faster O(|V |2|E|W )-Time Energy Algorithms and Energy-LatticeDecomposition for Optimal Strategy Synthesis in Mean Payoff Games

I Submitted.


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Summary of Results (III)

Not part of the dissertation, but for completeness:

Algorithms and Complexity for Computational Biology1 Sorting with Forbidden Intermediates

I 3rd International Conference on Algorithms for Computational Biology,AlCoB 2016, Trujillo, Spain, 2016

I Submitted to IEEE/ACM Transactions on Computational Biology andBioinformatics (TCBB), Special Issue of AlCoB 2016

2 An Improved Upper Bound on Maximal Clique Listing via RectangularFast Matrix Multiplication (i.e., from O(n2.3728639) to O(n2.093362) time-delay)

I Algorithmica, Springer-US (2017)


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Chapter 2Hyper Temporal Networks


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Simple Temporal Problems

Simple Temporal Networks (STNs) [Dechter, Meiri, Pearl, 1991]Represent a general framework for analyzing systems (conjunctions) ofdifference constraints on ordered pairs of temporal variables.An STN can be encoded by a weighted directed graph:

I a node represents a time-point variable (time-point);I an arc represents a temporal distance constraint:

F ua→ v stands for v − u ≤ a, a ∈ R;

F (v −a→ u stands for v − u ≥ a).

C

AB D

RepresentedConstraints:

4 ≤ A−D ≤ 56 ≤ B −A ≤ 71 ≤ C −A ≤ 4−1 ≤ B − C ≤ 1

1 ≤ C −D ≤ 3

11

−14 −13

7

−6 −4

5


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Consistency of Simple Temporal Problems

STN Consistency [Dechter, Meiri, Pearl 1991]

An STN 〈G = (V,E), `〉 is consistent if it admits a feasible schedulingfunction, i.e., we can assign a real value s(v) to each time-point v, suchthat all constraints are satisfied:

∃ s : V 7→ R such that:s(v) ≤ s(u) + `(u,v) ∀(u, v) ∈ E.

An STN is not consistent if it contains a negative cycle.

C

AB D


4 ≤ A−D ≤ 56 ≤ B −A ≤ 71 ≤ C −A ≤ 4−1 ≤ B − C ≤ 1

1 ≤ C −D ≤ 3

11

−14 −1

7 5

−6 −4

3


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Introduction - STNsSTN Consistency

An STN (G = (V,E), `) is consistent if it admits a feasible schedulingfunction, i.e., we can assign a real value s(v) to each time-point v, suchthat all constraints are satisfied:

∃s : V 7→ R such that:s(v) ≤ s(u) + `(u,v) ∀(u, v) ∈ E.


C

AB D


4 ≤ A−D ≤ 56 ≤ B −A ≤ 71 ≤ C −A ≤ 4−1 ≤ B − C ≤ 1

1 ≤ C −D ≤ 3

11

−14 −1

7 5

−6 −4

3

−1


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C

AB D


4 ≤ A−D ≤ 56 ≤ B −A ≤ 71 ≤ C −A ≤ 4−1 ≤ B − C ≤ 1

1 ≤ C −D ≤ 30

11

−14 −1

7 5

−6 −4

3

−1


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C

AB D


4 ≤ A−D ≤ 56 ≤ B −A ≤ 71 ≤ C −A ≤ 4−1 ≤ B − C ≤ 1

1 ≤ C −D ≤ 30

≤ −6

11

−14 −1

7 5

−6 −4

3

−1


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C

AB D


4 ≤ A−D ≤ 56≤B −A ≤ 71 ≤ C −A ≤ 4−1 ≤ B − C ≤ 1

1 ≤ C −D ≤ 30

≤ −6≤ −10

11

−14 −1

7 5

−6 −4

3

−1


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C

AB D


4≤A−D ≤ 51 ≤ C −A ≤ 4−1 ≤ B − C ≤ 1

1 ≤ C −D ≤ 30≤ −6

≤ −10

≤ −7

11

−14 −1

7 5

−6 −4

3

−1


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C

AB D


1 ≤ C −A ≤ 4−1 ≤ B − C ≤ 1

1 ≤ C −D≤30

≤ −6≤ −10

≤ −7

≤ −8

11

−14 −1

7 5

−6 −4

3

−1


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C

AB D

Negative circuits:`(ACB) = −1`(ADC) = −2

`(ADCB) = −60

≤ −6≤ −10

≤ −7

≤ −8

11

−14 −1

7 5

−6 −4

3

−1


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C

AB D


`(ADCB) = −60

≤ −6≤ −10

≤ −7

≤ −8≤ −10

11

−14 −1

7 5

−6 −4

3

−1


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C

AB D


`(ADCB) = −60

≤ −6≤ −10

≤ −7

≤ −8≤ −10≤ −12

11

−14 −1

7 5

−6 −4

3

−1


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HyTNs: an extension of STNs

A more general arc constraint

v1

v2

v3

tA

A,w1

A,w2A,w3

A hypergraph H is a pair (V,A), where V is the set of nodes, and A isthe set of hyperarcs. Each hyperarc A ∈ A has a distinguished node tA,called the tail of A, and a nonempty weighted set (HA, wA), whereHA ⊆ V \ tA contains the heads of A, and each head v ∈ HA isassociated with a weight wA(v) ∈ R.


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A more general arc constraint

v1

v2

v3

tA

A,w1

A,w2A,w3

An HyTNH = (V,A) is consistent if it admits a feasible schedulingfunction, i.e., we can assign a real value φ(v) to each time-point v, suchthat each hyperarc constraint is satisfied:

φ(tA) ≥ minv∈HA

φ(v)− wA(v) ∀A ∈ A. (1)


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Milestone events and joins

v1

v2

v3

hA

A,0

A, 0

A, 0

0

0

0

We may consider multi-tail hyperarcs as well multi-head ones.This allows to introduce, in any HyTN model, an event bound to occurprecisely when the last/first event of an associated set of events occurs.This allows to represent AND/OR join connectors from workflows.These could not be done before.


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Our findings on the two versions of HyTNs in a nutshell

A complexity map, algorithms, and characterizationsIf we allow both multi-tails arcs and multi-heads arcs theConsistency-Checking problem becomes NP-complete.

The multi-tails and the multi-heads versions of HyTNs are inter-reducible.(Direct reduction by reversing all arcs and inverting the time axis).

If we allow either only multi-tails arcs or only multi-heads arcs theConsistency problem can be efficiently solved in pseudo-polynomial time.

Both of these problems can be reduced to the Determinacy problem forEnergy and Mean Payoff Games.

In fact, both of these problems are equivalent to the Determinacy problemfor Mean Payoff Games.

HyTN’s Consistency-Checking is well characterized: we have introduceda notion of negative hyper-circuit as NO-certificate.


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Negative Cycles Certificates

Given a multi-head HyTNH = (V,A), a cycle is a pair (S, C) with S ⊆ V andC ⊆ A such that:

1 S =⋃A∈C(HA ∪ tA) and S 6= ∅;

2 ∀v ∈ S there exists an unique A ∈ C such that tA = v.

v1

v2

v3

v0

v4v5

v6

A0,wA

0(v1)

A0, wA0(v2)A

0 , wA

0 (v3 )

A1 , w

A1 (v4 ) A1,

wA1(v5)

A2,w A

2(v 5

)

A2 , w

A2 (v6 )

A4,wA

4(v

0)

A4, wA4(v5)

A6 ,w

A6 (v

5 )

A6, wA6(v3)

A3, wA3(v6)

A3 , w

A3 (v0 )

A5,w A

5(v 2

) A5 ,w

A5 (v

6 )


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Extensions of HyTNs: NP-Completeness proofs

Let ϕ(x1, . . . , xn) =∧mi=1(αi ∨ βi ∨ γi) be a 3-CNF formula,

z

[0]

xixi

1

00

−1

1

0

0

−1

(a) Gadget for a 3-SAT variable xi.

Cj

[1]

βjαj γj

z

[0]

+1

−1

00

0

(b) Gadget for a 3-SAT clauseCj = (αj ∨ βj ∨ γj) where each αj , βj , γj isa positive or negative literal.


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The local replacement reduction

h1

h2

h3

tA

A,w1

A,w2A,w

3

h1

h2

h3

AtA

w1

w2

w3

0

How an hyperarc (multi-head) gets replaced:

If H = (V,A) then GH = (V0 ∪ V1, E), where:

V1 = V ;

V0 = A;

E = (tA, A, 0) | A ∈ A ∪ (A, h,wA(h)) | A ∈ A, h ∈ HA.


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Mean Payoff Games: the model

A B

CD

−1

−2

+1

+2 −1 −9

−1

+1

+2

A Mean-Payoff Game (MPG) is a two-player infinite game played on anarena Γ = 〈V,E,w, (VMax, VMin)〉.GΓ = 〈V,E,w〉 is a finite weighted directed graph whose nodes arepartitioned in two classes, VMax and VMin.Every node has at least one outgoing arc.Weights are integers, i.e., w : E → Z.Nodes in Vp, where p ∈ Max, Min, are those under control of Player p.


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Mean Payoff Games: play example

A B

CD

−1

−9 −2

+1

+2 −1

−1

+1

+2

Each match starts with a pebble placed at some node v ∈ VMax ∪ VMin.

Here v = A ∈ VMax, the nodes controlled by Player Max.

Player Max chooses an arc e ∈ E exiting v and moves the pebble along e.


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A B

CD

start −1

−2

+1

+2 −1

−1

+1

+2

−9

A B

CD

−1

−2

+1

+2 −1

−1

+1

+2

−9

A B

CD

−1

−2

+1

+2 −1

−1

+1

+2

−9

A B

CD

−1

−2

+1

+2 −1

−1

+1

+2

−9

The two players move the pebble ad infinitum along the arcs...

The infinite sequence of encountered nodes, i.e.,π = v0v1 · · · vn · · · = ADC(DC)∗ is a play.


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A B

CD

−1

−9 −2

+1

+2 −1

−1

+1

+2

In order to play well, Player Max wants to maximize the limit inferior of thelong-run average weight, i.e., lim infn→∞ 1

n

∑n−1i=0 w(vi, vi+1).

and Player Min wants to minimize lim supn→∞ 1n

∑n−1i=0 w(vi, vi+1).


Carlo Comin 23/89


EC

B

A

D

F G0

0

0

0

+3+3

−5−5

−5

+3

Formally, the Player 0’s payoff of a play v0v1 . . . vn . . . in Γ is:

MP0(v0v1 . . . vn . . .) := lim infn→∞

1n

n−1∑i=0

w(vi, vi+1).

The value secured by a strategy σ0 ∈ Σ0 in a vertex v is:

valσ0(v) := infσ1∈Σ1

MP0(outcomeΓ(v, σ0, σ1)

),

The (optimal) value of a vertex v ∈ V in the MPG Γ turns out to be:

valΓ(v) = supσ0∈Σ0

valσ0(v) = infσ1∈Σ1

valσ1(v).


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Mean Payoff Games: main facts

Positional StrategyIn a positional strategy σp for Player p, she decides once and for all the onearc to use for exiting node v, for every v ∈ Vp. She forgets about all her otheroptions and lets the opponent play solitaire in the resulting arena GΓ

σp .

EC

B

A

D

F G0

0

0

0

+3+3

−5−5

−5

+3

Determinacy Theorem (Ehrenfeucht, Mycielski 1979)

For every node v ∈ VMax ∪ VMin, there exists a unique value valΓ(v) ∈ Qsuch that Player Max has a positional strategy which secures game payoff atleast valΓ(s) in any match starting from s, whereas Player Min has apositional strategy which secures game payoff at most valΓ(s) in any matchstarting from s.


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MPG ProblemsThe DECISION problem asks, given a node v ∈ V , to decide whether or notvalΓ(v) ≥ 0. The setsWMax = v ∈ V | valΓ(v) ≥ 0 andWMin = V \WMax arecalled Winning Regions.

I The DECISION problem lies in NP ∩ coNP (Zwick, Paterson 1996).It is recognizable with Unambiguous Polynomial-Time Non-DeterministicTuring Machines (Jurdzinski 1998), i.e., it is in UP ∩ coUP.

The VALUE problem asks to compute the values of all nodes.

The STRATEGY SYNTHESIS problem asks to compute, for each player, a positionalstrategy securing her payoffs corresponding to the values of the starting node.


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Pseudopolynomial-time algorithms for MPGs:

In 1996, Zwick and Paterson gave a O(|V |3|E|W ) time algorithm for the ValueProblem and a O(|V |4|E|W log(|E|/|V |)) time algorithm for OptimalStrategy Synthesis.

In 2011, Brim, Raskin, et al. gave a O(|V |2|E|W log(|V |W )) time algorithmboth for the Optimal Strategy Synthesis and for the Value Problem.


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Energy Games: the model

In EGs, given an initial credit c ∈ N and a node v ∈ V , Player Max winsthe game starting from v if she can maintain the sum of the encounteredweights always non-negative, i.e., if it holds on the play vi∞i=0 that:

c+j∑i=0

w(vi, vi+1) ≥ 0, for all j ≥ 0;

otherwise, the winner is Player Min.

The decision problem for EGs asks whether there exists an initial credit c∗

for which Player Max wins from a given starting position node v. Thecorresponding winning regions are also denotedWMax andWMin.

EGs are log-space equivalent to MPGs [Patricia Bouyer (2008)].

EGs can be solved in O(|V ||E|W ) time with the Value-IterationAlgorithm [Brim, Raskin, et. al, 2011]


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The Consistency-Checking algorithm

Feasible Scheduling ComputationIf every node in the corresponding game GH is winning for Player 0:

I we compute a positional winning strategy π0 for GH ;I the graph Gπ0 induced by π0 is conservative, hence, we compute a

potential function p : Gπ0 → R;I p is a feasible scheduling of H .

Negative Cycle ComputationIf there exist nodes in GH which are winning for Player 1:

I let G[W1] be the subgraph of GH obtained by removing all nodes whichare winning for Player 0;

I on G[W1] we compute a positional winning strategy π1 for Player 1;I let W1 = W1 ∩ V1 and consider the family of hyperarcsC = π1(v)v∈W1

;I (W1, C) is a negative cycle of H .


Carlo Comin 29/89

Algorithmics of HyTNs: Upper Bound

Theorem (Comin, Posenato, Rizzi, TIME2014 and Constraints, 2016 )

The following propositions hold on HyTNs.1 There exists an O((|V |+ |A|)mAW ) pseudo-polynomial time algorithm

for checking HyTN-Consistency;2 There exists an O((|V |+ |A|)mAW ) pseudo-polynomial time algorithm

such that, given in input any consistent HyTNH = (V,A), it returns asoutput a feasible scheduling φ : V → R ofH;

Here, W , maxA∈A,v∈HA |wA(v)|.

The approach was shown to be robust by experimental evaluations, e.g. (randomly generated)HyTNs of size up to |V | ∼ 106 and W ∼ 103 were solved.


Carlo Comin 30/89

Algorithmics of HyTNs: Lower Bound

v5

v4 v3

v0

v1

v2

A0, 0A0, 0

A1, 0A1, 0 A2,−1

A3,−WA4, 0A5, 0

A HyTN which requires Θ(W ) computation time.


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Chapter 3Conditional Hyper Temporal Networks and Dynamic Consistency Checking


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Conditional Simple Temporal Networks: the Model

CSTN Γ = 〈V,A, L,O,OV, P 〉 [Tsamardinos, Vidal, Pollack (2003) /Hunsberger, Posenato, Combi (2012)

The CSTN formalism extends STNs in that:1 some of the nodes are called observation events and to each of them is

associated a boolean variable, to be disclosed only at execution time;2 labels (i.e. conjunctions over the literals) are attached to all nodes and

constraints, to indicate the situations in which each of them is required.

A B C

Op

p?Op

p?

Oq

q?Oq

q?

10

−10

3, p ∧ ¬q3, p ∧ ¬q 2, q2, q0

50

90

10

1,¬p1,¬p


Carlo Comin 33/89


ScenarioA scenario s over a set P of boolean variables is a truth assignment:

s : P → >,⊥.

An example CSTN Γ = 〈V,A, L,O,OV, P 〉:

A B C

Op

p?

Oq

q?

10

−10

3, p ∧ ¬q 2, q0

50

90

10

1,¬p


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s : P → >,⊥.

Let s(p) = > and s(q) = ⊥...

A B C

Op

s(p) = >

Oq

s(q) = ⊥

10

−10

3, p ∧ ¬q 2, q0

50

90

10

1,¬p


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s : P → >,⊥.

Let s(p) = > and s(q) = ⊥, then Γ becomes:

A B C

Op

Oq

10

−10

30

50

90

10


Carlo Comin 36/89

CSTNs: Consistency

CSTN’s ConsistenciesThree notions of consistency arise for CSTNs: weak, strong, and...

dynamic consistency.

A B C

Op

p?

Oq

q?

10

−10

3, p ∧ ¬q 2, q0

50

90

10

1,¬p


Carlo Comin 37/89

CSTNs: Consistency

CSTN’s Dynamic ConsistencyDynamic Consistency (DC) requires the existence of conditional plans wheredecisions about precise timing of actions are postponed until exec. time,but it guarantees that all the relevant constraints will be ultimately satisfied.

A B C

Op

p?

Oq

q?

10

−10

3, p ∧ ¬q 2, q0

50

90

10

1,¬p


Carlo Comin 37/89

CSTNs: Dynamic Consistency

Difference Set ∆(s1; s2)

∆(s1; s2) ,Op ∈ V +

s1 ∩ OV | s1(p) 6= s2(p).

Dynamic Execution StrategyLet σ ∈ SΓ be an execution strategy. Then, σ is dynamic if and only if thefollowing implication holds for every s1, s2 ∈ ΣP , u ∈ V +

s1,s2 :( ∧v∈∆(s1;s2)

[σ(s1)]u ≤ [σ(s1)]v)⇒ [σ(s1)]u = [σ(s2)]u


Carlo Comin 38/89

CSTNs: Dynamic Consistency

Execution StrategyAn Execution Strategy for Γ is a mapping σ : ΣP → ΦV such that, for anyscenario s ∈ ΣP , the domain of the scheduling σ(s) ∈ ΦV is V +

s .

φ(A) = 0

φ(Op) = 1

φ(B) = 8

φ(Oq) = 9

φ(C) = 10

s(q) = >,⊥

φ(Oq) = 2

φ(B) = 3

φ(C) = 10

φ(B) = 8

φ(C) = 10

s(q) = > s(q) = ⊥

s(p) = > s(p) = ⊥


Carlo Comin 39/89

Main Results at TIME2015

Our Contribution:Lower-Bound: coNP-hard.

Upper-Bound: NE ∩ coNE ∩ pseudo-E.

A (pseudo) Singly-Exponential Time DC-Checking for CSTNs.There exists an

O(|ΣP |3|A|2|V |+ |ΣP |4|A||V |2|P |+ |ΣP |5|V |3|P |)W

(pseudo) singly-exponential time algorithm for checking DC on any inputCSTN Γ = 〈V,A,L,O,OV, P 〉.

I Here, W , maxa∈A |wa| and |ΣP | ≤ 2|P |.

In particular, given any dynamically-consistent CSTN Γ, the algorithmreturns a viable and dynamic execution strategy.

(here above, E is deterministic singly-exponential time, NE is nondeterministicsingly-exponential time)Complexity in Infinite Games on Graphs and Temporal Constraint Networks

Carlo Comin 40/89


Checking DC of CSTNs via MPGsMost importantly, we unveil a connection between the problem of checking DCin CSTNs and that of determining Mean Payoff Games (and Energy Games).


Carlo Comin 41/89


ε-Dynamic Consistency and the Reaction Time ε(Γ)In order to analyze the algorithm, we introduce a novel and refined notionof dynamic-consistency, named ε-dynamic-consistency;

I Consider ε(Γ) = supε ∈ R>0 | Γ is ε-dynamically-consistent.F ε(Γ) is the reaction time of Γ.

I We provide a sharp lower bounding analysis of the critical value of thereaction time ε(Γ) where the CSTN Γ transits from being, to not being,dynamically-consistent.

I This clarifies the role of the reaction time ε in the DC-checking of CSTNs.


Carlo Comin 42/89

Sketch of the reduction from CSTN-Dynamic-Consistency toHyTN-Consistency

Sketch of the reductionIntroduce ε-dynamic-consistency.

Prove that any execution strategy σ is dynamic iff σ is ε-dynamic for somereal number ε ∈ (0,+∞).Consider ε(Γ) = supε ∈ R>0 | Γ is ε-dynamically-consistent.

I ε(Γ) is the reaction time of Γ.

Prove that for any dynamically-consistent CSTN Γ, where V is the set ofevents and ΣP is the set of scenarios, it holds ε(Γ) ≥ |ΣP |−1|V |−1.

Devise an algorithm for checking ε-dynamic-consistency by reducing thatproblem to the consistency checking of HyTNs.Also, we proved that the bound ε(Γ) ≥ |ΣP |−1|V |−1 is optimal.


Carlo Comin 43/89

ε-dynamic-consistency, for some small real ε > 0

ε-dynamic-consistency

Given any CSTN 〈V,A,L,O,OV, P 〉 and any real number ε ∈ (0,+∞), anexecution strategy σ ∈ SΓ is ε-dynamic if it satisfies all the Hε-constraints,namely, for any two scenarios s1, s2 ∈ ΣP and any event u ∈ V +

s1,s2 , theexecution strategy σ satisfies the following constraint, denoted Hε(s1; s2;u):

[σ(s1)]u ≥ min(

[σ(s2)]u∪[σ(s1)]v + ε | v ∈ ∆(s1; s2)

)A CSTN Γ is ε-DC if it admits σ ∈ SΓ which is both viable and ε-dynamic.

v1

v2us1

us2

s1

s2

A,−ε

A,−ε

A,0

Figure: An Hε(s1; s2;u) constraint, modeled as a hyperarc.Complexity in Infinite Games on Graphs and Temporal Constraint Networks

Carlo Comin 44/89


Lemma 1If Γ is ε-dynamically-consistent, for some ε > 0, then Γ isε′-dynamically-consistent for every ε′ ∈ (0, ε].


Carlo Comin 45/89


Lemma 2Let σ be a dynamic execution strategy for the CSTN Γ. Then, there exists asufficiently small real number ε ∈ (0,+∞) such that σ is ε-dynamic.


Carlo Comin 45/89


Lemma 3Let σ be an ε-dynamic execution strategy for the CSTN Γ, for someε ∈ (0,+∞). Then, σ is dynamic.

⇒ Dynamic-Consistency of CSTNs is expressible with Max-Plus (or Min-Plus)constraints.


Carlo Comin 45/89

Solving ε-dynamic-consistency: Expansion of a CSTN

A B C

Op

p?

Oq

q?

10

−10

3, p ∧ ¬q 2, q0

50

90

10

1,¬p


Carlo Comin 46/89

Solving ε-dynamic-consistency: Expansion of a CSTN

As4Bs4Cs4Os4p

ps4 = >

Os4q qs4 = >

Os1q qs1 = ⊥ Os1p

ps1 = ⊥

Cs1 Bs1 As1

[10, 10]

2 0

[0, 5][0, 9]

10

[10, 10]

0

[0, 5]

[0, 9]

10

1

An excerpt of the expansion of the CSTN Γ with two scenarios s1 and s4.Complexity in Infinite Games on Graphs and Temporal Constraint Networks

Carlo Comin 47/89

As4Bs4Cs4Os4p

ps4 = >

Os4q qs4 = >

Os1q qs1 = ⊥ Os1p

ps1 = ⊥

Cs1 Bs1 As1

[10, 10]

2 0

[0, 5][0, 9]

10

[10, 10]

0

[0, 5]

[0, 9]

10

1


Carlo Comin 48/89

As4Bs4Cs4Os4p

ps4 = >

Os4q qs4 = >

Os1q qs1 = ⊥ Os1p

ps1 = ⊥

Cs1 Bs1 As1

[10, 10]

2 0

[0, 5][0, 9]

10

[10, 10]

0

[0, 5]

[0, 9]

10

1

0

−ε

−ε


Carlo Comin 48/89

As4Bs4Cs4Os4p

ps4 = >

Os4q qs4 = >

Os1q qs1 = ⊥ Os1p

ps1 = ⊥

Cs1 Bs1 As1

[10, 10]

2 0

[0, 5][0, 9]

10

[10, 10]

0

[0, 5]

[0, 9]

10

1

0

−ε

−ε0

−ε

−ε


Carlo Comin 48/89

As4Bs4Cs4Os4p

ps4 = >

Os4q qs4 = >

Os1q qs1 = ⊥ Os1p

ps1 = ⊥

Cs1 Bs1 As1

[10, 10]

2 0

[0, 5][0, 9]

10

[10, 10]

0

[0, 5]

[0, 9]

10

1

0

−ε

−ε0

−ε

−ε

0

−ε

−ε


Carlo Comin 48/89

As4Bs4Cs4Os4p

ps4 = >

Os4q qs4 = >

Os1q qs1 = ⊥ Os1p

ps1 = ⊥

Cs1 Bs1 As1

[10, 10]

2 0

[0, 5][0, 9]

10

[10, 10]

0

[0, 5]

[0, 9]

10

1

0

−ε

−ε0

−ε

−ε

0

−ε

−ε

0

−ε


Carlo Comin 48/89

As4Bs4Cs4Os4p

ps4 = >

Os4q qs4 = >

Os1q qs1 = ⊥ Os1p

ps1 = ⊥

Cs1 Bs1 As1

[10, 10]

2 0

[0, 5][0, 9]

10

[10, 10]

0

[0, 5]

[0, 9]

10

1

0

−ε

−ε0

−ε

−ε

0

−ε

−ε

0

−ε

0

−ε


Carlo Comin 48/89

As4Bs4Cs4Os4p

ps4 = >

Os4q qs4 = >

Os1q qs1 = ⊥ Os1p

ps1 = ⊥

Cs1 Bs1 As1

[10, 10]

2 0

[0, 5][0, 9]

10

[10, 10]

0

[0, 5]

[0, 9]

10

1

0

−ε

−ε0

−ε

−ε

0

−ε

−ε

0

−ε

0

−ε

0

−ε

−ε


Carlo Comin 48/89

As4Bs4Cs4Os4p

ps4 = >

Os4q qs4 = >

Os1q qs1 = ⊥ Os1p

ps1 = ⊥

Cs1 Bs1 As1

[10, 10]

2 0

[0, 5][0, 9]

10

[10, 10]

0

[0, 5]

[0, 9]

10

1

0

−ε

−ε0

−ε

−ε

0

−ε

−ε

0

−ε

0

−ε

0

−ε

−ε0

−ε

−ε


Carlo Comin 48/89

As4Bs4Cs4Os4p

ps4 = >

Os4q qs4 = >

Os1q qs1 = ⊥ Os1p

ps1 = ⊥

Cs1 Bs1 As1

[10, 10]

2 0

[0, 5][0, 9]

10

[10, 10]

0

[0, 5]

[0, 9]

10

1

0

−ε

−ε0

−ε

−ε

0

−ε

−ε

0

−ε

0

−ε

0

−ε

−ε0

−ε

−ε

0

−ε

−ε


Carlo Comin 48/89

As4Bs4Cs4Os4p

ps4 = >

Os4q qs4 = >

Os1q qs1 = ⊥ Os1p

ps1 = ⊥

Cs1 Bs1 As1

[10, 10]

2 0

[0, 5][0, 9]

10

[10, 10]

0

[0, 5]

[0, 9]

10

1

0

−ε

−ε0

−ε

−ε

0

−ε

−ε

0

−ε

0

−ε

0

−ε

−ε0

−ε

−ε

0

−ε

−ε

0

−ε


Carlo Comin 48/89

As4Bs4Cs4Os4p

ps4 = >

Os4q qs4 = >

Os1q qs1 = ⊥ Os1p

ps1 = ⊥

Cs1 Bs1 As1

[10, 10]

2 0

[0, 5][0, 9]

10

[10, 10]

0

[0, 5]

[0, 9]

10

1

0

−ε

−ε0

−ε

−ε

0

−ε

−ε

0

−ε

0

−ε

0

−ε

−ε0

−ε

−ε

0

−ε

−ε

0

−ε

0

−ε


Carlo Comin 48/89

Solving ε-DC: reducing from CSTN Γ to HyTNHε(Γ)

Theorem (CSTNs and HyTNs)

Given any CSTN Γ = 〈V,A,L,O,OV, P 〉, there exists a sufficiently smallreal number ε ∈ (0,+∞) such that Γ is dynamically-consistent if and only ifHε(Γ) is consistent.

Moreover,Hε(Γ) has at most |VHε | ≤ |ΣP | |V | nodes,|AHε | = O(|ΣP | |A|+ |ΣP |2|V |) hyperarcs,and it has size at most mAHε = O(|ΣP | |A|+ |ΣP |2|V | |P |).


Carlo Comin 49/89

The bound ε(Γ) ≥ |ΣP |−1 |V |−1 is sharp.

A natural question is whether the lower bound ε(Γ) ≥ |ΣP |−1 |V |−1 canbe improved up to ε(Γ) = Ω(|V |−1)...

... this would improve the time complexity by a factor |ΣP |.However, the following theorem shows that this is not the case byexhibiting a CSTN for which ε(Γ) = 2−Ω(|P |).

This proves that the bound ε(Γ) ≥ |ΣP |−1 |V |−1 is (almost) sharp.

Theorem

For each n ∈ N0 there exists a CSTN Γn such that

ε(Γn) < 2−n+1 = 2−|Pn|/3+1,

where Pn is the set of boolean variables of Γn.


Carlo Comin 50/89

How Γn looks like

Y1

Y1?

X10

X1?

Z1

Z1?1, X1Y1

[2],¬X1 [2],¬Y1


Carlo Comin 51/89

How Γn looks like

Y1

Y1?

X10

X1?

Z1

Z1?

Y2

Y2?

X2

X2?

Z2

Z2?

1, X1Y1

[2],¬X1 [2],¬Y1

[5], Z1 [5],¬Z1X2Y2

[5],¬Z1 [5], Z1X2Y2

[2],¬X2 [2],¬Y2


Carlo Comin 51/89

How Γn looks like

Y1

Y1?

X10

X1?

Z1

Z1?

Y2

Y2?

X2

X2?

Z2

Z2?

Yn

Yn?

Xn

Xn?

Zn

Zn?

1, X1Y1

[2],¬X1 [2],¬Y1

[5], Z1 [5],¬Z1X2Y2

[5],¬Z1 [5], Z1X2Y2

[2],¬X2 [2],¬Y2

[5],¬Z2 [5], Z2X3Y3

[5], Z2 [5],¬Z2X3Y3

[2],¬Xn [2],¬Yn

[5], Zn−1 [5],¬Zn−1XnYn

[5],¬Zn−1 [5], Zn−1XnYn


Carlo Comin 52/89

Chapter 4Instantaneous-Reaction Time in DC-Checking of CSTNs


Carlo Comin 53/89

Instantaneous-Reaction Time (TIME2016)

Instantaneous Reaction-TimeThe ε-DC notion is interesting per se, and the ε-DC-Checking algorithm(TIME2015) rests on the assumption that ε > 0;

I i.e., leaving unsolved the question of what happens when ε = 0.

Then, we introduced and studied π-DC, a sound notion of DC with aninstantaneous reaction-time.

I i.e., one in which the planner can react to any observation at the sameinstant of time in which the observation is made.


Carlo Comin 54/89

Instantaneous-Reaction Time vs 0-DC, TIME2016

Consider the following CSTN Γ2.

>[1]

BB?

A

A?C

C?

⊥[0]

+1

−1

0,¬B0

0, C

0, B¬C

0,¬A

00,¬C

0, AC

0, A

00, B

0,¬A¬B

PropositionThe CSTN Γ2 is 0-DC.

Still ...Complexity in Infinite Games on Graphs and Temporal Constraint Networks

Carlo Comin 55/89

Instantaneous-Reaction Time vs 0-DC (TIME2016)

>1

C1

B0

A0

⊥0

ΣP2

000; 001

>1

C

0

B

0

A

1

⊥0

010; 110

>1

C0

B0

A0

⊥0

011; 100

>1

C

0

B

1

A

0

⊥0

101; 111

0

0

[1]

0

[1]

00

0

0

0

[1]

0

0

0

[1]

0


Carlo Comin 56/89

π-DC

We need to take explicitly into account an additional ordering between theobservation events scheduled at the same execution time.

π-Execution-StrategyAn ordered-Execution-Strategy (π-ES) for Γ is a mapping:

σ : s 7→([σ(s)]t, [σ(s)]π

),

s ∈ ΣP , [σ(s)]t ∈ ΦV and [σ(s)]π : OV +s 1, . . . , |OV +

s | is bijective.

RemarkWe require positions to be coherent w.r.t. execution times, i.e.,∀(Op,Oq ∈ OV +

s ) if [σ(s)]tOp < [σ(s)]tOq then [σ(s)]πOp < [σ(s)]πOq .


Carlo Comin 57/89

π-DC

π-History

Let σ ∈ SΓ, s ∈ ΣP , and let τ ∈ R and ψ ∈ 1, . . . , |V |.The ordered-history π-Hst(τ, ψ, s, σ) of τ and ψ in the scenario s, under theπ-ES σ, is defined as: π-Hst(τ, ψ, s, σ) ==(p, s(p)

)∈ P × 0, 1 | Op ∈ OV +

s , [σ(s)]tOp ≤ τ, [σ(s)]πOp < ψ

.

π-Dynamic-ConsistencyAny σ ∈ SΓ is called π-dynamic when, for any two scenarios s1, s2 ∈ ΣP andany event u ∈ V +

s1,s2 , if τ , [σ(s1)]tu and ψ , [σ(s1)]πu, then:

π-Hst(τ, ψ, s1, σ) ∧ s2 ⇒ [σ(s2)]tu = τ, [σ(s2)]πu = ψ.

We say that Γ is π-dynamically-consistent (π-DC) if it admits σ ∈ SΓ which isboth viable and π-dynamic. The problem of checking whether a given CSTN isπ-DC is named π-DC-Checking.


Carlo Comin 58/89

π-DC

RemarkDue to the strict inequality “[σ(s)]πOp < ψ" in the definition of π-Hst(·), in aπ-dynamic π-ES, there must be exactly one Op′ ∈ OV , for some p′ ∈ P ,which is executed at first (w.r.t. both execution time and position) under allpossible scenarios s ∈ ΣP (i.e., a root).

a

bc

cd

d

cd

bd

d

0

0

10

11 0

10

1

An example of a permutation-scenario-tree (ps-tree) over P = a, b, c, d.Complexity in Infinite Games on Graphs and Temporal Constraint Networks

Carlo Comin 59/89

π-DC

In summary, the following chain of implications hold on the various DCs:

(ε-DC, ε > 0)⇔ DC6⇐⇒ π-DC

6⇐⇒ (ε-DC, ε = 0)


Carlo Comin 60/89

Main Result

Main Algorithmic Result [CCR, TIME2016]

The time-complexity of π-DC-Checking problem remains (pseudo)Singly-Exponential in the number of propositional letters P of the input CSTN.


Carlo Comin 61/89

Chapter 5Linear Time Algorithm for Update Gamesvia Strongly-Trap-Connected Components


Carlo Comin 62/89

Update Games: the model

AB C

ArenaRecall an arena is a finite directed graph whose vertices are divided intotwo classes, say V = V ∪ V#.

a play is thus an infinite path π = v0v1v2 . . . ∈ V ω such that(vi, vi+1) ∈ A for every i ∈ N.

Given two strategies σ ∈ ΣA and σ# ∈ ΣA#, and some vs ∈ V , theoutcome play,

ρA(vs, σ, σ#)

is the (unique) play that starts at vs and is consistent with both σ, σ#.


Carlo Comin 63/89

Update Games: the model

AB C

Update Games [Dinneen, Khoussainov, 1999]

Let Inf(π) be the set of all and only those vertices v ∈ V that appearinfinitely often in π; namely,

Inf(π) ,v ∈ V | ∀j∈N ∃k∈N such that k > j and vk = v

.

Player wins the UG A iff ∃(σ ∈ ΣA) ∀(σ# ∈ ΣA#):

∀vs∈V Inf(ρA(vs, σ, σ#)

)= V.

An O(|V ||A|) time algorithm was given in [Dinneen, Khoussainov, 1999].


Carlo Comin 63/89

C B

AH

D G

FE

(a) An Arena A.

A 1

B 2

C 3

D 4

E 5

F 6

G 7

H 8

tree

tree

tree

tree

tree

tree tree

frond

frond

frond

(b) The rev-palm-treegenerated by rev-DFS,with indices of vertices.

1.(B,A)2.(C,B)3.(D,C)4.(E,D)5.(F,E)6.(A,F )

7.(G,F )8.(D,G)9.(H,E)10.(C,H)11.(G,B)12.(H,A)

(c) The order of arcs’exploration.

Figure: An arena (a), and a rev-palm-tree (b), generated by rev-DFS (c).


Carlo Comin 64/89

Alphabet of a playFor any finite (or infinite) path p ∈ V ∗ (or p ∈ V ω), the alphabet of p is

Ξ(p) , v ∈ V | v appears in p.

Trap-ReachabilityGiven an arena A on vertex set V , let U ⊆ V and u, v ∈ U . We say that v isU -trap-reachable from u when there exists σ ∈ ΣA (i.e., σ = σ(u, v))such that for every σ# ∈ ΣA#:

[reachability] v ∈ Ξ(ρA(u, σ, σ#)

); and,

[entrapment] Ξ(ρA(u, σ, σ#, v)

)⊆ U .


Carlo Comin 65/89

TR-Palm-Tree

C B

AH

E D

GF

(a) An arena A.

A

1

B 2

C 7 D 3

E 4 G 5

F 8

H6

tree

petiole tree

treepetiole

petiole

tree

frond

petiole

tree

petiole

petiole

tree

tree

(b) The tr-palm-tree generated bytr-DFS rooted at A, with indices ofvertices and labelled arcs.

1.(B,A)2.(D,B)3.(E,D)4.(C,E)5.(F,E)6.(G,D)

7.(A,G)8.(F,G)9.(C,B)

10.(H,A)11.(C,H)12.(F,H)


Figure: An arena (a), and a tr-palm-tree (b), generated by tr-DFS (c).


Carlo Comin 66/89

TR-Depth-First-Search

C B

AH

E D

GF

(a) An arena A.

A

1

B 2

C 7 D 3

E 4 G 5

F 8

H6

tree

petiole tree

treepetiole

petiole

tree

frond

petiole

tree

petiole

petiole

tree

tree


1.(B,A)2.(D,B)3.(E,D)4.(C,E)5.(F,E)6.(G,D)

7.(A,G)8.(F,G)9.(C,B)

10.(H,A)11.(C,H)12.(F,H)




Carlo Comin 67/89


C B

AH

E D

GF

(a) An arena A.

A

1

B 2

C 7 D 3

E 4 G 5

F 8

H6

tree

petiole tree

treepetiole

petiole

tree

frond

petiole

tree

petiole

petiole

tree

tree


1.(B,A)2.(D,B)3.(E,D)4.(C,E)5.(F,E)6.(G,D)

7.(A,G)8.(F,G)9.(C,B)

10.(H,A)11.(C,H)12.(F,H)




Carlo Comin 68/89


C B

AH

E D

GF

(a) An arena A.

A

1

B 2

C 7 D 3

E 4 G 5

F 8

H6

tree

petiole tree

treepetiole

petiole

tree

frond

petiole

tree

petiole

petiole

tree

tree


1.(B,A)2.(D,B)3.(E,D)4.(C,E)5.(F,E)6.(G,D)

7.(A,G)8.(F,G)9.(C,B)

10.(H,A)11.(C,H)12.(F,H)




Carlo Comin 69/89


C B

AH

E D

GF

(a) An arena A.

A

1

B 2

C 7 D 3

E 4 G 5

F 8

H6

tree

petiole tree

treepetiole

petiole

tree

frond

petiole

tree

petiole

petiole

tree

tree


1.(B,A)2.(D,B)3.(E,D)4.(C,E)5.(F,E)6.(G,D)

7.(A,G)8.(F,G)9.(C,B)

10.(H,A)11.(C,H)12.(F,H)




Carlo Comin 70/89


C B

AH

E D

GF

(a) An arena A.

A

1

B 2

C 7 D 3

E 4 G 5

F 8

H6

tree

petiole tree

treepetiole

petiole

tree

frond

petiole

tree

petiole

petiole

tree

tree


1.(B,A)2.(D,B)3.(E,D)4.(C,E)5.(F,E)6.(G,D)

7.(A,G)8.(F,G)9.(C,B)

10.(H,A)11.(C,H)12.(F,H)




Carlo Comin 71/89


C B

AH

E D

GF

(a) An arena A.

A

1

B 2

C 7 D 3

E 4 G 5

F 8

H6

tree

petiole tree

treepetiole

petiole

tree

frond

petiole

tree

petiole

petiole

tree

tree


1.(B,A)2.(D,B)3.(E,D)4.(C,E)5.(F,E)6.(G,D)

7.(A,G)8.(F,G)9.(C,B)

10.(H,A)11.(C,H)12.(F,H)




Carlo Comin 72/89


C B

AH

E D

GF

(a) An arena A.

A

1

B 2

C 7 D 3

E 4 G 5

F 8

H6

tree

petiole tree

treepetiole

petiole

tree

frond

petiole

tree

petiole

petiole

tree

tree


1.(B,A)2.(D,B)3.(E,D)4.(C,E)5.(F,E)6.(G,D)

7.(A,G)8.(F,G)9.(C,B)

10.(H,A)11.(C,H)12.(F,H)




Carlo Comin 73/89


C B

AH

E D

GF

(a) An arena A.

A

1

B 2

C 7 D 3

E 4 G 5

F 8

H6

tree

petiole tree

treepetiole

petiole

tree

frond

petiole

tree

petiole

petiole

tree

tree


1.(B,A)2.(D,B)3.(E,D)4.(C,E)5.(F,E)6.(G,D)

7.(A,G)8.(F,G)9.(C,B)

10.(H,A)11.(C,H)12.(F,H)




Carlo Comin 74/89


C B

AH

E D

GF

(a) An arena A.

A

1

B 2

C 7 D 3

E 4 G 5

F 8

H6

tree

petiole tree

treepetiole

petiole

tree

frond

petiole

tree

petiole

petiole

tree

tree


1.(B,A)2.(D,B)3.(E,D)4.(C,E)5.(F,E)6.(G,D)

7.(A,G)8.(F,G)9.(C,B)

10.(H,A)11.(C,H)12.(F,H)




Carlo Comin 75/89


C B

AH

E D

GF

(a) An arena A.

A

1

B 2

C 7 D 3

E 4 G 5

F 8

H6

tree

petiole tree

treepetiole

petiole

tree

frond

petiole

tree

petiole

petiole

tree

tree


1.(B,A)2.(D,B)3.(E,D)4.(C,E)5.(F,E)6.(G,D)

7.(A,G)8.(F,G)9.(C,B)

10.(H,A)11.(C,H)12.(F,H)




Carlo Comin 76/89


C B

AH

E D

GF

(a) An arena A.

A

1

B 2

C 7 D 3

E 4 G 5

F 8

H6

tree

petiole tree

treepetiole

petiole

tree

frond

petiole

tree

petiole

petiole

tree

tree


1.(B,A)2.(D,B)3.(E,D)4.(C,E)5.(F,E)6.(G,D)

7.(A,G)8.(F,G)9.(C,B)

10.(H,A)11.(C,H)12.(F,H)




Carlo Comin 77/89


C B

AH

E D

GF

(a) An arena A.

A

1

B 2

C 7 D 3

E 4 G 5

F 8

H6

tree

petiole tree

treepetiole

petiole

tree

frond

petiole

tree

petiole

petiole

tree

tree


1.(B,A)2.(D,B)3.(E,D)4.(C,E)5.(F,E)6.(G,D)

7.(A,G)8.(F,G)9.(C,B)

10.(H,A)11.(C,H)12.(F,H)




Carlo Comin 78/89

TR-Jungle

C B

AG

H D

EF

A

1∗

B 2

C

n.a.

D n.a.

E 3

Fn.a.

G4

tree tree

tree

A

1∗

B 2

C

n.a.

H

5∗

D n.a.

E 3

F n.a.

G4

tree tree

tree

A

1∗

B 2

C

6∗

H

5∗

D 7∗

E 3

F 8∗

G4

tree tree

tree

A

1∗

B 2

C

6∗

H

5∗

D 7∗

E 3

F 8∗

G4

tree tree

tree


Carlo Comin 79/89

Strongly-Trap-Connectedness

We say that U ⊆ V is strongly-trap-connected when for every (u, v) ∈ U × Uthere exists some σ ∈ ΣA (i.e., σ = σ(u, v)) such that σ : u U; v.

Strongly-Trap-Connected ComponentsThe binary relation ∼stc on V is defined as follows:

∼stc,

(u, v) ∈ V × V | ∃U⊆V s.t. U is STC and u, v ⊆ U.

It holds that ∼stc is an equivalence relation on V .Equivalence classes of ∼stc are strongly-trap-connected components of A.


Carlo Comin 80/89

Main Results on UGs

UGs and STCThere is a linear time algorithm for decomposing any given arena into itsstrongly-trap-connected components.

UGs and STCAn UG A is winning for Player if and only if V is strongly-trap-connected.

Deciding UGs in Linear TimeDeciding whether or not a given UG A is winning for Player takesΘ(|V |+ |A|) time.


Carlo Comin 81/89

Application to Explicit McNaughton-Muller Games

Explicit McNaughton-Muller Games

The most straightforward way to represent a Müller winning condition F ⊆ 2Vis to provide an explicit list of subsets of vertices:

F = Fi ⊆ V | 1 ≤ i ≤ `, for some ` ∈ N.

A play ρ ∈ V ω is winning for Player if and only if Inf(ρ) ∈ F .

Polynomial Time Algorithm [Horn 2008]Explicit MMGs can be solved in polynomial time, where anO(|F| · (|A|+ |F|)2) time algorithm is given in [Horn 2008].

Quadratic Time Algorithm for Explicit MMGs

We can solve Explicit MMGs in O(|F| · (|A|+ |F|)

)quadratic time.


Carlo Comin 82/89

Chapter 6An Improved Upper Bound on Value Problem and Optimal Strategy Synthesis

in MPGs


Carlo Comin 83/89

Table: Complexity of the main Algorithms for solving MPGs.

Algorithm Value ProblemOptimalStrategySynthesis

Note

This work Θ(|V |2|E|W ) Θ(|V |2|E|W ) Determ.Brim11 O(|V |2|E|W log(|V |W )) O(|V |2|E|W log(|V |W )) Determ.

ZP96 Θ(|V |3|E|W ) Θ(|V |4|E|W log |E||V |) Determ.

LP07 O(2|V | |V | |E| logW ) n/a Determ.

An06 O(|V |2|E| e

2

√|V | ln

(|E|√

|V |

)+O(√

|V |+ln |E|))same complexity Random.


Carlo Comin 84/89

An O(|V |2|E|W ) Improved Upper Bound for OSS in MPGs

Our Contribution: Improved Upper Bound for MPGs [CR, Algorithmica, 2016]

There exists an algorithm for solving the STRATEGY SYNTHESIS and theVALUE problem in:

O(|V |2|E|W ) time and O(|V |) space.

Particularly, the time complexity is bounded as follows:

Θ(|V |2|E|W +

∑v∈V

degΓ(v) · `0Γ(v)),

where `0Γ(v) ≤ (|V | − 1)|V |W denotes the total number of times that anenergy-lifting operator δ(·, v) is applied to any v ∈ V .

So, the best previously known upper bound [Brim, et al. 2011] is improved by afactor O(log(|V |W )). Here, W = maxe∈E |we|.


Carlo Comin 85/89

Surprisingly, in this problem, a Linear Search turns out to be faster than a Binary Search...

The O(|V |2|E|W log(|V |W )) algorithm of Brim, et al. (2011) is basedon a binary search procedure that reduces the VALUE and the STRATEGY

SYNTHESIS problems to multiple resolutions of the DECISION problem.Can the binary search be avoided ?

I The answer is YES.

Our algorithm also reduces to multiple resolutions of the DECISIONproblem, but it succeeds in amortizing the cost of these.

I Actually, each DECISION problem generated gets solved by thecomputation of a Small Energy Progress Measure (SEPM) into thecorresponding Energy Game (EG).


Carlo Comin 86/89

Chapter 7A Faster O(|V |2|E|W ) Time Energy Algorithm for MPGs


Carlo Comin 87/89

A Faster O(|V |2|E|W ) Algorithm for MPGs which is not Ω(|V |2|E|W )

A more technical but “truly" O(|V |2|E|W ) algorithm for MPGs

The running time of our first solution turns out to be also Ω(|V |2|E|W ),the actual time complexity being

Θ(|V |2|E|W +

∑v∈V

degΓ(v) · `0Γ(v)).

We introduce a novel algorithmic scheme, named Jumping, which tackleson some further regularities of the problem, thus reducing the estimate onthe pseudo-polynomial time complexity of MPGs to:

O(|E| log |V |) + Θ( ∑v∈V

degΓ(v) · `1Γ(v)).

I `1Γ ≤ (|V | − 1)|V |W (worst-case; but experimentally, `1Γ `0Γ)I the working space is Θ(|V |+ |E|).


Carlo Comin 88/89

Thank you

Thank you for the attention


Carlo Comin 89/89

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