=Paper=
{{Paper
|id=Vol-2272/short2
|storemode=property
|title=Preliminary Results on Modeling Interdependent Scheduling Games via Answer Set Programming
|pdfUrl=https://ceur-ws.org/Vol-2272/short2.pdf
|volume=Vol-2272
|authors=Giovanni Amendola
|dblpUrl=https://dblp.org/rec/conf/aiia/Amendola18a
}}
==Preliminary Results on Modeling Interdependent Scheduling Games via Answer Set Programming==
https://ceur-ws.org/Vol-2272/short2.pdf
Preliminary Results on Modeling Interdependent
Scheduling Games via Answer Set Programming
Giovanni Amendola
Department of Mathematics and Computer Science, University of Calabria, Italy
amendola@mat.unical.it
Abstract. Recently, in the context of planning and coordinating large-scale in-
frastructures, a model of Interdependent Scheduling Games (ISG) has been pro-
posed. This problem requires interdependent services among players, that con-
trol only a limited number of services and schedule independently. A player can
schedule his services at any time. However, each service starts to be profitable
for the player when all the services on which it depends have been activated. In
this paper we approach the ISG by means of Answer Set Programming (ASP).
ASP is an established logic-based programming paradigm which has been suc-
cessfully applied for solving complex combinatorial problems arising in many
areas of knowledge. We show how the ISG can be elegantly encoded by means
of an ASP program.
1 Introduction
Recently, in the context of planning and coordinating large-scale infrastructures, a model
of Interdependent Scheduling Games (ISG) in which each player controls a set of ser-
vices that they schedule independently has been proposed [1]. This problem requires
interdependent services among players that control only a limited number of services
and schedule independently. A player can schedule his services at any time. However,
each service starts to be profitable for the player when all the services on which it de-
pends have been activated. A relevant task is to find a schedule that maximize the most
profitable services activated for the longest amount of time.
Complex combinatorial optimization problems, like the one just described, are usu-
ally the target for the application of formalisms developed in the area of Artificial Intel-
ligence. Among these, Answer Set Programming (ASP) [19, 18], a well-known declar-
ative programming paradigm which has been proposed in the field of logic program-
ming and non-monotonic reasoning, is an ideal candidate. Indeed, ASP combines a
comparatively high knowledge-modeling power [19, 3] with a robust solving technol-
ogy [4, 21, 25, 24, 31, 27, 26, 10, 14–16, 34, 33], that can also be responsive in case of
incoherent logic programs [7, 8]. For these reasons ASP has become an established
logic-based programming paradigm with successful applications to complex problems
in Artificial Intelligence [30, 29, 9, 6], Databases [32], Game Theory [13], Information
Extraction [2], and many other fields of knowledge.
The goal of this paper is to provide an assessment of the applicability of ASP to the
ISG as a starting point for a more in-depth investigation to address this problem and its
variants in comparison with other evaluation approaches.
2 G. Amendola
2 Answer Set Programming
Answer Set Programming (ASP) [19] is a programming paradigm developed in the
field of logic programming and nonmonotonic reasoning. In this section, we overview
the language of ASP. The reader is referred to [17] for a more detailed introduction.
Syntax. Variables are strings starting with uppercase letter and constants are non-
negative integers or strings starting with lowercase letters. A term is either a variable
or a constant. A standard atom is an expression p(t1 , . . . ,tn ), where p is a predicate
of arity n and t1 , . . . ,tn are terms. An atom p(t1 , . . . ,tn ) is ground if t1 , . . . ,tn are con-
stants. A ground set is a set of pairs of the form hconsts : con ji, where consts is a list of
constants and con j is a conjunction of ground standard atoms. A symbolic set is a set
specified syntactically as {Terms1 : Con j1 ; · · · ; Termst : Con jt }, where t > 0, and for
each i = 1, . . . ,t, Termsi is a list of terms such that |Termsi | > 0, and each Con ji is a
conjunction of standard atoms. A set term is either a symbolic set or a ground set. An
aggregate functionis of the form f (S), where S is a set term, and f is an aggregate func-
tion symbol. Basically, aggregate functions map multisets of constants to a constant. In
the rest of the paper, we will use the following common functions implemented in ASP
systems: #count, number of terms; and #sum, sum of integers. An aggregate atom is
of the form f (S) ≺ T , where f (S) is an aggregate function, ≺ ∈ {<, ≤, >, ≥, =, 6=} is
a comparison operator, and T is a term called guard. An aggregate atom f (S) ≺ T is
ground if T is a constant and S is a ground set. An atom is either a standard atom or an
aggregate atom. A rule r is of the form:
a1 | . . . | an ← b1 , . . . , bk , not bk+1 , . . . , not bm .
where a1 , . . . , an are standard atoms, b1 , . . . , bk are atoms, bk+1 , . . . , bm are standard
atoms, and n, k, m ≥ 0. A literal is either a standard atom a or its negation not a.
The disjunction a1 | . . . |an is the head of r, while the conjunction b1 , . . . , bk , not bk+1 ,
. . . , not bm is its body. A rule is a fact if its body is empty (← is omitted), whereas it is
a constraint if its head is empty. A variable appearing uniquely in set terms of a rule r
is said to be local in r, otherwise it is global in r. An ASP program is a set of safe rules.
A rule r is safe if both the following conditions hold: (i) for each global variable X of
r there is a positive standard atom ` in the body of r such that X appears in `; (ii) each
local variable of r appearing in a symbolic set {Terms :Con j} also appears in Con j.
A weak constraint [20] ω is of the form:
b1 , . . . , bk , not bk+1 , . . . , not bm . [w@l]
where w and l are the weight and level of ω. (Intuitively, [w@l] is read “as weight
w at level l”, where weight is the “cost” of violating the condition in the body of w,
whereas levels can be specified for defining a priority among preference criteria). An
ASP program with weak constraints is Π = hP,W i, where P is a program and W is a set
of weak constraints. A standard atom, a literal, a rule, a program or a weak constraint is
ground if no variable appears in it.
Preliminary Results on Modeling Interdependent Scheduling Games via ASP 3
Semantics. Let P be an ASP program. The Herbrand universe UP and the Herbrand
base BP of P are defined as usual [17]. The ground program GP is the set of all the
ground instances of rules of P obtained by substituting variables with constants from
UP . An interpretation I for P is a subset I of BP . A ground atom a is true with respect to I
if a ∈ I, and false otherwise. Literal not a is true in I if a is false in I, and true otherwise.
An aggregate atom is true with respect to I if the evaluation of its aggregate function
(i.e., the result of the application of f on the multiset S) with respect to I satisfies the
guard; otherwise, it is false. A ground rule r is satisfied by I if at least one atom in
the head is true with respect to I whenever all conjuncts of the body of r are true with
respect to I. A model is an interpretation that satisfies all the rules of a program. Given
a ground program GP and an interpretation I, the reduct [23] of GP with respect to I is
the subset GIP of GP obtained by deleting from GP the rules in which a body literal is
false with respect to I. An interpretation I for P is an answer set (or stable model [28])
for P if I is a minimal model (under subset inclusion) of GIP [23]. We denote the set
of all answer sets of an ASP program P by AS(P). A program having an answer set is
called coherent, otherwise it is incoherent [5, 12, 11].
Given a program with weak constraints Π = hP,W i, the semantics of Π extends
from the basic case defined above. Thus, let GΠ = hGP , GW i be the instantiation of Π ;
a constraint ω ∈ GW is violated by I if all the literals in ω are true with respect to I.
An optimum answer set O for Π is an answer set of GP that minimizes the sum of the
weights of the violated weak constraints in a prioritized way. We denote by OAS(Π )
the set of all optimum answer sets of Π .
3 Interdependent Scheduling Games via ASP
Recently, a model of Interdependent Scheduling Games (ISGs) in which each player
controls a set of services that they schedule independently has been proposed [1]. An
ISG with n players is a tuple of the form ((T1 , . . . , Tn ), G, r). Each Ti represents a set of
services which player i has to schedule, and sets of services are pairwise disjoint and of
the same size, i.e., for each i 6= j, Ti ∩ T j = 0/ and |T1 | = · · · = |Tn |. Moreover, let T be
the union of all set of services, for each service u ∈ T , there is a (non-negative) reward
r(u) ≥ 0, representing payment received in each time period that the service is active.
Finally, there is a dependency relation among services. Indeed, to activate a service,
it and all its prerequisites must be deployed. This is formalized by a transitive acyclic
directed graph G = (T, E), where (u, v) ∈ E if v will generate a reward only after u has
been deployed.
Example 1. Consider an ISG with 2 players, say p1 and p2 , and to each player is as-
sociated a set of services: T1 = {u1 , u2 , u3 } and T2 = {v1 , v2 , v3 }, respectively. Hence,
T = {u1 , u2 , u3 , v1 , v2 , v3 }. Moreover, the rewards of each service is as follows: r(u1 ) =
10, r(u2 ) = r(u3 ) = r(v1 ) = 1, and r(v2 ) = r(v3 ) = 100. Finally, there are dependency
relations from u1 to v1 ; from u2 to u3 ; from u2 to v2 ; and from u2 to v3 . Hence,
E = {(u1 , v1 ), (u2 , u3 ), (u2 , v2 ), (u2 , v3 )}.
A tuple π = (π1 , . . . , πn ), where each πi is a permutation of the services Ti of player
i (i.e., πi : Ti → {1, . . . , |Ti |}), is a schedule of all services in T . Intuitively, the posi-
tion k ∈ {1, . . . , |Ti |} of a service u ∈ Ti in the permutation πi denotes the time when
4 G. Amendola
u is deployed. A service u is active during a time step if itself and all services v such
that (v, u) ∈ E are deployed at or before that time step. The time when a service u be-
comes active is denoted by a(u), that is a(u) = max{π(v) : v = u or (v, u) ∈ E}. Given
a schedule π = (π1 , . . . , πn ), the utility of player i is
|Ti |
Ri (π) = ∑ ∑ r(u).
t=1 u∈Ti , t≥a(u)
Finally, the welfare of π is defined as ∑ni=1 Ri (π).
Example 2. Consider the ISG of the Example 1. Let π = (π1 , π2 ) be a schedule where
π1 (u1 ) = 1, π1 (u2 ) = 2, π1 (u3 ) = 3, and π2 (v1 ) = 1, π2 (v2 ) = 2, π2 (v3 ) = 3. Hence,
R1 (π) = 3r(u1 )+ 2r(u2 ) +r(u3 ) = 3 ·10 +2 ·1 +1 = 33, and R2 (π) = 3r(v1 ) +2r(v2 )+
r(v3 ) = 3 · 1 + 2 · 100 + 100 = 303, as u1 and v1 are active for three time steps, u2 and
v2 are active for two time steps, and u3 and v3 are active for one time step. Therefore,
the welfare of π is R1 (π) + R2 (π) = 33 + 303 = 336. Similarly, let π 0 = (π10 , π20 ) be
a schedule where π10 (u2 ) = 1, π10 (u3 ) = 2, π10 (u1 ) = 3, and π20 (v2 ) = 1, π20 (v3 ) = 2,
π20 (v1 ) = 3. Hence, R1 (π 0 ) = 3r(u2 ) + 2r(u3 ) + r(u1 ) = 3 · 1 + 2 · 1 + 10 = 15, and
R2 (π 0 ) = 3r(v2 ) + 2r(v3 ) + r(v1 ) = 3 · 100 + 2 · 100 + 1 = 501, as u2 and v2 are ac-
tive for three time steps, u3 and v3 are active for two time steps, and u1 and v1 are active
for one time step. Therefore, the welfare of π 0 is R1 (π 0 ) + R2 (π 0 ) = 15 + 501 = 516.
An important task is to find a schedule that maximize the welfare. The correspond-
ing decision problem, denoted by ISG W ELFARE, is stated as follows. Given an ISG
n R (π) ≥ w? This
((T1 , . . . , Tn ), G, r) and an integer w, is there a schedule π such that Σi=1 i
decision problem is known to be NP-complete [1]. Hence, it can be in theory solved via
ASP. We propose an encoding of the problem of computing a welfare greater than the
threshold w.
Given an ISG instance Γ = ((T1 , . . . , Tn ), G, r), and an integer w, the set of services
of each player and its corresponding rewards are encoded as facts of the logic program:
f1 : service(i, u, r(u)). if i ∈ {1, . . . , n} and u ∈ Ti .
It means that service u belongs to player i and has a reward r(u). Moreover, we need
facts for each dependency of a service from another one, and for the threshold w.
f2 : edge(u, v). if (u, v) ∈ E.
f3 : threshold(w).
Example 3. Considering the ISG of the Example 1, and a threshold w = 500. Then, we
obtain as facts of the logic program: service(1, u1 , 10), service(1, u2 , 1), service(1, u3 ,
1), service(2, v1 , 1), service(2, v2 , 100), service(2, v3 , 100), edge(u1 , v1 ), edge(u2 ,
u3 ), edge(u2 , v2 ), edge(u2 , v3 ), and threshold(500).
First, we consider the following three rules:
ρ1 : numbTotServ(L) ← #count{C : service(X,C, )} = L.
ρ2 : numbServForPlayer(X, L) ← #count{C : service(X,C, )} = L, service(X, , ).
ρ3 : maxValueServ(X,Y, M) ← service(X,Y, N), numbServForPlayer(X, L),
M = N ∗ L.
Preliminary Results on Modeling Interdependent Scheduling Games via ASP 5
They encode the total number of services, the number of services for each player, and
the possible maximum profit obtainable for each service, respectively.
Example 4. Considering Example 1, by instantiating rules ρ1 , ρ2 , and ρ3 , we will in-
fer the following set of ground atoms: numbTotServ(6), numbServForPlayer(1, 3),
numbServForPlayer(2, 3), maxValueServ(1, u1 , 30), maxValueServ(1, u2 , 3),
maxValueServ(1, u3 , 3), maxValueServ(2, v1 , 3), maxValueServ(2, v2 , 300), and
maxValueServ(2, v3 , 300).
Now, we encode a rule identifying all possible time values that can be used during
permutations:
ρ4 : pi(X,Y, 1..L) ← service(X,Y, K), numbServForPlayer(X, L).
Example 5. Considering again the Example 1, by instantiating rule (ρ4 ), we will infer
the following ground atoms: pi(1, u1 , 1), pi(1, u1 , 2), pi(1, u1 , 3), pi(1, u2 , 1), pi(1,
u2 , 2), pi(1, u2 , 3), pi(1, u3 , 1), pi(1, u3 , 2), pi(1, u3 , 3) for the player p1 , and pi(2,
v1 , 1), pi(2, v1 , 2), pi(2, v1 , 3), pi(2, v2 , 1), pi(2, v2 , 2), pi(2, v2 , 3), pi(2, v3 , 1), pi(2,
v3 , 2), pi(2, v3 , 3) for the player p2 .
The following set of rules guess a possible schedule, by identifying permutations of
services for each player.
ρ5 : inPi(X,Y, Z) | outPi(X,Y, Z) ← pi(X,Y, Z).
ρ6 : ← inPi(X,Y, Z), inPi(X,V, Z), Y > V.
ρ7 : ← inPi(X,Y, Z), inPi(X,Y,W ), Z > W.
ρ8 : ← #count{Y : inPi(X,Y, Z)} < L, numbTotServ(L).
Now, given two services S1 and S2, such that S2 depends on S1, it must not happen that
the execution time of S2, namely T 2, is strictly lower than the execution time of S1,
namely T 1. The following constraint encodes this condition.
ρ9 : ← inPi(P1, S1, T 1), inPi(P2, S2, T 2), T 2 < T 1, edge(S1, S2).
To compute the profit obtained by a player X on service Y , we have to multiply the
reward of the service Y , namely M, by the number of time steps for which Y has been
used, that is L + 1 − Z, where Z is the time step associated to Y with respect to the
permuation considered. Hence,
ρ10 : val(X,Y, K) ← inPi(X,Y, Z), numbServForPlayer(X, L),
service(X,Y, M), K = M ∗ (L + 1 − Z).
Finally, the welfare is obtained by the following rule:
ρ11 : welfare(N) ← #sum{M,Y : val(X,Y, M)} = N.
We denote by PΓ the ASP program so constructed, i.e. PΓ = { f1 , f2 , f3 } ∪ 11
i=1 {ρi }.
S
The previous enconding allows to easily compute different reasoning tasks. To com-
pute a welfare greater than the threshold w, we just add the following rule.
ρ12 : ← welfare(N), threshold(W ), N < W.
6 G. Amendola
Rule ρ12 is a constraint requiring that the welfare, namely N, must not be less than the
threshold, namely W . Therefore, we can formally prove the completeness and sound-
ness of the encoding proposed.
Theorem 1. Let Γ = ((T1 , . . . , Tn ), G, r) be an ISG, and let w be an integer. Then, there
is a schedule π such that ∑ni=1 Ri (π) ≥ w if, and only if, AS(PΓ ∪ {ρ12 }) 6= 0./
Note that, starting from PΓ we can easily compute other interesting reasoning tasks.
For instance, we can compute the maximum welfare for an ISG by adding the following
weak constraint.
ρ13 : val(X,Y, K), maxValueServ(X,Y, M), L = M − K. [L@1]
Indeed, ρ13 minimizes the difference between the possible maximum profit obtainable
for a given service, namely M, and the current profit of that service, namely K. Also in
this case, the completeness and soundness of the approach can be proved.
Theorem 2. Let Γ = ((T1 , . . . , Tn ), G, r) be an ISG. Then, µ is the maximum welfare for
Γ if, and only if, for each O ∈ OAS(hPΓ , {ρ13 }i), it holds that welfare(µ) ∈ O.
4 Conclusion and Future Work
In the paper, we showed that the ISG can be elegantly encoded by means of an ASP pro-
gram. In particular, we deal with the problem of maximizing the welfare (ISG WEL -
FARE ). However, another interesting question is known as ISG BEST RESPONSE [1].
Here, the goal is to find a best response for a given player by knowing the actions
of each other player. More specifically, given an ISG ((T1 , . . . , Tn ), G, r), a schedule
(π1 , . . . , πi−1 , πi+1 , . . . , πn ) for all players except for i, and an integer w, we ask for a per-
mutation πi0 of the services Ti of player i such that Ri (π1 , . . . , πi−1 , πi0 , πi+1 , . . . , pn ) ≥ w.
Moreover, as future work, we want to address the ISG and its possible variants (such
as extensions of the model including cyclic interdependencies [22], adding services or
misreporting utilities [35]) via ASP in comparison with other modeling approaches,
such as the integer linear programming formulation [1].
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