Optimization of Partial-Order Plans via MaxSAT
Christian Muise and Sheila A. McIlraith
Dept. of Computer Science
University of Toronto
Toronto, Canada. M5S 3G4
{cjmuise,sheila}@cs.toronto.edu
J. Christopher Beck
Dept. of Mechanical & Industrial Engineering
University of Toronto
Toronto, Canada. M5S 3G8
jcb@mie.utoronto.ca
Abstract
Partial-order plans (POPs) are attractive because of their least
commitment nature, providing enhanced plan flexibility at
execution time relative to sequential plans. Despite the ap-
peal of POPs, most of the recent research on automated plan
generation has focused on sequential plans. In this paper we
examine the task of POP generation by relaxing or modify-
ing the action orderings of a sequential plan to optimize for
plan criteria that promote flexibility in the POP. Our approach
relies on a novel partial weighted MaxSAT encoding of a se-
quential plan that supports the minimization of deordering or
reordering of actions. We further extend the classical least
commitment criterion for a POP to consider the number of
actions in a solution, and provide an encoding to achieve least
commitment plans with respect to this criterion. Our partial
weighted MaxSAT encoding gives us an effective means of
computing a POP from a sequential plan. We compare the
efficiency of our approach to a previous approach for POP
generation via sequential-plan relaxation. Our results show
that while the previous approach is proficient at producing the
optimal deordering of a sequential plan, our approach gains
greater flexibility with the optimal reordering.
1 Introduction
Partial-order planning reflects a least commitment strategy
(Weld 1994). Unlike a sequential plan that specifies a set of
actions and a total order over those actions, a partial-order
plan (POP) only specifies those action orderings necessary
to achieve the goal from the initial state. In doing so, a POP
embodies a family of sequential plans – a set of lineariza-
tions all sharing the same actions, but differing with respect
to the order of the actions.
The flexibility afforded by POPs makes them attrac-
tive for real-time execution, multi-agent taskability, and
a range of other applications that can benefit from their
least commitment nature (Veloso, Pollack, and Cox 1998;
Weld 1994). However, in recent years research on plan
generation has shifted away from partial-order planning to-
wards sequential planning, primarily due to the success of
heuristic-based forward-search planners. To regain the least
commitment nature of POPs while leveraging fast sequential
plan generation, it is compeling to examine the computation
of POPs via sequential planning technology. Indeed this ap-
proach has been realized in the planner POPF (Coles et al.
2010), which generated a POP by searching in a heuristic-
based forward-chaining manner.
Another possibility for leveraging the strengths of sequen-
tial planning is to generate a sequential plan with a state-of-
the-art planner, and subsequently relax the plan to a POP.
Removing ordering constraints from the sequential plan, re-
ferred to as a deordering, or allowing changes in the order-
ing constraints, referred to as a reordering, are approaches
that have been theoretically investigated (Ba¨ckstro¨m 1998).
Unfortunately, finding the optimal deordering or reordering
is NP-hard to solve, and difficult to approximate within a
constant factor. Nevertheless, with the advent of powerful
optimization techniques (such as MaxSAT), we can effec-
tively solve many problems in practice.
In this paper we focus on the optimization problem of
computing the minimum deordering and minimum reorder-
ing of a sequential plan treated as a POP. The minimum de-
ordering of a POP minimizes the number of ordering con-
straints between actions, so long as the POP remains valid
and no two actions have their order reversed. Similarly, a
minimum reordering minimizes the number of ordering con-
straints, but has no restriction on which ordering constraints
are forbidden. These two notions cover a natural aspect of
least commitment planning – minimizing the ordering con-
straints placed on a POP. We extend this characterization to
consider the number of actions in a plan. In the spirit of
least commitment planning, we argue that a POP should first
commit to as few actions as possible before committing to
as few ordering constraints as possible. With the various
notions of least commitment planning in mind (deordering,
reordering, etc), we propose a set of criteria against which
we evaluate our work. These include the number of actions
and ordering constraints found in the transitive closure of a
POP, and the number of linearizations the POP represents.
The criteria serve as a measure of the flexibility of a POP.
Our approach is to use a family of novel encodings for
partial weighted MaxSAT whose solution corresponds to
an optimal least commitment plan. Unlike typical SAT-
based planning techniques, we represent an action occur-
rence once, giving us a succinct representation for use with
a modern MaxSAT solver. We compare our approach to
an existing algorithm for relaxing a sequential plan, due to
Kambhampati and Kedar (Kambhampati and Kedar 1994),
and evaluate our approach empirically. We demonstrate the

efficiency of using our MaxSAT encoding to relax a sequen-
tial plan optimally, and demonstrate the strength of Kamb-
hampati and Kedar’s algorithm in computing a deording of
a sequential plan.
We find that the existing algorithm is extremely proficient
at computing the minimum deordering, matching the opti-
mal solution in every problem tested. However, we find that
the minimum reordering is usually far more flexible than the
minimum deordering (having fewer ordering constraints and
far more linearizations). Our approach to encoding the prob-
lem gives us the first technique, to the best of our knowledge,
for computing the optimal reordering of a POP. We further
see a benefit in the flexibility of a POP when we use the pro-
posed extension to least commitment planning that considers
the number of actions.
In the next section we provide background on the notation
we use throughout the paper. We follow with a description of
least commitment planning in Section 3, and then describe
our encoding for the various forms of optimization criteria
in Section 4. In Section 5 we describe the prior approach to
relaxing a plan, and we present the experimental results in
Section 6. We finish with a brief discussion and conclusion
in Sections 7 and 8.
2 Background
Propositional Planning
For the purposes of this document, we restrict ourselves to
STRIPS planning problems. In STRIPS, a planning problem
is a tuple Π = 〈F,O, I,G〉 where F is a finite set of fluents,
O is the set of operators, I ⊆ F is the initial state, and G ⊆
F is the goal state. We refer to a complete state (or just state)
as a subset of F . We interpret fluents that do not appear in a
complete state as being false in that state. We characterize an
operator o ∈ O by three sets: PRE(o), the fluents that must
be true in order for o to be executable; ADD(o), the fluents
that operator o adds to the state; and DEL(o), the fluents
that operator o deletes from the state. An action refers to
a specific instance of an operator, and it inherits the PRE,
ADD, and DEL sets of the matching operator. We say that
an action a is executable in state s iff PRE(a) ⊆ s. A
sequence of actions is a sequential plan for the problem Π
if the execution of each action in sequence, when starting in
state I , causes the goal to hold in the final state.
We will make use of two further items of notation with
respect to a set of actions A. We define adders(f ) to be
the set of actions in A that add the fluent f (i.e., {a | f ∈
ADD(a)}), and we define deleters(f ) to be the set of ac-
tions in A that delete the fluent f (i.e., {a | f ∈ DEL(a)}).
Partial-Order Plans
For this paper, we adopt the notation typically used by the
partial-order planning community. With respect to a plan-
ning problem Π = 〈F,O, I,G〉, a partial-order plan (POP)
is a tuple P = 〈A,O, C〉 where A is the set of actions
in the plan (all of which have a corresponding operator in
O), O is a set of orderings between the actions in A (e.g.,
(a1 ≺ a2) ∈ O), and C is a set of causal links between the
actions in A (Weld 1994). A causal link is an annotated or-
dering constraint where the annotation of the link is a fluent
from F that represents the reason for that link’s existence.
For a causal link (a1
f
≺ a2) ∈ C, we can assume that f ∈
ADD(a1) and f ∈ PRE(a2). The ordering constraints
found in C will always be a subset of the ordering constraints
in O, and we assume that O is transitively closed. Where
convenient, we will ignore the set C and simply use 〈A,O〉
to represent a POP. We refer to a total ordering of the actions
in A that respects O as a linearization of P . A POP provides
a compact representation for multiple linearizations.
Intuitively, a POP is valid for a planning problem if it is
able to achieve the goal. There are two related formal no-
tions of what a valid POP consists of. The first, only refer-
ring to A and O, says that a POP P is valid for a planning
problem Π iff every linearization of P is a plan for Π. While
simple and intuitive, this notion is rarely used to verify the
validity of a POP since there may be a prohibitively large
number of linearizations represented by the POP.
The second notion is slightly more involved, and refers
to open preconditions and threats. An open precondition is
a precondition p of an action a ∈ A that does not have an
associated causal link (i.e., ∄a′ ∈ A s.t. (a′ p≺ a) ∈ C).
If a precondition is not open, we say that it is supported,
and we refer to the associated action in the causal link as the
achiever for the precondition.
A threat in a POP refers to an action that can invalidate a
causal link due to the ordering constraints (or lack thereof).
Formally, if (a′
p
≺ a) ∈ C, we say that an action a′′ threatens
the causal link if the following is true:
• We can order a′′ between a′ and a
(i.e., {(a′′ ≺ a′), (a ≺ a′′)} ∩ O = ∅)
• The action a′′ can delete p (i.e., p ∈ DEL(a′′))
The existence of a threat means that a linearization may exist
that is not executable because one of the preconditions of an
action in the linearization is not satisfied.
We will typically add two special actions to the POP, aI
and aG, that encode the initial and goal states through their
add effects and preconditions (i.e., PRE(aG) = G and
ADD(aI) = I). With this modification, we say that a POP
P = 〈A,O, C〉 is a valid POP for the planning problem Π
iff it has no open preconditions and no causal link in the set
C has a threatening action in A. The two notions of POP va-
lidity are similar in the sense that if the second notion holds,
then the first follows. If the first notion holds for A and O,
then a set of causal links C exists such that the second notion
holds for P = 〈A,O, C〉. Further details on these notions
of validity can be found in (Russell and Norvig 2009).
Partial Weighted MaxSAT
In boolean logic, the problem of Satisfiability (SAT) is to
find a True/False setting of boolean variables such that a
logical formula referring to those variables evaluates to True
(Biere et al. 2009). Typically, we write problems in Con-
junctive Normal Form (CNF) which is made up of a con-
junction of clauses, where each clause is a disjunction of

literals. A literal is either a boolean variable or its negation.
A setting of the variables satisfies a CNF formula iff every
clause has at least one literal that the setting satisfies.
The MaxSAT problem is the optimization variant of the
SAT problem in which the goal is to maximize the num-
ber of satisfied clauses (Biere et al. 2009, Ch. 19). Adding
non-uniform weights to each clause allows for a more natu-
ral representation of the optimization problem, and we refer
to this as the weighted MaxSAT problem. If we wish to
force the solver to find a solution that satisfies a particular
subset of the clauses, we refer to clauses in this subset as
hard, while all other clauses in the problem are soft. When
we have a mix of hard and soft clauses, we have a partial
weighted MaxSAT problem (Biere et al. 2009, Ch. 19.6).
In a partial weighted MaxSAT problem, only the soft
clauses are given a weight, and a valid solution corresponds
to any setting of the variables that satisfies the hard clauses in
the CNF. An optimal solution to a partial weighted MaxSAT
problem is any valid solution that maximizes the sum of the
weights on the satisfied soft clauses.
3 Least Commitment Criteria
The aim of least commitment planning is to find flexible so-
lutions that allow us to defer decisions regarding the execu-
tion of the plan. Considering only the ordering constraints
of a POP, two appealing notions for least commitment plan-
ning are the deordering and reordering of a POP. Following
(Ba¨ckstro¨m 1998), we define these formally:
Definition 1. Let P = 〈A,O〉 and Q = 〈A′ ,O′〉 be two
POPs, and Π a planning problem. Then,
1. Q is a reordering of P wrt. Π iff P and Q are valid POPs
for Π, and A = A′
2. Q is a deordering of P wrt. Π iff P and Q are valid POPs
for Π, A = A′, and O′ ⊆ O.
Recall that we assume the ordering constraints of a POP
are transitively closed. We define the optimal deordering and
optimal reordering as follows:
Definition 2. Let P = 〈A,O〉 and Q = 〈A′ ,O′〉 be two
POPs, and Π a planning problem. Then,
1. Q is a minimum deordering of P wrt. Π iff
(a) Q is a deordering of P wrt. Π, and
(b) there is no deordering 〈A′′ ,O′′〉 of P wrt. Π s.t.
|O′′| < |O′|
2. Q is a minimum reordering of P wrt. Π iff
(a) Q is a reordering of P wrt. Π, and
(b) there is no reordering 〈A′′ ,O′′〉 of P wrt. Π s.t.
|O′′| < |O′|
Note that we use < rather than ⊂ for 1(b) and 2(b) since
the orderings in O′ and O′′ may only partially overlap. We
will equivalently refer to a minimum deordering (resp. re-
ordering) as an optimal deordering (resp. reordering). In
both cases, we prefer a POP that has the smallest set of
ordering constraints. In other words, no POP exists with
the same actions and fewer ordering constraints while re-
maining valid with respect to Π. The problem of finding
the minimum deordering or reordering of a POP is NP-hard,
and cannot be approximated within a constant factor unless
NP ∈ DTIME(npoly log n) (Ba¨ckstro¨m 1998).
While the notion of a minimum deordering or reordering
of a POP addresses the commitment of ordering constraints,
in the spirit of least commitment planning we would like
to commit to as few actions as possible. To this end, we
provide an extended criterion of what a least commitment
POP (LCP) should satisfy:
Definition 3. Let P = 〈A,O〉 and Q = 〈A′ ,O′〉 be two
POPs valid for Π. Q is a least commitment POP (LCP) of P
iff Q is the minimum reordering of itself and there is no valid
POP 〈A′′ ,O′′〉 for Π such that A′′ ⊆ A and |A′′| < |A′|.
Intuitively, we can compute the LCP of an arbitrary POP
by first minimizing the number of actions, and then mini-
mizing the number of ordering constraints.
It may turn out that preferring fewer actions causes us to
commit to more ordering constraints, simply due to the inter-
action between the actions we choose. However, in practice
we usually place a much greater emphasis on minimizing the
number of actions in a POP, as, in the standard interpretation
of a POP, every every action must be executed.
Following the above criteria we will evaluate the qual-
ity of a POP by the number of actions and ordering con-
straints it contains, as these metrics give us a direct measure
of the least commitment nature of a POP. Another property
of interest is a POP’s flexibility, which provides a measure
of the robustness inherent in the POP. We measure the flex-
ibility, whenever computationally feasible, as the number of
linearizations a POP represents.
As we have discussed earlier, verifying a POP’s validity
by way of the linearizations is not always practical. As such,
we will not attempt to compute POPs that maximize the
number of linearizations, but rather we will compute POPs
that adhere to one of the above criteria: minimum deorder-
ing, minimum reordering, or LCP.
4 A Partial Weighted MaxSAT Encoding
To generate a POP, we encode the problem of finding a
minimum deordering or reordering as a partial weighted
MaxSAT problem. Solutions to the default encoding corre-
spond to a LCP. That is, no POP exists with a proper subset
of the actions, or with a proper subset of the ordering con-
straints. We add further clauses to produce encodings that
correspond to optimal deorderings or reorderings.
We use two types of propositional variables: action vari-
ables and ordering variables. For every action a ∈ A, the
variable xa indicates whether or not the action a appears in
the POP. For every pair of actions a1, a2 ∈ A, the variable
κ(a1, a2) indicates an ordering constraint between action a1
and a2 in the POP.
In a partial weighted MaxSAT encoding there must be a
distinction between hard and soft clauses. We first present
the hard clauses of the encoding as boolean formulae which
we subsequently convert to CNF, and later describe the soft
clauses with their associated weight. We define the formu-
lae that ensure the POP generated is acyclic, and the order-
ing constraints produced include the transitive closure (here,

actions are universally quantified, and for formula (4) we
assume aI 6= ai 6= aG):
(¬κ(a, a)) (1)
(xaI ) ∧ (xaG) (2)
κ(ai, aj) → xai ∧ xaj (3)
xai → κ(aI , ai) ∧ κ(ai, aG) (4)
κ(ai, aj) ∧ κ(aj , ak) → κ(ai, ak) (5)
(1) ensures that there are no self-loops; (2) ensures that
we include the initial and goal actions; (3) ensures that if we
use an ordering variable, then we include both actions in the
POP; (4) ensures that an action cannot appear before the ini-
tial action (or after the goal); and (5) ensures that a solution
satisfies the transitive closure of ordering constraints. To-
gether, (1) and (5) ensure the POP will be acyclic, while the
remaining formulae tie the two types of variables together
and deal with the initial and goal actions.
In contrast to the typical SAT encoding for planning prob-
lems, we do not require the actions to be placed in a partic-
ular layer. Instead, we represent each action only once and
handle the ordering between actions through the κ variables.
Finally, we include the formulae needed to ensure that ev-
ery action has its preconditions met, and there are no threats
in the solution:
Υ(aj , ai, p) ≡
∧
ak∈deleters(p)
xak → κ(ak, aj) ∨ κ(ai, ak) (6)
xai →
∧
p∈PRE(ai)
∨
aj∈adders(p)
κ(aj , ai) ∧Υ(aj , ai, p) (7)
Intuitively, Υ(aj , ai, p) ensures that if aj is the achiever
of precondition p for action ai, then no deleter of p will be
allowed to occur between the actions aj and ai. Υ ensures
that every causal link remains unthreatened in a satisfying
assignment. Formula (7) ensures that if we include action
ai in the POP, then every precondition p of ai (the con-
junction) must be satisfied by at least one achiever aj (the
disjunction). κ(aj , ai) orders the achiever correctly, while
Υ(aj , ai, p) removes threats.
In order to achieve a POP that is least commitment, we
prefer solutions that first minimize the actions, and then min-
imize the ordering constraints. We achieve this by adding
a soft unit clause for every variable in our encoding. We
weight the κ variables with a unit cost and weight the action
variables high enough for the solver to focus on satisfying
them first:1
• w(¬κ(ai, aj)) = 1, ∀ai, aj ∈ A
• w(¬xa) = 1 + |A|2, ∀a ∈ A \ {aI , aG}
Note that the weight of any single action clause is greater
than the weight of all ordering constraint clauses. Since the
soft clauses are all unit clauses, we are able to use negation
1In domains with non-uniform action cost we could replace the
weight of 1 in the action clause with the cost of the action, allowing
us to minimize the total cost of the actions in the POP.
and solve the encoding with a MaxSAT procedure. A vio-
lation of any one of the unit clauses means that the solution
includes the action or ordering constraint corresponding to
the variable in the violated clause.
Proposition 1. Given a planning problem Π and a valid POP
P = 〈A,O〉, any variable setting that satisfies the formulae
(1)-(7) will correspond to a valid POP for Π where the or-
dering constraints are transitively closed.
Proof sketch. We have already seen that the POP induced
by a solution to the hard clauses will be a acyclic and tran-
sitively closed (due to formulae (1)-(5)). We can further see
that there will be no open preconditions since we include
aG, and the conjunction of (7) ensures that every precondi-
tion will be satisfied when the POP includes an action. Ad-
ditionally, there are no threats in the final solution because
of formula (6), which will be enforced every time a precon-
dition is met by formula (7). Since the POP corresponding
to any solution to the hard clauses will have no open pre-
conditions and no threats, the second notion of POP validity
allows us to conclude that the POP will be valid for Π. 
Proposition 2. Given a planning problem Π and a valid POP
P = 〈A,O〉, any valid POP Q = 〈A′ ,O′〉, where A′ ⊆ A
and O′ is transitively closed, has a corresponding variable
setting that satisfies formulae (1)-(7).
Proof sketch. The proposition follows from the direct en-
coding of the POP Q where xa = True iff a ∈ A′ and
κ(ai, aj) = True iff (ai ≺ aj) ∈ O′. If Q is a valid POP,
then it will be acyclic, include aI and aG, have all actions
ordered after aI and before aG, and be transitively closed
(satisfying (1)-(5)). We further can see that (6) and (7) must
be satisfied: if (7) did not hold, then there would be an action
a in the POP with a precondition p such that every potential
achiever of p has a threat that could be ordered between the
achiever and a. Such a situation is only possible when the
POP is invalid, which is a contradiction. 
Proposition 3. Given a planning problem Π and a valid POP
P = 〈A,O〉, a partial weighted MaxSAT solver will find a
solution to the soft clauses and formulae (1)-(7) that mini-
mizes the number of actions in the corresponding POP, and
subsequently minimizes the number of ordering constraints.
Proof sketch. With |A| actions, there can only be |A|2 order-
ing constraints. Since every soft clause that corresponds to
an ordering constraint has weight 1, the total sum of satisfy-
ing every ordering constraint clause will be |A|2. Since the
weight of satisfying any action clause is greater than |A|2,
the soft clauses corresponding to actions dominate the opti-
mization criteria. As such, there will be no valid POP for Π
which has a subset of the actions in P and has fewer actions
than a solution that satisfies formulae (1)-(7) while maxi-
mizing the weight of the satisfied soft clauses. 
Theorem 1 (Encoding Correctness). Given a planning prob-
lem Π, and a valid POP P for Π, a solution to our partial
weighted MaxSAT encoding is a LCP for P .
Proof sketch. This follows from propositions 1, 2, and 3. 
Observe that (1)-(7) never use the sequential plan. An op-
timal solution to the encoding will correspond to a LCP. To
enforce solutions that are minimum deorderings or reorder-
ings, we introduce two sets of hard clauses.

All Actions For optimal deorderings and reorderings, we
require every action to be a part of the POP. We consider
a formula that ensures we use every action (and so the opti-
mization works only on the ordering constraints). To achieve
this, we simply need to add each action as a hard clause:
(xa), ∀a ∈ A (8)
Deordering For a deordering we must forbid any explicit
ordering that contradicts the sequential plan. Assume our
sequential plan is [a0, · · · , ak]. We ensure that the computed
solution is a deordering by adding the following family of
hard unit clauses:
(¬κ(aj , ai)), 0 ≤ i < j ≤ k (9)
Intuitively, (9) simply forbids any ordering that contra-
dicts the orderings found in the transitive closure of the se-
quential plan, thus ensuring the solution is a deordering.
Due to space limitations, we refrain from proving the cor-
rectness of the two encoding extensions (8) and (9).
5 Relaxer Algorithm
We investigate the efficiency of an existing algorithm for re-
laxing a sequential plan to produce a deordering. Originally
due to Kambhampati and Kedar (1994), the algorithm oper-
ates by removing ordering constraints from a sequential plan
in a systematic manner. A heuristic guides the procedure,
and as pointed out in (Ba¨ckstro¨m 1998), the process does
not provide any guarantee that the resulting POP is mini-
mally constrained (that is, we may be able to remove further
ordering constraints and the POP remains valid). Despite
this lack of theoretical guarantee, we show later that the al-
gorithm produces excellent results.
The intuition behind the algorithm is to remove any
ordering (ai ≺ ak) from the sequential plan where ai
does not contribute to a precondition of ak and ai does
not threaten a precondition of ak (and vice versa). For
example, consider the case where our sequential plan is
[a1 · · · , ai, · · · , ak, · · · , an] and p ∈ PRE(ak). The al-
gorithm will keep the ordering (ai ≺ ak) only if leaving it
out would create a threat for a precondition of one of the ac-
tions, or if ai is the earliest action in the sequence where the
following holds:
1. p ∈ ADD(ai): ai is an achiever for p
2. ∀aj , i < j < k, p /∈ DEL(aj): p is not threatened.
Algorithm 1, which we will refer to as the Relaxer Algo-
rithm, presents this approach formally. We use index(a,~a)
to refer to the index of action a in the sequence ~a.
If ~a is a valid plan, line 11 will evaluate to true before
either line 8 evaluates to true or the for-loop at line 6 runs out
of actions. That is, we know that an unthreatened achiever
exists and the earliest such one is found.
The achiever is then added to the POP as a new causal link
(line 14), and the for-loop at line 17 adds all of the necessary
ordering constraints so the achiever remains unthreatened.
Note that for any deleter found in this for-loop, either line
18 or 20 must evaluate to true.
Algorithm 1: Relaxer Algorithm
Input: Sequential plan, ~a, including aI and aG
Output: Partial-order plan, 〈A,O, C〉
1 A = set(~a);
2 O = C = ∅;
3 foreach a ∈ A do
4 foreach f ∈ PRE(a) do
5 ach = Null;
6 for i = (index(a,~a)− 1) · · · 0 do
7 // Stop if we find a deleter of f
8 if f ∈ DEL(~a[i]) then
9 break;
10 // See if we have an earlier achiever
11 if f ∈ ADD(~a[i]) then
12 ach = ~a[i];
13 // Add the appropriate causal link
14 C = C ∪ {(ach
f
≺ a)};
15 O = O ∪ {(ach ≺ a)};
16 // Add orderings to avoid threats
17 foreach a′ ∈ deleters(f) \ {a} do
18 if index(a′,~a) < index(ach,~a) then
19 O = O ∪ {(a′ ≺ ach)};
20 if index(a′,~a) > index(a,~a) then
21 O = O ∪ {(a ≺ a′)};
22 return 〈A,O, C〉;
After going through the outer loop at line 3, every action
in the newly formed POP has an unthreatened causal link for
each of its preconditions. We are left with a valid POP, as
there are no open preconditions or causal threats.
6 Evaluation
We evaluate the effectiveness of using the partial weighted
MaxSAT solver, minimaxsat1.0 (Heras, Larrosa, and Oliv-
eras 2008), to optimally relax a plan using our proposed
encoding. To measure the quality of the POPs we gener-
ate, we consider the number of actions, ordering constraints,
and linearizations (whenever feasible to compute). Further,
we investigate the effectiveness of the Relaxer Algorithm to
produce a minimally constrained deordering.
For our analysis, we use six domains from the Interna-
tional Planning Competition (IPC)2 that allow for a par-
tially ordered solution: Depots, Driverlog, Logistics, TPP,
Rovers, and Zenotravel. These domains demonstrate both
the strengths and weaknesses of our approach. We con-
ducted the experiments on a Linux desktop with an eight-
core 3.0GHz processor. Each run was limited to 30 minutes
and 2GB of memory.
We generated an initial sequential plan by using the FF
planner (Hoffmann and Nebel 2001). The encodings pro-
vided in Section 4 were converted to CNF using simple dis-
tributive rules so they may be used with the minimaxsat1.0
solver. We investigated the possibility of using a starting
solution from the POPF planner (Coles et al. 2010), but
2http://ipc.icaps-conference.org/

Num FF Successfully
Domain Probs Solved Encoded
Depots 22 22 11
Driverlog 20 16 15
Logistics 35 35 30
TPP 30 30 7
Rovers 20 20 19
Zeno 20 20 15
ALL 147 143 97
Table 1: Number of instances successfully encoded. We in-
dicate the number problems per domain, the number of prob-
lems for which FF finds a sequential plan, and the number
of problems that our approach can encode successfully.
found that POPF failed to solve as many of the problems
as FF (albeit POPF handles a far richer set of planning prob-
lems). Of the problems that were mutually solved by FF
and POPF, we found that the POPs produced by the POPF
planner were quite over-constrained, having many more or-
dering constraints than necessary. Once relaxed (by way of a
similar encoding), the optimal deorderings, optimal reorder-
ings, and LCPs of both FF and POPF plans were very sim-
ilar. That is to say, there is usually little difference between
the POPs generated by relaxing a sequential FF plan and the
POPs generated by relaxing the POP found by POPF. For
this reason, we only present the results for FF.
With two extensions for the encoding (All Actions and
DeOrdering) we have 4 possible variations for every in-
stance. We will be primarily concerned with the following
combinations for the settings:
• ¬ AA, ¬ DO: No additional clauses correspond to the de-
fault encoding where an optimal solution gives us a least
commitment POP. We denote this setting as LCP.
• AA, ¬ DO: When we require all of the actions, solu-
tions correspond to a minimum reordering of the sequen-
tial plan. We denote this setting as MR.
• AA, DO: When we require all of the actions, and a de-
ordering, solutions correspond to a minimum deordering
of the sequential plan. We denote this setting as MD.
In the following evaluation, we only report on the prob-
lems where FF was able to find a sequential plan. We found
that many problems in the TPP and Depots domains are too
large to encode in CNF. In these cases, we found that the
combinatorial explosion for converting formulae (6) and (7)
to CNF caused the encoding size to become too large. In Ta-
ble 1 we present the number of problems per domain, solved
by FF, and successfully encoded. Below we discuss a poten-
tial solution to dealing with this drawback.
POP Quality We begin by examining the relative qual-
ity of the POPs produced with different optimization crite-
ria (LCP, MR, and MD), as well as the Relaxer Algorithm
(abbreviated as RX). We report the number of actions and
ordering constraints in the generated POP. Since the number
of actions for RX, MR, and MD are equal to the number of
actions in the sequential plan, we report the value only for
RX and LCP. Table 2 shows the results for all six domains on
|A| |O|
Domain RX LCP RX MD MR LCP
Depots (10) 34.2 30.9 451.8 451.8 407.9 339.1
Driverlog (15) 27.5 26.5 332.6 332.6 326.9 297.3
Logistics (25) 59.8 59.3 906.3 906.3 883.5 894.0
TPP (5) 13.4 13.4 74.8 74.8 74.8 74.8
Rovers (17) 30.6 30.1 214.3 214.3 208.8 200.2
Zeno (15) 19.8 19.8 137.1 137.1 136.6 136.6
ALL (87) 36.0 35.2 439.5 439.5 425.8 414.1
Table 2: Mean number of actions and ordering constraints
for the various approaches. Numbers next to the domain
indicate the number of instances solved by all methods (and
included in the mean).
the problems for which every approach succeeded in finding
a solution (87 of the 97 successfully encoded problems).
There are a few items of interest to point out. First,
columns 4 and 5 coincide perfectly. It turns out, perhaps
surprisingly, that the Relaxer Algorithm is able to produce
the optimal deordering in every case, even though it is not
guaranteed to do so. Since the algorithm can only produce
deorderings, this is the best we could hope for from the algo-
rithm. Second, we see the number of ordering constraints for
the LCP approach is greater than those for the MR approach
(on average) in the Logistics domain. The reason for this
is because POPs in the Logistics domain require more or-
dering constraints for a solution with slightly fewer actions.
For example, in prob15 the MR solution has 100 actions and
2278 ordering constraints, while the LCP solution reduces
the number of actions in the POP to 96 at the expense of
requiring 2437 ordering constraints.
In general, the LCP has fewer actions and ordering con-
straints than the optimal reordering, which in turn has fewer
ordering constraints than the optimal deordering. If the LCP
has the same number of actions as the sequential plan, then
the LCP and minimum reordering coincide. We can see this
effect in the TPP and Zenotravel where the number of ac-
tions and ordering constraints for LCP and MR are equal.
Finally, we note that in 4 problems (1 from Logistics,
and 3 from Rovers), we found that the number of actions
and constraints for either the LCP or MR POP matched that
of the Relaxer POP, but the number of linearizations dif-
fered. Further, the differences in linearizations were not in
the same direction (some better, and some worse). While the
number of ordering constraints in a POP (for a given number
of actions) usually gives an indication of the number of lin-
earizations for that POP, these 4 problems indicate that this
is not always the case.
Encoding Difficulty To measure the difficulty of solving
the encoded problems, we computed the average time min-
imaxsat1.0 required to find an optimal solution (since the
timing results had such a high standard deviation, we include
both the mean and median values). It should be noted that
an initial solution was consistently produced almost imme-
diately by minimaxsat1.0’s pre-processing step (a stochastic
local search that satisfies every hard clause and serves as a
lower bound on the optimal solution). Table 3 shows the
average solving time for each domain given a particular set-

Figure 1: Ratio of Linearizations. The y-axis represents the
number of linearizations induced by the POP for the optimal
reordering divided by the number of linearizations induced
by the POP for the optimal deordering. The x-axis ranges
over all problems where the number of linearizations dif-
fered (∼40%), and is sorted based on the y-axis value.
ting. The largest solving time recorded was just under 500
seconds for a problem in the Rovers domain with the LCP
setting. For comparison, we additionally include the average
time the Relaxer Algorithm required to find a deordering.
For at least one of MR and LCP, 10 of the 97 problems
successfully encoded proved too difficult for minimaxsat1.0
to solve, causing the solver to time out. We found that MD
was on average easier to solve, but overall the majority of
the problems were readily handled by minimaxsat1.0: 74%
being solved in under 5 seconds. Being a polynomial al-
gorithm, Relaxer consistently found a solution very quickly.
The maximum time for a problem was just over 4 seconds.
Reordering Flexibility We have already seen that the Re-
laxer Algorithm is capable of producing the minimum de-
ordering. To further evaluate the flexibility of the optimal
deordering, we compare the number of linearizations in-
duced by the optimal deordering with the number of lin-
earizations induced by the optimal reordering. We found
that of the 78 problems we could successfully compute the
linearizations for, approximately 40% of the problems ex-
hibited a difference between the optimal deordering and op-
timal reordering. Figure 1 shows the number of lineariza-
tions for the optimal reordering divided by the number of
linearizations for the optimal deordering. For readability,
we omit the 47 instances where the linearizations matched.
The ratio of linearizations ranges from 0.9 (an anomaly
discussed below) to over two million. While the Relaxer Al-
gorithm is proficient at finding the optimal deordering, this
result demonstrates that there are still significant gains to be
had in terms of flexibility by using the optimal reordering.
7 Discussion
The results paint an overall picture of how the optimization
criteria compare to one another. We find that the Relaxer Al-
gorithm is extremely adept at finding the optimal deordering,
despite its lack of theoretical guarantee. In contrast, in many
of the domains we see gains in terms of flexibility of the POP
if we compute the optimal reordering or a least commitment
POP. The encoding for the optimal deordering is easier for
minimaxsat1.0 to solve compared to the optimal reordering
or LCP, but the majority of problems for all optimization
criteria were readily handled by minimaxsat1.0.
Whenever possible, we used the number of linearizations
a POP represents as a measure of the POP’s flexibility (this
may not always be possible to compute due to the struc-
ture and size of the POP). We found that there are prob-
lems where the number of actions and ordering constraints
in two POPs are equal, while the number of linearizations is
not. Since the objective function of the encoding includes
only the number of actions and ordering constraints, there
is no guarantee on the number of linearizations that will re-
sult from a computed POP. An optimal reordering may even
have fewer linearizations than an optimal deordering (which
was observed in one case, as seen in Figure 1).
For a concrete example, consider two POPs on four ac-
tions A = {a1, a2, a3, a4}. Ignoring causal links, Figure 2
shows the structure of the POPs P1 and P2. Both POPs have
the same number of actions and ordering constraints, but the
number of linearizations differ: P1 has 6 linearizations while
P2 only has 5. These POPs serve as a basic example of how
the LCP criteria does not fully capture the notion of POP
flexibility. However, we should point out that fewer order-
ing constraints usually indicates more linearizations.
a1
a2
a3
a4
(a) P1
a1 a2
a3 a4
(b) P2
Figure 2: Two POPs with the same number of actions and
ordering constraints, but different number of linearizations.
For the Depots and TPP domains, the encoding size be-
came prohibitive. The larger formulae (i.e., (6) and (7)) are
not too large in themselves, but when converting to CNF
the theory becomes large. In future work, we plan to use
the Tseitin encoding to convert the theory (Tseitin 1970).
The Tseitin encoding will allow us to avoid the exponen-
tial blow-up in theory size, at the expense of introducing
more variables. We also hope to investigate versions of par-
tial weighted MaxSAT solvers tailored to problems in which
only unit clauses are soft (as is the case with our encod-
ing). There are other optimization techniques we plan on
investigating, including constraint programming encodings,
mixed-integer programming models, and a restricted form
of partial-order causal link (POCL) search.
The encoding technique we have presented differs signif-
icantly from the standard SAT-based planning encodings. In
particular, we avoid the need to encode an action for every
layer in a planning graph by appealing to the fact that we
already know the (superset of) actions that will be in the so-
lution. The core of our encoding follows an approach similar

Mean Time Median Time Mean Mean # Clauses
Domain RX MD MR LCP RX MD MR LCP # Vars MD MR LCP
Depots 0.54 89.63 55.63 19.72 0.44 1.26 1.55 1.71 1346.55 398205.91 397566.36 397532.64
Driverlog 0.58 1.35 38.36 6.02 0.55 0.77 0.70 0.87 943.87 39082.60 38638.20 38610.67
Logistics 1.16 9.40 127.66 105.02 0.80 3.13 5.69 4.98 7378.00 927265.59 923648.79 923576.59
TPP 0.87 3.58 171.71 171.68 0.61 0.44 0.50 0.50 1024.57 110964.86 110477.00 110452.57
Rovers 1.14 3.88 72.78 81.21 1.00 1.90 1.97 2.34 1428.74 111052.00 110371.11 110337.63
Zeno 0.60 0.86 0.85 0.87 0.46 0.42 0.53 0.43 541.73 18728.53 18477.47 18457.67
ALL 0.89 14.48 77.99 63.65 0.61 1.16 1.28 1.38 2972.67 364842.46 363397.60 363356.12
Table 3: Average time for the MaxSAT encoding to be solved by minimaxsat1.0, average time for the Relaxer Algorithm to
compute a deordering, and average number of clauses and variables in the encoding. Mean and median values of run time are
given in seconds. The mean is used to compute the average number of variables and clauses.
to Variant-II of Robinson et al. (Robinson et al. 2010). We
similarly encode the ordering between any pair of actions
as a variable (κ(ai, aj)), but rather than encoding a relaxed
planning graph, we encode the formulae that need to hold
for a valid POP. There are also similarities between our work
and that of (Do and Kambhampati 2003). In particular, the
optimization criteria for minimizing the number of ordering
constraints coincide, as does the optional use of constraints
to force a deordering. However, while Do and Kambhampati
focus on temporal relaxation in the context of action order-
ing, we take the orthogonal view to minimize the number of
actions required.
It is natural to consider the impact the choice of initial
plan has on the final POP. As was mentioned earlier, the
choice of initial solution between FF and POPF makes little
difference in the quality of the optimally relaxed POP. The
question remains open, however, on how to best compute an
initial set of actions for our encoding.
8 Conclusion
In this paper we proposed a practical method for computing
the optimal deordering and reordering of a sequential plan.
Despite the theoretical complexity of computing the opti-
mal deordering or reordering being NP-hard, we are able
to compute the optimal solution by leveraging the power of
modern MaxSAT solvers. We further propose an extension
to the classical least commitment criterion that considers the
number of actions in a solution, and demonstrate the added
flexibility of a POP that satisfies this criterion.
Our approach uses a family of novel encodings for partial
weighted MaxSAT where a solution corresponds to an opti-
mal POP satisfying one of the three criteria we investigate
(minimal deordering, minimal reordering, and our proposed
least commitment POP). We solve the encoding with a state-
of-the-art partial weighted MaxSAT solver, minimaxsat1.0,
and find that the majority of problems are readily handled
by minimaxsat1.0. We do, however, find that two domains
present a problem for the encoding phase of our approach.
In TPP and Depots, we find that the encoding size becomes
too large to handle, which limits the applicability of our cur-
rent encoding technique. In the future, we hope to employ
the Tseiten encoding to limit this drawback.
We also investigated an existing algorithm for deordering
sequential plans, and discovered that it successfully com-
putes the optimal deordering in every problem we tested (de-
spite its lack of theoretical guarantee). Since the algorithm
is polynomial, and quite fast in practice, it is well suited for
relaxing a POP if we require a deordering. However, if a
reordering or least commitment POP is acceptable, then we
can produce a far more flexible POP by using one of the
proposed encodings.
Acknowledgements
The authors gratefully acknowledge funding from the On-
tario Ministry of Innovation and the Natural Sciences and
Engineering Research Council of Canada (NSERC). We
would also like to thank the anonymous referees for useful
feedback on earlier drafts of the paper.
References
Ba¨ckstro¨m, C. 1998. Computational aspects of reordering plans.
Journal of Artificial Intelligence Research 9(1):99–137.
Biere, A.; Heule, M.; van Maaren, H.; and Walsh, T. 2009. Hand-
book of Satisfiability, Frontiers in Artificial Intelligence and Appli-
cations, vol. 185.
Coles, A.; Coles, A.; Fox, M.; and Long, D. 2010. Forward-
Chaining Partial-Order Planning. In Twentieth International Con-
ference on Automated Planning and Scheduling (ICAPS 10).
Do, M., and Kambhampati, S. 2003. Improving the temporal flexi-
bility of position constrained metric temporal plans. In AIPS Work-
shop on Planning in Temporal Domains (AIPS 03).
Heras, F.; Larrosa, J.; and Oliveras, A. 2008. MiniMaxSAT: An ef-
ficient weighted Max-SAT solver. Journal of Artificial Intelligence
Research 31(1):1–32.
Hoffmann, J., and Nebel, B. 2001. The FF planning system: Fast
plan generation through heuristic search. Journal of Artificial In-
telligence Research 14(1):253–302.
Kambhampati, S., and Kedar, S. 1994. A unified framework for
explanation-based generalization of partially ordered and partially
instantiated plans. Artificial Intelligence 67(1):29–70.
Robinson, N.; Gretton, C.; Pham, D. N.; and Sattar, A. 2010.
Partial weighted maxsat for optimal planning. In Proceedings of
the 11th Pacific Rim International Conference on Artificial Intelli-
gence, Daegu, Korea, August 30 - September 02, 2010.
Russell, S., and Norvig, P. 2009. Artificial intelligence: a modern
approach. Prentice hall.
Tseitin, G. 1970. On the complexity of proofs in propositional
logics. In Seminars in Mathematics, volume 8, 1967–1970.
Veloso, M.; Pollack, M.; and Cox, M. 1998. Rationale-based mon-
itoring for planning in dynamic environments. In Artificial Intelli-
gence Planning Systems, 171–179.
Weld, D. 1994. An introduction to least commitment planning. AI
Magazine 15(4):27.