Original: A ∗ Search and Bound-Sensitive Heuristics for Oversubscription Planning
Revised: A ∗ Search and Bound-Sensitive Heuristics for Oversubscription Planning

Original: Anonymous

Original: Abstract
Revised: Abstract

Original: Oversubscription planning (OSP) is the problem of ﬁnding plans that maximize the utility value of their end state while staying within a speciﬁed cost bound.
Revised: Oversubscription planning (OSP) is the problem of ﬁnding plans that maximize the utility value of their end state while staying within a speciﬁed cost bound.

Original: Recently, it has been shown that OSP problems can be reformulated as classical planning problems with multiple cost functions but no utili- ties.
Revised: Recently, it has been shown that OSP problems can be reformulated as classical planning problems with multiple cost functions but no utilities.

Original: Here we take advantage of this reformulation to show that OSP problems can be solved optimally using the A ∗ search algorithm, in contrast to previous approaches that have used variations on branch-and-bound search.
Revised: Here we take advantage of this reformulation to show that OSP problems can be solved optimally using the A ∗ search algorithm, in contrast to previous approaches that have used variations on branchand-bound search.

Original: This allows many powerful techniques developed for classical planning to be applied to OSP problems.
Revised: This allows many powerful techniques developed for classical planning to be applied to OSP problems.

Original: We also introduce novel boundsensitive heuristics, which are able to reason about the pri- mary cost of a solution while taking into account secondary cost functions and bounds, to provide superior guidance com- pared to heuristics that do not take these bounds into account.
Revised: We also introduce novel bound-sensitive heuristics, which are able to reason about the primary cost of a solution while taking into account secondary cost functions and bounds, to provide superior guidance compared to heuristics that do not take these bounds into account.

Original: We implement two such bound-sensitive variants of existing classical planning heuristics, and show experimentally that the resulting search is signiﬁcantly more informed than com- parable heuristics that do not consider bounds.
Revised: We implement two such bound-sensitive variants of existing classical planning heuristics, and show experimentally that the resulting search is significantly more informed than comparable heuristics that do not consider bounds.

Original: Introduction
Revised: Introduction

Original: Oversubscription planning (OSP) problems are a family of deterministic planning problems.
Revised: Oversubscription planning (OSP) problems are a family of deterministic planning problems.

Original: In contrast to classical planning, in which a set of hard goals is speciﬁed and the planner searches for a minimal (or low) cost plan that reaches a state in which all of the goals are made true, oversubscription planning speciﬁes a utility function that describes the beneﬁt associated with achieving different possible states, and asks for a plan whose cost does not exceed a set bound and achieves as high a utility as possible (Smith 2004).
Revised: In contrast to classical planning, where a set of hard goals is speciﬁed and the planner searches for a minimal (or low) cost plan that reaches a state in which all of the goals are made true, oversubscription planning speciﬁes a utility function that describes the beneﬁt associated with achieving different possible states and asks for a plan whose cost does not exceed a set bound and achieves as high a utility as possible [Smith, 2004].

Original: While domain-independent classical planning approaches have increasingly standardized around variations on A ∗ search and heuristics that are automatically extracted from the problem description (Bonet and Geffner 2001; Keyder and Geffner 2008; Haslum and Geffner 2000; Edelkamp 2001;Helmert et al. 2014; Helmert and Domshlak 2009),
Revised: While domain-independent classical planning approaches have increasingly standardized around variations on A ∗ search and heuristics that are automatically extracted from the problem description [ Bonet and Geffner, 2001; Keyder and Geffner, 2008; Haslum and Geffner, 2000; Edelkamp, 2001;

Original:  OSP has generally been solved with branch-and-bound algorithms and heuristics that compute an admissible (in this context non-under) estimate of the utility achievable from a state.
Revised: Helmert et al. , 2014; Helmert and Domshlak, 2009], OSP has generally been solved with branch-and-bound algorithms and heuristics that compute an admissible (in this context nonunder) estimate of the utility achievable from a state.

Original: In order to obtain these estimates, recent approaches often adapt classical planning techniques such as landmarks (Mirkis and Domshlak 2014; Muller and Karpas 2018) or abstractions (Mirkis and Domshlak 2013), and enhance them with reasoning that is speciﬁc to the context of OSP, such as the knowledge that there always exists an optimal plan that ends with a utility-increasing action, or that the cost bound for the problem can be reduced under speciﬁc conditions to aid the search algorithm in detecting that improving over the currently achieved utility is impossible.
Revised: In order to obtain these estimates, recent approaches often adapt classical planning techniques such as landmarks [Mirkis and Domshlak, 2014; Muller and Karpas, 2018] or abstractions [ Mirkis and Domshlak, 2013 ] , and enhance them with reasoning that is speciﬁc to the context of OSP, such as the knowledge that there always exists an optimal plan that ends with a utility-increasing action, or that the cost bound for the problem can be reduced under speciﬁc conditions to aid the search algorithm in detecting that improving over the currently achieved utility is impossible.

Original: In contrast to these approaches, our aim here is to show that general methods from classical planning, including A ∗ search, can be used in the OSP setting nearly as is.
Revised: In contrast to these approaches, our aim here is to show that general methods from classical planning, including A ∗ search, can be used in the OSP setting nearly as is.

Original: This previously turned out to be the case for the related netbeneﬁt planning problem, where classical planners solving a compilation were shown tobe outperform planners designed speciﬁcally for that task (Keyder and Geffner 2009).
Revised: This previously turned out to be the case for the related net-beneﬁt planning problem, where classical planners solving a compilation were shown to outperform planners designed speciﬁcally for that task [ Keyder and Geffner, 2009 ] .

Original: Here, we use a similar, recently proposed compilation that converts OSP problems into classical planning problems with multiple cost functions but no utilities (Katz et al. 2019a).
Revised: Here, we use a similar, recently proposed compilation that converts OSP problems into classical planning problems with multiple cost functions but no utilities [ Katz et al. , 2019a ] .

Original: In addition, we demonstrate that existing classical planning heuristics can be used to guide the search for optimal plans.
Revised: In addition, we demonstrate that existing classical planning heuristics can be used to guide the search for optimal plans.

Original: While these heuristics are typically uninformative out-ofthe-box, they require only minor modiﬁcations (and no speciﬁc reasoning about utilities) to render them sensitive to the secondary cost functions and bounds that are introduced by the compilation.
Revised: While these heuristics are typically uninformative out-of-the-box, they require only minor modiﬁcations (and no speciﬁc reasoning about utilities) to render them sensitive to the secondary cost functions and bounds that are introduced by the compilation.

Original: Our experiments with A ∗ and the newly introduced estimators that we refer to as bound-sensitive heuristics show that they lead to informed searches that are competitive with, and in some cases outperform, the state of the art for optimal OSP.
Revised: Our experiments with A ∗ and the newly introduced estimators that we refer to as bound-sensitive heuristics show that they lead to informed searches that are competitive with, and in some cases outperform, the state of the art for optimal OSP.

Revised: One related area of research in the classical setting is that of bounded-cost planning , where the planner looks for any plan with (primary) cost below a given bound, similar to the treatment of the secondary cost in the OSP setting.

Revised: Approaches proposed for this setting include dedicated search algorithms [Stern et al. , 2011] and heuristics that take into account accumulated cost and plan length at the current search node [Thayer and Ruml, 2011; Haslum, 2013;

Revised: Dobson and Haslum, 2017 ] .

Revised: These approaches work by preferentially expanding nodes in areas of the search space that are likely to have a solution under the cost bound.

Revised: Optimal

Original:  We now brieﬂy review the various ﬂavors of planning that we consider in this work, and introduce the formalisms by which we describe them.
Revised: OSP, however, requires expanding all nodes that potentially lie on a path to state with maximal utility. Furthermore, it cannot be assumed that solutions necessarily achieve all soft goals. Heuristics that are able to take into account bounds on secondary cost functions have also been investigated in the stochastic shortest path setting, where they were used as additional constraints in an LP-based heuristic to consider limitations on fuel or time resources [Trevizan et al. , 2017]. We now brieﬂy review the various ﬂavors of planning that we consider in this work, and introduce the formalisms by which we describe them.

Original: Background
Revised: Background

Original: We describe planning problems in terms of extensions to the
Revised: We describe planning problems in terms of extensions to the

Original: SAS + formalism (B¨ackstr¨om and Nebel 1995).
Revised: SAS + formalism [B¨ackstr¨om and Nebel, 1995].

Original: A classical planning task Π
Revised: A classical planning task Π

Original: = (cid:104) V , O ; s I , G, C (cid:105) is given by a set of variables V , with each variable v ∈ V having a ﬁnite do- main dom ( v ) , a set of actions O , with each action o ∈ O described by a pair (cid:104) pre ( o ) , eﬀ ( o ) (cid:105) of partial assignments to 
Revised: = (cid:104) V , O ; s I , G, C (cid:105) is given by a set of vari- ables V , with each variable v ∈ V having a ﬁnite domain dom ( v ) , a set of actions O , with each action o ∈ O described by a pair (cid:104) pre ( o ) , eﬀ ( o ) (cid:105) of partial assignments to V , called the precondition and effect of o , respectively,

Original: V , called the precondition and effect of o , respectively, initial state s I and goal condition G , which are full and par- tial assignments to V , respectively, and the cost function 
Revised: initial state s I and goal condition G , which are full and partial assignments to V , respectively, and the cost function C : O

Original: C : O → R 0+ .
Revised: → R 0+ .

Original: A state s is given by a full assignment to V .
Revised: A state s is given by a full assignment to V .

Original: An action is said to be applicable in a state s if pre ( o ) ⊆
Revised: An action is said to be applicable in a state s if pre ( o ) ⊆

Original: s , and s (cid:74) o
Revised: s , and s (cid:74) o

Original: (cid:75) denotes the result of applying o in s , where the value of each v ∈ V is given by eﬀ ( o )
Revised: (cid:75) denotes the result of applying o in s , where the value of each v ∈ V is given by eﬀ ( o )

Original: [ v ] if deﬁned and s [ v ] otherwise.
Revised: [ v ] if deﬁned and s [ v ] otherwise.

Original: An operator sequence π = (cid:104) o 1 , . . .
Revised: An operator sequence π = (cid:104) o 1 , . . .

Original: , o k (cid:105) is applicable in s if there exist states s 0 , · · · , s k such that (i) s 

Revised: , o k (cid:105) is applicable in s if there exist states s 0 , · · · , s k such that (i) s

Original: = s , and (ii) for each 1 ≤ i ≤ k , 
Revised: = s , and (ii) for each≤ i ≤ k , o

Original: o i is applicable in s i − 1 and s i
Revised: i is applicable in s i − 1 and s i

Original: = s i − 1 (cid:74) o
Revised: = s i − 1 (cid:74) o

Original: i (cid:75) .
Revised: i (cid:75) .

Original: We refer to the state s k by s (cid:74) π (cid:75) and call it the end state of π .
Revised: We refer to the state s k by s (cid:74) π (cid:75) and call it the end state of π .

Original: An operator sequence π is a plan for a classical planning problem if it is applicable in s I and
Revised: An operator sequence π is a plan for a classical planning problem if it is applicable in s I and

Original: G ⊆
Revised: G ⊆

Original: s I (cid:74) π (cid:75) .
Revised: s I (cid:74) π (cid:75) .

Original: The cost of a  π is given by C ( π ) 
Revised: The cost of a plan π is given by C ( π ) =

Original: = (cid:80) o ∈ π C
Revised: (cid:80) o ∈ π C

Original: ( o )
Revised: ( o )

Original: ; the goal of optimal classical planning is to ﬁnd a plan with minimal cost.
Revised: ; the goal of optimal classical planning is to ﬁnd a plan with minimal cost.

Revised: We refer to a pair of variable v and its value ϑ ∈ dom ( v ) as a fact and denote it by (cid:104) v, ϑ (cid:105) .

Revised: We sometimes abuse notation and treat partial assignments as sets of facts.

Original: An oversubscription planning (OSP) problem is given by Π OSP =
Revised: An oversubscription planning (OSP) problem is given by Π OSP =

Original: (cid:104) V , O, s I , C , u , B (cid:105) , where V , O , s I , and C are as in classical planning, u : (  v, ϑ  ) → R 0+ is a non-negative valued utility function over variable assignments, and B is a cost bound for the plan, imposing the additional requirement that only plans π such that C ( π ) ≤ B are valid.
Revised: (cid:104) V , O, s I , C , u , B (cid:105) , where V , O , s I , and C are as in classical planning, u : ( (cid:104) v, ϑ (cid:105) ) → R 0+ is a non-negative valued utility function over variable assignments (facts), and B is a cost bound for the plan, imposing the additional requirement that only plans π such that C ( π ) ≤ B are valid.

Original: The utility of a plan π is given by (cid:80)
Revised: The utility of a plan π is given by (cid:80)

Original: ( v,ϑ ) ∈
Revised: [Equation]

Original: s I (cid:74) π (cid:75) u ( v, ϑ ) ; the objective of OSP problems is to ﬁnd valid plans with maximal utility.
Revised: ) ; the ob- jective of OSP problems is to ﬁnd valid plans with maximal utility.

Original: A multiple cost function (MCF) problem is given by Π MCF = (cid:104) V, O, s I , G, C 0 , C (cid:105) , where V , O , s I , and C 0 are as in classical planning,
Revised: A multiple cost function (MCF) problem is given by Π MCF = (cid:104) V, O, s I , G, C 0 , C (cid:105) , where V , O , s I , and C 0 are as in classical planning,

Original: C 0 is the primary cost function , and C = {(cid:104) C
Revised: C 0 is the primary cost function , and C = {(cid:104) C

Original: i , B i (cid:105) | 1 ≤ i ≤ n } is a set of secondary cost functions
Revised: i , B i (cid:105) | 1 ≤ i ≤ n } is a set of secondary cost functions

Original: C i : O → R 0+ , and bounds , both non-negative.
Revised: C i : O → R 0+ , and bounds , both non-negative.

Original: Valid plans for MCF planning problems fulﬁll the condition 
Revised: Valid plans for MCF planning problems fulﬁll the condition C

Original: C i ( π ) ≤ B i for all secondary cost functions, and optimal plans for MCF planning have minimal primary cost
Revised: [Equation] ( π [Equation] ≤ B i for all secondary cost functions, and optimal plans for MCF planning have minimal primary cost

Original: C 0 ( π ) .
Revised: [Equation]

Original: In this paper we only consider MCF problems with a single secondary cost function, i.e. n = 1 .
Revised: In this paper we only consider MCF problems with a single secondary cost function, i.e. n = 1 .

Original: Reformulating OSP Problems
Revised: Reformulating OSP Problems

Original: It has recently been shown that an OSP problem can be compiled into an MCF planning problem with a single secondary cost function that corresponds to the cost function C of the original problem, and is constrained to not exceed the speciﬁed bound B (Katz et al. 2019a).
Revised: It has recently been shown that an OSP problem can be compiled into an MCF planning problem with a single secondary cost function that corresponds to the cost function C of the original problem, and is constrained to not exceed the speciﬁed bound B [Katz et al. , 2019a].

Original: The primary cost function for the problem, or the cost function to be optimized, results from compiling the utilities from the original problem into costs.
Revised: The primary cost function for the problem, or the cost function to be optimized, results from compiling the utilities from the original problem into costs.

Original: Two different compilations have been proposed for this task: (i) the soft goals compilation , which adds for each variable v that has some value ϑ ∈ dom ( v ) for which a utility is speciﬁed, a hard goal, along with actions that are able to achieve this hard goal at different costs, and (ii) the state delta compilation which encodes in the cost of each action the change in state utility that results from applying it.
Revised: Two different compilations have been proposed for this task: (i) the soft goals compilation , which adds for each variable v that has some value ϑ ∈ dom ( v ) for which a utility is speciﬁed, a hard goal, along with actions that are able to achieve this hard goal at different costs, and (ii) the state delta compilation which encodes in the cost of each action the change in state utility that results from applying it.

Original: Here we consider only (i), as (ii) introduces negative action costs that A ∗ and existing classical planning heuristics are not designed to handle.
Revised: Here we consider only (i), as (ii) introduces negative action costs that A ∗ and existing classical planning heuristics are not designed to handle.

Original: Note, however, that our methods do not depend on the speciﬁc choice of compilation, as long as they remove utilities from the problem and do not introduce negative action costs.
Revised: Note, however, that our methods do not depend on the speciﬁc choice of compilation, as long as they remove utilities from the problem and do not introduce negative action costs.

Original: The soft goals compilation was originally introduced in the context of net-beneﬁt planning, which is similar to oversubscription planning but does not specify a bound on plan cost, having instead as an objective the minimization of the difference between the achieved utility and the cost of the plan (Keyder and Geffner 2009).
Revised: The soft goals compilation was originally introduced in the context of net-beneﬁt planning, which is similar to oversubscription planning but does not specify a bound on plan cost, having instead as an objective the minimization of the difference between the achieved utility and the cost of the plan [ Keyder and Geffner, 2009 ] .

Original: It can be applied in the OSP setting to result in an MCF planning problem as follows:
Revised: It can be applied in the OSP setting to result in an MCF planning problem as follows:

Original: Deﬁnition 1 Let Π OSP =
Revised: Deﬁnition 1 Let Π OSP =

Original: (cid:104) V , O, s I , C , u , B (cid:105) be an over- subscription planning task.
Revised: (cid:104) V , O, s I , C , u , B (cid:105) be an over- subscription planning task.

Original: The soft goals reformulationΠ
Revised: The soft goals reformulation

Original: [Equation]
Revised: [Equation]

Revised: , s I , G (cid:48) , C 0 , {(cid:104) C (cid:48) , B (cid:105)}(cid:105) of Π

Original: Π OSP is an MCF planning task ,where
Revised: OSP is an MCF planning task ,

Original:  • V (cid:48)
Revised: where • V (cid:48)

Original: = { v (cid:48) | v ∈ V } , with dom ( v (cid:48) )
Revised: = { v (cid:48) | v ∈ V } , with dom ( v (cid:48) )

Original: = (cid:26) dom ( v ) ∪ {v g  } u max ( v ) >
Revised: = (cid:26) dom ( v ) ∪ { g v } u max ( v ) >

Original: dom ( v ) otherwise , • O (cid:48)
Revised: dom ( v ) otherwise , • O (cid:48)

Original: =
Revised: =

Original: O ∪ { o v,ϑ = (cid:104){ v = ϑ } , { v = v g }(cid:105) | ϑ ∈ dom ( v ) , v ∈ V, u max ( v ) > 0 } • G (cid:48) 
Revised: O ∪ { o v,ϑ = (cid:104){(cid:104) v, ϑ (cid:105)} , {(cid:104) v, g v (cid:105)}(cid:105) | ϑ ∈ dom ( v ) , v ∈ V, u max ( v ) > 0 } • G (cid:48) =

Original: =

Original: { v g | v ∈ V, u max ( v ) > 0 } , • C 0 ( o ) =
Revised: {(cid:104) v, g v (cid:105)| v ∈ V, u max ( v ) > 0 } , • C 0 ( o ) =

Original: (cid:26)o ∈ O u max ( v ) − u
Revised: (cid:26)o ∈ O u max ( v ) − u

Original: (cid:104) v, ϑ (cid:105) )
Revised: (cid:104) v, ϑ (cid:105) )

Original: o = o v,ϑ , • C (cid:48)
Revised: o = o v,ϑ , • C (cid:48)

Original: ( o )
Revised: ( o )

Original: =
Revised: =

Original: (cid:26) C ( o ) o ∈ Ootherwise ,
Revised: (cid:26) C ( o ) o ∈ Ootherwise ,

Original: with
Revised: with

Original: u max ( v ) := max ϑ ∈ dom ( v ) u
Revised: u max ( v ) := max ϑ ∈ dom ( v ) u

Original: ( (cid:104) v, ϑ (cid:105) 
Revised: ( (cid:104) v, ϑ (cid:105) )

Original: ) denoting the maxi- mum utility over the values of the variable v .
Revised: denoting the maxi- mum utility over the values of the variable v .

Original: In the reformulated problem, only the o v,ϑ actions for which ϑ is not the maximum utility value of v have positive primary costs.
Revised: In the reformulated problem, only the o v,ϑ actions for which ϑ is not the maximum utility value of v have positive primary costs.

Original: These actions make explicit that a particular utility will not be achieved, and that the plan has instead chosen to achieve the associatedv g  by accepting the associated cost penalty.
Revised: These actions make explicit that a particular utility will not be achieved, and that the plan has instead chosen to achieve the associated g v by accepting the associated cost penalty.

Original: The primary cost of a plan π for the reformulated problem is then given by (cid:80) v ∈ 
Revised: The primary cost of a plan π for the reformulated problem is then given by (cid:80) v ∈ V u max ( v ) − (cid:80)

Original: [Equation]
Revised: [Equation]

Original: v, θ [Equation]
Revised: Note that this compilation assumes that utilities are deﬁned for single facts. The more general case, in which utilities are instead deﬁned for logical formulae ϕ , can be handled as in the soft goals compilation by introducing a new variable v ϕ , and two actions that achieve its goal value with cost 0 and precondition ϕ , and cost u ( ϕ ) and precondition ∅ , respectively [Keyder and Geffner, 2009]. Since we consider only single fact utilities here, we do not discuss this case in detail.

Original: While this compilation is sound as stated, two further optimizations can be made to reduce the state space of the resulting compiled problem.
Revised: While this compilation is sound as stated, two further optimizations can be made to reduce the state space of the resulting compiled problem.

Original: First, an arbitrary ordering can be introduced over V to ensure that thev g  are achieved in a ﬁxed sequence, to avoid searching over different orderings.
Revised: First, an arbitrary ordering can be introduced over V to ensure that the g v values are achieved in a ﬁxed sequence, to avoid searching over different orderings.

Original: Second, a new precondition fact that is deleted by the o v,ϑ actions can be added to the original domain actions to ensure that o v,ϑ actions happen only at the end of the plan and are not interleaved with the original domain actions.
Revised: Second, a new precondition fact that is deleted by the o v,ϑ actions can be added to the original domain actions to ensure that o v,ϑ actions happen only at the end of the plan and are not interleaved with the original domain actions.

Original: We make use of both of these optimizations here.
Revised: We make use of both of these optimizations here.

Original: A ∗ for MCF Planning Problems
Revised: A ∗ for MCF Planning Problems

Original: The A ∗ algorithm extends blind search techniques such as Dijkstra’s algorithm by allowing the incorporation of admissible (non-overestimating) heuristics (Hart et al. 1968).
Revised: The A ∗ algorithm extends blind search techniques such as Dijkstra’s algorithm by allowing the incorporation of admissible (non-overestimating) heuristics [ Hart et al. , 1968 ] .

Original: In each iteration of its main loop, A ∗ picks a node n to expand with minimal f ( n ) =
Revised: In each iteration of its main loop, A ∗ picks a node n to expand with minimal f ( n ) =

Original: g ( n ) + h ( n ) value, where g ( n ) is the cost of the path to n , and h ( n ) is an admissible estimate of the remaining cost to the goal.
Revised: g ( n ) + h ( n ) value, where g ( n ) is the cost of the path to n , and h ( n ) is an admissible estimate of the remaining cost to the goal.

Original: An optimal solution to the problem is found when the node n with minimal f ( n ) value is a goal state.
Revised: An optimal solution to the problem is found when a node n with minimal f ( n ) value is a goal node.

Original: To adapt A ∗ to the MCF planning setting, we store at each node n a set of accumulated path costs g i ( n ) resulting from each of the secondary cost functions C 1 , . .
Revised: To adapt A ∗ to the MCF planning setting, we store at each node n a set of accumulated path costs g i ( n ) resulting from each of the secondary cost functions C 1 , . .

Original: . , C n , in addi- tion to the accumulated primary cost g 0 ( n ) .
Revised: C n , in addition to the accumulated primary cost g 0 ( n ) .

Original: When a node isllu ( visited ( l 1 )) : lu ( visited ( l 2 )) : taken from the priority queue and expanded, generated successor nodes for which any g i ( n ) >
Revised: When a node is taken from the priority queue and expanded, generated successor nodes for which any g i ( n ) >

Original: B i can be immediately pruned, as all C i are assumed to be non-negative, and they cannot constitute valid preﬁxes for solution paths.
Revised: B i can be immediately pruned, as all C i are assumed to be non-negative, and they cannot constitute valid preﬁxes for solution paths.

Original: One key optimization used in modern A ∗ implementations in the classical setting is duplicate detection, which allows states that are rediscovered during search to be discarded, if the new g value exceeds the cost of the path to the state that was previously found, or to be updated with a new parent, if the cost of the new path is less.
Revised: One key optimization used in modern A ∗ implementations in the classical setting is duplicate detection, which allows states that are rediscovered during search to be discarded, if the new g value exceeds the cost of the path to the state that was previously found, or to be updated with a new parent, if the cost of the new path is less.

Original: In the MCF setting, care must be taken to ensure that newly discovered nodes are discarded (or replace existing nodes), only when they are dominated by (or dominate), the existing node in all cost dimensions.
Revised: In the MCF setting, care must be taken to ensure that newly discovered nodes are discarded (or replace existing nodes), only when they are dominated by (or dominate), the existing node in all cost dimensions.

Original: While the only necessary property of the open list from a correctness perspective is that it order nodes by increasing primary f ( n ) value, the choice of a secondary ordering heuristic plays a role here: an ordering that causes a dominating node to be generated ﬁrst and enables subsequently generated nodes to be immediately discarded as dominated results in superior performance.
Revised: While the only necessary property of the open list from a correctness perspective is that it order nodes by increasing primary f ( n ) value, the choice of a secondary ordering heuristic plays a role here: an ordering that causes a dominating node to be generated ﬁrst and enables subsequently generated nodes to be immediately discarded as dominated results in superior performance.

Original: In our implementation of the algorithm, we therefore use an open list that orders nodes by increasing g i ( n ) value when their primary f ( n ) values are the same. 
Revised: In our implementation of the algorithm, we therefore use an open list that orders nodes by increasing g i ( n ) value when their primary f ( n ) values are the same. [Figure] u ( visited ( l 1 )) [Figure]

Revised: [Figure]

Revised: u ( visited ( l 2 )) :

Original: Bound-Sensitive Heuristics
Revised: Bound-Sensitive Heuristics

Original: While any admissible heuristic can be used to guide search in MCF planning, classical planning heuristics that ignore bounds entirely are typically extremely uninformative.
Revised: While any admissible heuristic can be used to guide search in MCF planning, classical planning heuristics that ignore bounds entirely are typically extremely uninformative.

Original: Consider the problem shown in Figure 1: the agent is initially at l 0 , and can obtain a utility of 10 by visiting each of the locations l 1 and l 2 .
Revised: Consider the problem shown in Figure 1: the agent is initially at l 0 , and can obtain a utility of 10 by visiting each of the locations l 1 and l 2 .

Original: The costs of the actions move ( l 0 , l 1 ) and move ( l 1 , l 2 ) are both 1 .
Revised: The costs of the actions move ( l 0 , l 1 ) and move ( l 1 , l 2 ) are both 1 .

Original: In the compiled MCF version of this problem, an optimal but naive heuristic that ignores the bound will give an estimate for the primary cost of 0 , as both visited ( l 1 ) and visited ( l 2 ) can be made true, and the associated 0 -primary cost o visited ( l ∗ ) actions applied to reach the newly introduced hard goals corresponding to each utility.
Revised: In the compiled MCF version of this problem, an optimal but naive heuristic that ignores the bound will give an estimate for the primary cost of 0 , as both visited ( l 1 ) and visited ( l 2 ) can be made true, and the associated 0 -primary cost o visited ( l ∗ ) actions applied to reach the newly introduced hard goals corresponding to each utility.

Original: If, however, B = 1 , the optimal C 0 cost at l 0 is 10 , since l 2 cannot be reached at cost ≤ B and the agent must use the o not-visited ( l 2 ) action to achieve the associated hard goal with a cost of 10 .
Revised: If, however, B = 1 , the optimal C 0 cost at l 0 is 10 , since lcannot be reached at cost ≤ B and the agent must use the o not-visited ( l 2 ) action to achieve the associated hard goal with a cost of 10 .

Original: Similarly, if B = 0 , the C 0 cost of the optimal plan is 20 , since the value of C 1 for all available actions exceeds the bound B .
Revised: Similarly, if B = 0 , the C 0 cost of the optimal plan is 20 , since the value of C 1 for all available actions exceeds the bound B .

Original: In practice, it turns out that the OSP versions of a large number classical planning problems have similar behavior: their state spaces are strongly connected, so any variable assignment can be achieved from any state, and classical planning heuristics that ignore bounds are no more informed than blind search.
Revised: In practice, it turns out that the OSP versions of many classical planning problems have similar behavior: their state spaces are strongly connected, so any variable assignment can be achieved from any state, and classical planning heuristics that ignore bounds are no more informed than blind search.

Original: In order to obtain estimates that take secondary cost bounds into account and can guide heuristic search towards feasible solutions, we therefore introduce bound-sensitive heuristics .
Revised: In order to obtain estimates that take secondary cost bounds into account and can guide heuristic search towards feasible solutions, we therefore introduce bound-sensitive heuristics .

Original: In the following, we use b to denote a budget vector of non-negative reals that indicate the unused component of each of the secondary cost bounds B i at a given search node.
Revised: In the following, we use b to denote a budget vector of nonnegative reals that indicate the unused component of each of the secondary cost bounds B i at a given search node.

Original: Deﬁnition 2 (Optimal bound-sensitive heuristic)
Revised: Deﬁnition 2 (Optimal bound-sensitive heuristic)

Original: Given an MCF planning problem Π MCF = (cid:104) V, O, s I , G, C 0 , C (cid:105) , the optimal bound-sensitive heuristic h ∗ ( s, b ) for a state s and budget vector b is given by the minimal primary cost C 

Revised: Given an MCF planning problem Π MCF = (cid:104) V, O, s I , G, C 0 , C (cid:105) , the optimal bound-sensitive heuristic h ∗ ( s, b ) for a state s and budget vector b is given by the minimal primary cost C

Original: ( π ) of a plan π for s such that C i ( π ) ≤ b i for i = 1 , . . . , n .
Revised: ( π ) of a plan π for s such that C i ( π ) ≤ b i for i = 1 , . . . , n .

Original: By analogy with standard admissible heuristics, an admissible bound-sensitive heuristic is a non-overestimating bound-sensitive heuristic:
Revised: By analogy with standard admissible heuristics, an admissible bound-sensitive heuristic is a non-overestimating bound-sensitive heuristic:

Original: Deﬁnition 3 (Admissible bound-sensitive heuristic)
Revised: Deﬁnition 3 (Admissible bound-sensitive heuristic)

Original: Given an

Original: MCF planning problem Π MCF = (cid:104) V, O

Original: [Equation]

Original: [Equation] [Equation] [Equation] (cid:105) , an admissible bound-sensitive heuristic h ( s, b ) for a state s and budget vector b is a heuristic h
Revised: Given an MCF planning problem Π MCF = (cid:104) V, O, s I , G, C 0 , C (cid:105) , an admissible bound-sensitive heuristic h ( s, b ) for a state s and budget vector b is a heuristic h

Original: such that h ( s, b ) ≤ h ∗ ( s, b ) for all s , b .
Revised: such that h ( s, b ) ≤ h ∗ ( s, b ) for all s , b .

Original: Any classical planning heuristic that completely ignores 
Revised: Any classical planning heuristic that completely ignores C i and B

Original: C i and B i can be thought of as an admissible bound-sensitive heuristic that assumes b = ∞ .
Revised: i can be thought of as an admissible bound-sensitive heuristic that assumes b = ∞ .

Original: As the value of b decreases, the value of h ∗ ( s, b ) can only increase.
Revised: As the value of b decreases, the value of h ∗ ( s, b ) can only increase.

Original: In general, it is useful to keep in mind the following property:
Revised: In general, it is useful to keep in mind the following property:

Original: Theorem 1 Given a state s and budget vectors b , b’ such that b ≤ b’ (where ≤ is interpreted as a pairwise compari- son [Equation]
Revised: Theorem 1 Given a state s and budget vectors b , b’ such that b ≤ b’ (where ≤ is interpreted as a pairwise comparison), h ∗ ( s, b ) ≥ h ∗ ( s, b’ ) .

Original: Proof: This follows from the fact that any plan π for s such that C i ( π ) ≤ b i also has the property that C i ( π ) ≤ b’ i for i = 1 , . . . , n since b ≤ b’ , yet the opposite is not the case.
Revised: Proof sketch: This follows from the fact that any plan π for s such that C i ( π ) ≤ b i also has the property that C i ( π ) ≤ b’ i for i = 1 , . . . , n since b ≤ b’ , yet the opposite is not the case.

Original: Theorem 1 applied to MCF planning problems obtained as the soft goals compilations of OSP problems states that for any s , decreasing b increases h ∗ ( s, b ) , and decreases the achievable utility, since the primary cost here indicates the utility that the plan must declare unachievable through o v,ϑ actions with C 0 ( o v,ϑ ) ≥ 0 .
Revised: Theorem 1 applied to MCF planning problems obtained as the soft goals compilations of OSP problems states that for any s , decreasing b increases h ∗ ( s, b ) , and decreases the achievable utility, since the primary cost here indicates the utility that the plan must declare unachievable through o v,ϑ actions with C 0 ( o v,ϑ ) ≥ 0 .

Original: Bound-Sensitive h max
Revised: Bound-Sensitive h max

Original: The admissible classical heuristic h max estimates the cost of a set of facts F as the cost of the most expensive fact (cid:104) v, θ (cid:105) ∈ F , and applies this approximation recursively to action preconditions in order to obtain the cost of the goal (Bonet and Geffner 2001):
Revised: The admissible classical heuristic h max estimates the cost of a set of facts F as the cost of the most expensive fact f ∈ F , and applies this approximation recursively to action preconditions in order to obtain the cost of the goal [ Bonet and Geffner, 2001 ] :

Original: [Equation]
Revised: [Equation]

Original: where h max
Revised: where h max

Original: C denotes the value of h max computed with a cost function C , and achievers ( (cid:104) v, θ (cid:105) , s ) denotes the set of ac- tions o for which [Equation] v, θ
Revised: C denotes the value of h max computed with a cost function C , and achievers ( f, s ) denotes the set of actions o for which f ∈ eﬀ ( o ) . Note that the h max cost of a fact f

Original: [Equation] ∈ eﬀ ( o ) . Note that the h max cost of a fact (cid:104) v, θ (cid:105) that is not present in s is computed by choosing an action o from this set that achieves it with minimum possible cost.
Revised: that is not present in s is computed by choosing an action o from this set that achieves it with minimum possible cost.

Original: Given a set of secondary cost functions and bounds C = {(cid:104) C 1 , B 1 (cid:105) , . . .
Revised: Given a set of secondary cost functions and bounds C = {(cid:104) C 1 , B 1 (cid:105) , . . .

Original: , (cid:104) C n , B n (cid:105)} , a bound-sensitive version of h max can easily be obtained by replacing the set of achievers used to compute h max C 0 with achievers ( 
Revised: , (cid:104) C n , B n (cid:105)} , a bound-sensitive version of h max can easily be obtained by replacing the set of achievers used to compute h max C 0 with achievers ( f,

Original: [Equation]
Revised: [Equation]

Original: v,

Original: [Equation]
Revised: [Equation]

Original: where actions o for which any estimate h max C i ( o, s ) exceeds B i are not considered.
Revised: where actions o for which any estimate h max C i ( o, s ) exceeds B i are not considered.

Original: Note that due to the admissibility of h max , this restriction of the set of achievers is sound but not complete: it is guaranteed that any action removed from the set of achievers cannot be used in a valid plan, but there may be additional actions that cannot be achievers but are not pruned by the heuristic.
Revised: Note that due to the admissibility of h max , this restriction of the set of achievers is sound but not complete: it is guaranteed that any action removed from the set of achievers cannot be used in a valid plan, but there may be additional actions that cannot be achievers but are not pruned by the heuristic.

Original: In general, any admissible estimate h max C i ( o, s ) could be used to compute achievers ( (cid:104) v, θ (cid:105) , s ) C 0 , but we have chosen h max here for simplicity.
Revised: In general, any admissible estimate h max C i ( o, s ) could be used to compute achievers ( f, s ) C 0 , but we have chosen h max here for simplicity.

Original: Theorem 2 Bound-sensitive h max C 0 is an admissible boundsensitive heuristic.
Revised: Theorem 2 Bound-sensitive h max C 0 is an admissible boundsensitive heuristic.

Original: Proof: This follows from the admissibility of the heuristic used to compute achievers ( (cid:104) v, θ (cid:105) , s ) C 0 .
Revised: Proof sketch: This follows from the admissibility of the heuristic used to compute achievers ( f, s ) C 0 .

Original: Bound-Sensitive Merge-and-shrink
Revised: Bound-Sensitive Merge-and-shrink

Original: Merge-and-shrink heuristics are a family of abstraction heuristics that incrementally build a representation of the full state space of a problem (Helmert et al. 2014).
Revised: Merge-and-shrink heuristics are a family of abstraction heuristics that incrementally build a representation of the full state space of a problem [Helmert et al. , 2014].

Original: The construction process begins with the set of transition systems induced over each state variable; at each step, two transition systems are selected to be merged and replaced with their synchronized product.
Revised: The construction process begins with the set of transition systems induced over each state variable; at each step, two transition systems are selected to be merged and replaced with their synchronized product.

Original: Since the transition systems need to be represented explicitly in memory, before the merge a shrinking step is perfomed on the two selected transition systems to enforce a user-speciﬁed threshold on the size of the synchronized product.
Revised: Since the transition systems need to be represented explicitly in memory, before the merge a shrinking step is perfomed on the two selected transition systems to enforce a user-speciﬁed threshold on the size of the synchronized product.

Original: This step is performed by abstracting multiple states in the current representation into a single state (and thereby losing optimality).
Revised: This step is performed by abstracting multiple states in the current representation into a single state (and thereby losing optimality).

Original: The ﬁnal output of the algorithm consists of a single abstract transition system in which multiple states and actions from the original task are mapped to a single state or transition, respectively.
Revised: The ﬁnal output of the algorithm consists of a single abstract transition system in which multiple states and actions from the original task are mapped to a single state or transition, respectively.

Original: h MS ( s ) is then given by the cost of a shortest path from the abstract state representing s to the closest abstract goal state in the ﬁnal transition system.
Revised: h MS ( s ) is then given by the cost of a shortest path from the abstract state representing s to the closest abstract goal state in the ﬁnal transition system.

Original: This estimate is admissible by deﬁnition.
Revised: This estimate is admissible by deﬁnition.

Original: Toadapt merge-and-shrink to the MCF setting, we maintain for each transition in the abstract state space the minimum C i cost for i = 1 , . .
Revised: To adapt merge-and-shrink to the MCF setting, we maintain for each transition in the abstract state space the minimum C i cost for i = 1 , . .

Original: . , n among all of the transitions from the original task represented by that transition.
Revised: . , n among all of the transitions from the original task represented by that transition.

Original: The  C idistance between any two abstract states s, s (cid:48)C i then represents a non-overestimate of the secondary cost of reaching s (cid:48) from s .
Revised: The distance C i between any two abstract states s, s (cid:48) then represents a non-overestimate of the secondary cost of reaching s (cid:48) from s .

Original: A bound-sensitive heuristic value for a state s can be computed as the minimum C 0 cost of a path π from s to an abstract goal state s g whose C i cost in the abstract state space does not exceed B i , for any i .
Revised: A bound-sensitive heuristic value for a state s can be computed as the minimum C 0 cost of a path π from s to an abstract goal state s g whose C i cost in the abstract state space does not exceed B i , for any i .

Original: The C 0 cost of such such a path can be computed with a modiﬁed version of Dijkstra’s algorithm that stores secondary cost information for each node and discards nodes for which C i > B i for any i .
Revised: The C 0 cost of such such a path can be computed with a modiﬁed version of Dijkstra’s algorithm that stores secondary cost information for each node and discards nodes for which C i > B i for any i .

Original: Theorem 3 Bound-sensitive h MS is an admissible boundsensitive heuristic.
Revised: Theorem 3 Bound-sensitive h MS is an admissible boundsensitive heuristic.

Original: Proof: This follows from the fact that the secondary costs used in the abstract state space are the minimums of the secondary costs
Revised: Proof sketch: This follows from the fact that the secondary costs used in the abstract state space are the minimums of the secondary costs

Original: C i of the represented transitions in the original problem, and the proof of admissibility of standard h MS .
Revised: C i of the represented transitions in the original problem, and the proof of admissibility of standard h MS .

Original: While the ms b heuristic can be implemented by running Dijkstra’s algorithm in the abstract state space for each heuristic computation, an important optimization when a single secondary cost function is present (which is the case in the compiled OSP problems that we consider) is to run Dijkstra only once during preprocessing, and compute the primary cost in the presence of different bounds on the secondary cost.
Revised: While the ms b heuristic can be implemented by running Dijkstra’s algorithm in the abstract state space for each heuristic computation, an important optimization when a single secondary cost function is present (which is the case in the compiled OSP problems that we consider) is to run Dijkstra only once during preprocessing, and compute the primary cost in the presence of different bounds on the secondary cost.

Original: This information can then be stored as a sequence of pairs (cid:104)(cid:104) b 0 , c 0 (cid:105) , . . .
Revised: This information can then be stored as a sequence of pairs (cid:104)(cid:104) [Equation]

Original: , (cid:104) b n , c n (cid:105)(cid:105) , where b 0 , . . .
Revised: (cid:104) [Equation] n , c n (cid:105)(cid:105) , where b 0 , . . .

Original: , b n is strictly increasing and c 0 , . . .
Revised: , b n is strictly increas- ing and c 0 , . . .

Original: , c n is strictly decreasing (recall Theorem 1).
Revised: , c n is strictly decreasing (recall Theorem 1).

Original: h MS ( s, b ) is then given by the ﬁrst c i such that b i ≤ b .
Revised: h MS ( s, b ) is then given by the ﬁrst c i such that b i ≤ b .

Original: Experiments
Revised: Experiments

Original: We implemented our approach in the Fast Downward planner (Helmert 2006), and evaluated it on a set of publically  available OSP benchmarks (Katz et al. 2019b).
Revised: We implemented our approach in the Fast Downward planner [Helmert, 2006], and evaluated it on a set of publically [Table] [Figure] available OSP benchmarks [Katz et al. , 2019b].

Original: The set of benchmarks is taken from the International Planning Competitions of recent years, in which goal facts are replaced with utilities, and the bound set at 25%, 50%, 75%, or 100% of the cost of the optimal or best known solution to each problem.
Revised: The set of benchmarks is taken from the International Planning Competitions of recent years, in which goal facts are replaced with utilities, and the bound set at 25%, 50%, 75%, or 100% of the cost of the optimal or best known solution to each problem.

Original: The baseline for our comparison is a blind branchand-bound search, currently the best available conﬁguration for oversubscription planning thatwe know of (Katz et al.
Revised: The baseline for our comparison is a blind branch-and-bound search, currently the best available conﬁguration for oversub scription planning that

Original: 2019a)
Revised: we know of [Katz et al. , 2019a].

Original: We compare this baseline to our suggested approach of A ∗ search on the MCF compilation of the OSP task.
Revised: We compare this baseline to our proposed approach of A ∗ search on the MCF compilation of the OSP task.

Original: Since the compilation introduces intermediate states at which some but not all of the o v,ϑ have been applied, we use a further optimization that avoids generating these nodes and applies all of the o v,ϑ actions in a single step, reducing the state space to that of theoriginal OSP task.
Revised: Since the compila tion introduces intermediate states at which some but not all of the o v,ϑ have been applied, we use a further optimization that avoids generating these nodes and applies all of the o v ϑ actions in a single step, reducing the state space to that of the

Original:  We experiment with blind A ∗ search, and A ∗ using classical h max and h MS , as well as the two heuristics’ bound-sensitive variants introduced here.
Revised: original OSP task. We experiment with blind A ∗ search, and A ∗ using classical h max and h MS , as well as the two heuristics’ bound-sensitive variants introduced here.

Original: For h MS , we used exact bisimulation with an abstract state space threshold of 50000 states andan exact generalized label reduction (Sievers et al. 2014).
Revised: For h MS , we used exact bisimulation with an abstract state space threshold of 50000 states and exact generalized label reduction Siev- ers et al. , 2014] The

Original: The experiments were performed on Intel(R)
Revised: experiments were performed on Intel R

Original: Xeon(R) CPU E78837 @2.67GHz machines, with
Revised: Xeon(R) CPU E7-8837 @2.67GHz machines, with

Original: time 
Revised: time and

Original: and memory limits of 30min and 3.5GB, respectively.
Revised: memory limits of 30min and 3.5GB

Original: Per-
Revised: , respectively.

Original: domain and overall coverage, as well as per- task node expansions for the various conﬁgurations 
Revised: Per-domain and overall coverage, as well as per-task node expansions for the various conﬁgurations and problem suites are shown in Table 1 and Figure 2, respectively.

Original: and
Revised: We

Original: problem suites are shown in Table 1 and Figure 2, respectively.
Revised: now

Original: We now report some observations from our results
Revised: report some observations from our results.

Original: • Blind A ∗ search consistently outperforms blind branchand-bound in terms of coverage.
Revised: • Blind branch-and-bound search usually slightly outperforms blind A ∗ in terms of coverage, except for the 100% suite.

Original: Since we expect blind variants of A ∗ and branch-and-bound to do roughly the 
Revised: The difference between the two may come down to the fact that A ∗ must do extra work in ordering the priority queue, while the variant of branch and bound search that we consider uses no ordering heuristic and can use a simple stack as its search queue.

Original: [Table]
Revised: Alternately it may be due to small differences in implementation.

Original: • Bound-sensitive heuristics are much more informative than their classical variants on OSP problems, sometimes decreasing expansions by orders of magnitude.
Revised: • Bound-sensitive heuristics are much more informative than their classical variants on OSP problems, sometimes decreasing expansions by orders of magnitude.

Original: They also almost always result in better coverage, compared to nonbound-sensitive heuristics.
Revised: Compared to non-bound-sensitive heuristics, they also almost always result in better coverage.

Original: • Blind search dominates informed search in terms of coverage when bounds are low, but the effect diminishes as the bound increases and it becomes intractable to explore the full state space under the bound.
Revised: • Blind search dominates informed search in terms of coverage when bounds are low, but the effect diminishes as the bound increases and it becomes intractable to explore the full state space under the bound.

Original: For the 25% suite of problems, heuristic conﬁgurations solve an average ofinstances fewer than the baseline, compared to approximately 10 instances fewer on the 100% suite.
Revised: For the 25% suite of problems, heuristic conﬁgurations solve an average of approximately 100 instances fewer than the baseline, compared to approximately 15 instances fewer on the 100% suite.

Original: Notably, bound-sensitive h MS has the best coverage in the 100% suite, solving 22 problems more than the baseline (though only 6 more than blind A ∗ ).
Revised: Notably, bound-sensitive h MS has the best coverage in the 100% suite, solving 13 problems more than the baseline, and 6 more than blind A ∗ .

Original: • Coverage on several domains beneﬁts from more informed search schemes.
Revised: • Coverage on several domains beneﬁts from more informed search schemes.

Original: On B LOCKSWORLD  and M I - [Figure] CONIC , bound-sensitive h MS solves the largest number of problems, and this is also the case for bound-sensitive h max on S OKOBAN and P ARC -P RINTER  .
Revised: On B LOCKSWORLD , D RIVERLOG , and M ICONIC , bound-sensitive h MS solves the largest number of problems, and this is also the case for bound-sensitive h max on F LOORTILE , P ARC -P RINTER , and S OKOBAN .

Original: • h MS often times out during construction  and before search has begun, on average in approximately 300 problems per suite, or 1200 problems total.
Revised: • h MS often times out in the construction phase and before search has begun. This occurs on average in approximately 300 problems per suite, or 1200 problems total.

Original: This is especially pronounced in  T IDYBOT , TETRIS , and P IPESWORLDN O T ANKAGE .
Revised: This is especially pronounced in the T IDYBOT , TETRIS , and P IPESWORLD -N O T ANKAGE domains.

Revised: This suggests a hybrid approach that combines the strengths of blind search and h MS : setting an upper bound on the time allotted to heuristic construction, and running blind search instead if construction does not terminate within this bound.

Revised: Using this conﬁguration with a value ofminutes for the upper bound results in a planner that outperforms blind A ∗ by +11 , +16 , +37 , and +38 instances for the 25%, 50%, 75%, and

Revised: 100% suites, respectively.

Original:  h MS schemes that are less expensive  but maintain informativeness in this setting remain a further subject of research.
Revised: This makes h MS schemes that are less expensive to construct but maintain informativeness in this setting an appealing future subject of research.

Original: Conclusions and Future Work
Revised: Conclusions and Future Work

Original: We have shown that a previously introduced compilation to multiple cost function classical planning allows the A ∗ algorithm to be used to solve oversubscription planning problems, and introduced a family of bound-sensitive heuristics that are much more informed than their classical counterparts in this setting.
Revised: We have shown that a previously introduced compilation to multiple cost function classical planning allows the A ∗ algorithm to be used to solve oversubscription planning problems, and introduced a family of bound-sensitive heuristics that are much more informed than their classical counterparts in this setting.

Original: Our experiments show that this approach results in a state-of-the-art method for some bound settings and domains.
Revised: Our experiments show that this approach results in a state-of-the-art method for some bound settings and domains.

Original: One future research direction we would like to explore that builds on the methods introduced here is the use of non-admissible heuristics for satisﬁcing OSP.
Revised: One future research direction we would like to explore that builds on the methods introduced here is the use of non-admissible heuristics for satisﬁcing OSP.

Original: The method by which bound-sensitive h max is obtained is fairly general and should be equally applicable for h add or general relaxed plan heuristics (Keyder and Geffner 2008).
Revised: The method by which bound-sensitive h max is obtained is fairly general and should be equally applicable for h add or general relaxed plan heuristics [Keyder and Geffner, 2008].

Original: A second direction is the use of these heuristics in other planning settings in which tradeoffs must be made between different cost functions, e.g. minimizing fuel use in the presence of bounds on time or vice versa in logistics problems.
Revised: A second direction is the use of these heuristics in other planning settings in which tradeoffs must be made between different cost functions, e.g. minimizing fuel use in the presence of bounds on time or vice versa in logistics problems.

Original: Finally, our methods may be applicable to numeric planning problems in which the variables describe resources that are strictly decreasing and can be expressed in terms of secondary cost functions and associated bounds.
Revised: Finally, our methods may be applicable to numeric planning problems in which the variables describe resources that are strictly decreasing and can be expressed in terms of secondary cost functions and associated bounds.

Original: Boundsensitive heuristics could provide a principled way of reasoning about numeric variables in this context.
Revised: Boundsensitive heuristics could provide a principled way of reasoning about numeric variables in this context.