Abstraction Methods for Resource Allocation Berthe Y. Choueiry, Boi Faltings and Guevara Noubir Arti cial Intelligence Laboratory Industrial Computing Laboratory Swiss Federal Institute of Technology (EPFL) IN-Ecublens, CH-1015 Lausanne, Switzerland E-mail: fchoueiry j faltings j noubir
[email protected] .ch Abstract. The main diculty in resource allocation is the enormous number of tasks and resources which have to be handled. This makes resource allocation a good candidate for proving the usefulness of abstraction methods, which summarize large amount of details and simplify problem solving. In this paper, we give a heuristic for nding groups of interchangeable tasks and resources, and operators for grouping them into common abstractions. We show how these abstractions simplify problem solving to the point where the user can intervene in the solution of the problem. These techniques have been implemented, but this paper is restricted to describing the methodology.
1. Introduction and motivation In many real-world applications, planning , scheduling and resource allocation intervene at di erent time scales and are decoupled (Fig. 1 left). They can be organized in a hierarchy where decisions taken at a given time granularity are fed as constraints to the next level. Often, unforeseen external events create con icts among the various decisions, and reactive strategies are applied to propose amendments and cope with change, Fig. 1. We identify two major motivations for the use of abstractions in this application domain: 1. Reduction of problem complexity. Frequently, the size of an application is so large that enumerative methods for nding a solution or for adapting it to change are computationally too complex. 2. Pertinent support to users. It is desirable to keep the users in the decision loop, since they consider more decision criteria than can be explicitly modeled. However, the amount of detail often overwhelms users, therefore they should be presented with compact and pertinent summaries. 1
Long term
Strategic Planning
Strategic Planning
Amendments
User or Reactive Agent
Making large investment decisions, purchasing airplanes, hiring personnel, etc..
Problem space reformulation Feedback by Concept Formation
Constraints Mid term
Reactive strategies
Scheduling
Conflicts
Scheduling
Setting flight timetables, regular maintenance work, working shifts for personnel, etc.. Problem space reformulation
Resource Allocation
Constraints Short term Resource Allocation Allocation of airplanes, crew, maintenance teams, etc.
Interchan− geability Subproblems sets
Conflicts Amendments
Reduction of problem size Problem description
Figure 1: Left: In an airline company, for example, the tasks of planning, scheduling and resource allocation are decoupled. Right: Introducing abstraction techniques in the classical planning framework.
In this paper, we introduce three abstraction methods for computing such summarized information and extend the classical framework with the components shown in bold in Fig. 1 (right): 1. Discovery of interchangeability sets by sub-problem identi cation structures the solution space of a resource allocation problem in a compact manner and identi es equivalent choices. 2. Reduction of problem size decreases the complexity of problem solving. 3. Concept formation builds compact representations of critical situations. In this paper, we discuss only the resource allocation problem, but we expect that these techniques can be adapted for the other levels of the framework in Fig 1. In resource allocation, execution times for individual tasks have already been set (e.g. by a preceding scheduling phase), and the problem is to assign the available resources Ri to them so that no resource is assigned to more than one task at the same time. Fig. 2 (left). T1 T2 T3
{R1,R3}
{R1,R3} {R1,R3} T5 T7
T1 {R1,R3,R4}
T1 {R1,R3} T2
T7
T3
T6
Time
T6 {R2,R4}
T2
T4
{R1,R2,R3} {R2,R4} T6 {R2,R4}
T4 {R1,R2,R3}
T4
T5
{R1,R3} T3 {R1,R3}
T7 {R2,R4}
{R1,R2,R3} T5
Figure 2: Left: Interval orders: a schedule of seven tasks whose start time and duration are xed. For each task, a set of possible resources is shown. Center: Interval graph. Right: Constraint graph.
This problem can be represented by an intersection graph where a node represents an interval and an arc links two nodes whose intervals overlap in time, (Fig. 2 center). Such 2
a graph is called an interval graph . Resource allocation is a problem of list coloring in interval graphs. Graph coloring of interval graphs is linear in time [22, 19], but list coloring is NP-complete [2]. Resource allocation is an instance of the discrete Constraint Satisfaction Problem (CSP) [28], where all constraints among variables denote mutual exclusion with respect to the values. The nodes of the constraint graph represent tasks to be executed, labels are sets of resources, and arcs link nodes that intersect in time and have at least one resource in common (Fig. 2 right). Note that the constraint graph is a subgraph of the corresponding interval graph. In [11, 12], we introduce a heuristic called the Value-Assignment Delay heuristic (VAD) for decomposing a resource allocation problem into sub-problems. The VAD tries to solve sub-problems independently and isolates the con icts whenever they may occur. In [11], we discuss the tradeo s of using this decomposition scheme. As currently stated, the VAD applies to problems of reusable/non-sharable resources and to discrete CSPs with constraints of mutual exclusion only. It is our goal to extend the idea in the future to deal with more general problems. Comparison to related work in the scheduling and CSP literature as well as some results from a real-world case study can be found in [10, 11, 12]. In this paper, we identify three types of abstractions usable within the framework created by the VAD (Section 2): interchangeability (Section 3), summarization (Section 4.1) and generalization (Section 4.2). By decomposing a problem, the VAD implicitly generates equivalence classes of resources and tasks. We identify these various abstraction classes and characterize them in terms of interchangeability sets [17]. Summarizations are then used to reduce the size of the sub-problems identi ed by the VAD (Section 5). Finally, both summarization and generalization are applied to the con icts isolated by the VAD in a concept formation procedure (Section 6). The generated abstract information is a compact description of the types of missing resources, and serves as feedback to the user or to an automatic reactive scheduler. 1
2. The Value-Assignment Delay heuristic (VAD) The idea of the VAD heuristic as stated in [12] is as follows: \Delay the assignment of the most solicited values in a list coloring problem and try to solve the simpli ed problem without them. Distribute the delayed values to those variables that really cannot do without them." The application of this heuristic to a simple example is shown in Fig. 3. First, the set of values fa; bg is delayed, (i.e. set aside for attribution later in the process). The values whose assignments are delayed are called delayed values. The initial graph breaks up into two components, because N has no value in common and thus no link to the rest of the graph. This process is repeated iteratively, and yields a clustering tree in 2
List coloring di ers from ordinary graph coloring , also called minimal coloring , in that the set of colors allowed for each node is restricted, and may be di erent for each node. In the literature, list coloring is also called restricted coloring [26] and feasible coloring [21]. 1
3
N5 {a,b,g} N1 {a,b,c,d,g} N3 {a.b,g}
N2 {a,b,e,f}
{a,b,g}
Conflict over {a,b}
{a,b}
{a,b} Initial constraint graph
N4 N2 {e,f}
Delay {a,b}
{g}
Solved
Conflict over {g}
{g}
Cluster 2 Cluster 1 N2 {e,f} d= {a,b}
Solved
N1 {c,d,g} d={a,b} N3 {g} d={a,b}
{g} N5 d={a,b} {g} N4 d={a,b}
N1 {c,d}
Delay {g}
N3 { } N4 { } N5 { }
Solved
N3 { } N4 { } N5 { }
Cluster 4 Cluster 3 N1 {c,d} d={a,b,g}
Solved
Leaf node of the clustering tree Trunk−branch of the clustering tree Branching node of the clustering tree
N3 { } d={a,b,g} N4 { } d={a,b,g} N5 { } d={a,b,g}
Claiming delayed values
Figure 3: Left: Applying the Value-Assignment Delay heuristic to a simple resource allocation problem. Center: Clustering tree Right: Isolating con icts in the same problem.
which the leaves represent simpli ed sub-problems, the nodes are sets of delayed values and the branches show how these components relate (Fig. 3 center). Some of the subproblems can be easily solved in isolation without any e ect on the rest of the problem. For example, nodes in Cluster and Cluster can be assigned any element in fe; f g and fc; dg respectively. Such commitments are safe and never need to be undone. All the other parts need to claim delayed resources for their solution. A con ict is de ned around each value set claimed by several nodes in the tree leaves. Con icts may be resolved according to some heuristic and domain dependent preferences or by interactive users' intervention. Users may decide to relax constraints and \borrow" any of the values which remain unassigned in the leaves. The tree generated by the heuristic de nes a useful structure for interactive decision making. 1
3
3. Discovering interchangeability In [17], Freuder introduces the concept of interchangeability and de nes various kinds of value interchangeability. In this section, we quickly recall the de nitions of interest to us then illustrate how the decomposition of a list coloring problem by the VAD heuristic implicitly nds sets of interchangeable values and variables. The notation (i; j ) satisfies C means that the two values i and j for two variables linked by a constraint C are consistent with respect to this constraint.
De nition 1 Full interchangeability: A value b for a CSP variable V is fully inter-
changeable with a value c for V i every solution to the CSP which contains b remains a solution when c is substituted for b in V and vice versa.
4
De nition 2 Neighborhood interchangeability: A value b for a CSP variable V is neighborhood interchangeable with a value c for V i for every constraint C on V :
fi j (b; i) satis es Cg = fi j (c; i) satis es Cg
De nition 3 Substitutability: Given two values b and c for a CSP variable V , b is
substitutable for c i substituting b in any solution involving c yields another solution.
De nition 4 Neighborhood Substitutability: For two values b and c, for a CSP variable V , b is neighborhood substitutable for c i for every constraint C on V :
fi j (b; i) satis es Cg fi j (c; i) satis es Cg
De nition 5 Partial Interchangeability: Two values are partially interchangeable with respect to a subset S of variables, i any solution involving one implies a solution involving the other with possibly di erent values for S . 2
In [17], Freuder argues that interchangeable values are redundant and their removal simpli es the problem space. He shows that neighborhood interchangeability is a stronger condition than full interchangeability (i.e., not all full interchangeable sets are neighborhood interchangeable), but is easier to compute. Substitutability is a `oneway' interchangeability and \partial interchangeability captures the idea that values for variables may di er among themselves, but be fully interchangeable with respect to the world."
Interchangeability found by the VAD
We use a simple example taken from a case study to illustrate the various kinds of interchangeability discovered during decomposition by the VAD heuristic. The problem domain is the allocation of operating rooms and technicians to hospital operations. In Fig. 4, we show the tree structure generated by application of the VAD heuristic and the resources delayed at the branching node. For each leaf cluster, we display the tasks (surgical operations) but not the links between the tasks as in a constraint graph. The proposition of a surgical operation Opx to be allocated a resource of type garcon has the following format (garcon Opx missing). The set of resources following the proposition is the set of values that have not been delayed and are still directly available. In this example, the set fHenri, Erikag was delayed and the initial problem broke down into eight leaves. Fig. 5 shows the interchangeability relations discovered in one of the leaf clusters of Fig. 4. For a variable Vi in a leaf cluster, fvig denotes the set of values still available for Vi . For instance, for variable Op in cluster C in Fig. 4, fvig =fMobwete, Fernando, Giorgio, Alberto, Rolf, Martineg. We claim that: 10
2
In graph coloring, switching [25] of colors is often used to generate a coloration (a solution) from another one. This operation can be thought of as a special case of partial interchangeability where is the whole graph. 2
S
5
Figure 4: A simple resource allocation problem after decomposition by the VAD heuristic.
\
Lemma 1 For every variable Vi in a leaf cluster C , all values in the set fvj g are Vj 2C partially interchangeable with respect to S = fVj 2 C g. \ Proof: Let T = fvj g. Since the values in T have not been delayed, they are not Vj 2C
needed by any other variable in the problem, within the time window of the cluster C , except the ones in C . Thus no con ict with any other variable outside C is possible, and any decision about the assignment of elements of T can only a ect variables in C . Example from Fig. 5: In C-2, fMobwete, Fernando, Giorgio, Albertog are partially interchangeable for fOp , Op , Op g. 10
15
3
For lemmas 2 and 3, let Ui denote the set of values uniquely available to variable Vi [ in cluster C . Ui is given by the formula Ui = fvig ? fvj g. Vj 2C;j 6=i
Lemma every variable Vi in a leaf cluster C , all values in the set Ui = fvig ? [ fv2g For are neighborhood interchangeable. j Vj 2C;j 6=i
Proof: Ui may not create any con ict outside C for the same reason as in the proof of
Lemma 1. Within C , all values in Ui are possible only for Vi, by construction. Thus no con ict is possible with any other variable fVj 2 C; j 6= ig. Example from Fig. 5: In C-2, fRolf, Martineg are neighborhood interchangeable for Op , fMurat, Arthurg are neighborhood interchangeable for Op . 10
3
Lemma 3 For every variable Vi in a leaf cluster C , each value in Ui = fvig?
[ fv g j
Vj 2C;j 6=i
is neighborhood substitutable for every other value in the label of Vi. In particular, each 6
Lemma 3: N−Sub for every s in S3 If (Op10) claims (S3)
Lemma 2: N−Int for (Op10)
Lemma 1: P−Int for (S1)
C−2
S3
Op10
{ Mobwete Fernando Giorgio Alberto
Op15
{ Mobwete Fernando Giorgio Alberto }
Op3
{ Mobwete Fernando Giorgio Alberto
Rolf Martine
}
Murat Arthur
}
S1
{ Henri Erika }
C−1
C−8 Lemma 4: P−Int for (S2) if claimed by every Opx in S2
Lemma 3: N−Sub for (Op 3)
Op9 Op2.. Op25 Op26
Lemma 2: N−Int for (Op3)
N−Int = Neighborhood Interchangeable N−Sub = Neighborhood Substitutable P−Int = Partially Interchangeable
S2
Figure 5: Illustration of various types of interchangeabilities discovered by the VAD heuristic.
\
value in Ui is neighborhood substitutable for every value in fvj g and for every value Vj 2C in every delayed set that Vi may claim.
Proof: Since no value in Ui was delayed, this means that Ui is needed only for Vi during
Vi and assigning any element of Ui to Vi rules out no choices for the rest of the problem. Therefore, the set of values for nodes adjacent to Vi , for the case where Vi is assigned an element of Ui, will be a superset of that for the case where it is assigned any other value in its label. Example from Fig. 5: In C-2, each of fRolf, Martineg are neighborhood substitutable for each of fHenri, Erika, Mobwete, Fernando, Giorgio, Albertog for Op . 10
Lemma 4 All values in a given set of delayed values D are partially interchangeable with respect to the set S of variables located downstream that claim D.
Proof: The set of delayed values is choosen at each step as the set of values simultaneously claimed by a set of variables [11]. Thus, if a variable claims any element of the set of delayed values, it also claims all the others, by construction. Example from Fig. 4 and 5: fHenri, Erikag are partially interchangeable for all the nodes in the leaf clusters C : : : to C since they are admissible for all these nodes (although this information is not explicitly shown). 1
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In [17], Freuder suggests that interchangeability can also be computed dynamically during problem resolution, (i.e. dynamic interchangeability). An example of this is shown in Fig. 6. Suppose that a con ict resolution procedure (or the user), decides to assign b to Task K . The delayed value fag is now fully interchangeable with fe; f g for variable Task L. 7
Task K Task L
a
{a, b, c} Task M
{a, b, e, f}
b
{a }
{a, b, c} Task N
Task K {c} Task N {c}
a Task M
Task M
Task L{e,f} Two non−adjacent nodes (allies)
Task K {b} Task L{e,f} Task N {c}
Two non−adjacent nodes (allies)
Figure 6: Full interchangeability discovered dynamically.
Lemma 5 Any delayed value claimed only by a set N of non-adjacent nodes is fully interchangeable with all fvig for each Vi 2 N . 3
Proof: Since the values in the delayed set are claimed only by non-adjacent nodes,
whether they are assigned to two or more nodes simultaneously cannot a ect the problem in any way. In particular, each of them is neighborhood substitutable for all fvig in each Vi. Since all fvig can be proven neighborhood substitutable for each of these values using Lemma 3, they are fully interchangeable.
The advantages of nding interchangeable sets
In Fig. 4, it is easy to see that the numerous solutions to this example form compact families of partial solutions: many solutions can be generated simply by switching partially interchangeable values for tasks in the same leaf. This switching operation a ects only the nodes in the leaf, while keeping the rest of the `world' unchanged. Classical enumerative methods fail to organize the solution space in such a compact manner and thus fail to identify the boundaries within which the e ect of a change remains local. In interactive problem solving , interchangeable sets help the human decision maker to view alternative choices and participate in the solution process with the possibility of integrating domain dependent knowledge and unquanti able criteria. In search , the main advantages of discovering interchangeability sets are (1) compacting the solution space representation by grouping families of solutions which are equivalent, thus speeding up the search process (2) supporting backtracking [24], and (3) reducing the search e ort by limiting it to one representative solution. In concept formation , interchangeability identi es groups of objects as the basic components for abstraction, see Section 4. According to the criterion used for selecting the values delayed at each step of the clustering procedure introduced in [11], the VAD may decompose a problem in di erent ways. Whether the VAD discovers all maximal sets of interchangeable values is still an open question: the sets discovered by the VAD may be only subsets of the largest possible ones, and some possible sets may simply be missed. Two non-adjacent variables are called allies since, when they claim the same delayed set, they may be assigned values from this set simultaneously. 3
8
We have tested our algorithm on data from a hospital case study and compared the results to those obtained by the algorithm for computing neighborhood interchangeability proposed by Freuder in [17]. No set of neighborhood interchangeable values was missed. The Freuder algorithm is applicable to all types of constraints, however, it only nds neighborhood interchangeabilities (and a generalized version nds k-interchangeability). The VAD heuristic is only applicable to constraints of mutual exclusion; but in addition to the neighborhood interchangeable sets, it also determines other types of interchangeability, for which Freuder provides no algorithm. In the case of partial interchangeability, the VAD also determines the sets of nodes with respect to which the values are interchangeable. In many practical examples, we noticed that the occurrence of neighborhood interchangeability are rather rare and the other types of interchangeability de ned above occur very often . We still have to prove theoretically whether or not the discovered sets are maximal and test these methods on randomly generated examples. This task is currently in progress. 4
5
4. Abstraction operators Abstraction is a promising technique to reduce the complexity of problem solving in many domains. Abstraction can be de ned as follows, see also [18]: \Abstraction is the mapping of a problem representation into a simpler one that satis es some desirable properties in order to reduce the complexity of reasoning. The problem is solved in a simpli ed abstract space and the solution is then mapped back to the more complex ground space." Concrete Space Pre−image
Abstract Space
Concrete Space Domain
Image Domain
Abstract Space Co−Domain Image
TD−Abstractions preserve downward solution property
TI−Abstractions preserve upward solution property
Figure 7: Illustration of Theorem-Decreasing and Theorem-Increasing Abstractions.
The properties of abstraction techniques are presented and formalized in [32, 18]. Fig. 7 illustrates two main properties: (1) The downward solution property guarantees that a solution found in the abstract space can be mapped back to a feasible solution In terms of CSP measures, the tests show the following order parameters [3, 24]: h i= h22 72, 5 31, 0 037 0 108, 0 376 0 570i. is the number of variables, the maximumdomain tuplesj e?emin size, = jforbidden jall tuplesj the constraint tightness, = emax ?emin the constraint density, the number of constraints, min = ( ? 1), and max = n(n2?1) . 5 The possibility of discovering interchangeable sets increases in structured domains with some semantics. 4
n; a; t; d
:::
:::
:
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:
:
:::
:
n
d
t
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in the ground space. Theorem-Decreasing abstractions satisfy this property. (2) The upward solution property guarantees that any solution in the ground space is preserved in the abstract space. Theorem-Increasing abstractions satisfy this property. In our approach, abstractions operate on temporal information which is represented in terms of time tokens [13]. Time tokens are structures associating a proposition to a time interval and abstractions are themselves temporal tokens of higher level of granularity. We propose two operators for building temporal abstractions: summarization and generalization . These operators serve to construct concise descriptions of the interchangeable sets found in Section 3, see Sections 5 and 6. 4.1. Summarization
Summarization is the replacement of a collection of time tokens by a single one. The interval of the summarization is the smallest interval that encompasses all intervals of the component tokens. The property of the summarization is a combination of the properties of the constituent tokens, see Fig. 8. (location (op1 op2 op3) Room−1)
TMS−dependency links (location op1 Room−1)
(location op2 Room−1)
(location op3 Room−1)
Figure 8: Summarization of temporal intervals by interval union. In the domain of allocating operation rooms to surgical operations, operation-1 takes place in room-1 followed by operations 2 and 3 in the same room. In summary, all these operations are taking place in room-1 over the entire time interval.
Summarizations are TD-abstractions: after summarization, the abstract space comprises a subset of the initial set of possible solutions and any solution found in the abstract space can be safely mapped back to a concrete solution. In the example of Fig. 8, the tasks have been grouped so that they will be assigned the same resource, for instance a surgeon. Alternative solutions, for example assigning a di erent surgeon to Op and Op , have been deliberately ruled out and dropped from the solution space. 2
3
4.2. Generalization
Generalization does not a ect the temporal extension of a time token, instead, it operates only on its propositional expression or property. It provides a way to replace a disjunction of terms within one proposition by a single more general term. Generalization is based on domain dependent background knowledge in the form of a concept hierarchy (Fig. 9). It corresponds to using the climbing the generalization tree structure heuristic described by Dietterich and Michalski [14]. Generalizations can be characterized as TI-abstractions: they suppress the detailed descriptions of speci c entities by mapping them to more generalized categories. Thus decisions made on the basis of the general categories alone might violate constraints that refer to the speci c entities. 10
(Instrument−tech Hospital−Section−1 missing) Hospital−Section1
Hospital−Section2 TMS−dependency
urology
pediatrics
(Instrument−tech (cardiology urology) missing)
cardiology
TMS−dependency op2
op3
op1
op14
op20
(Instrument−tech (op1 op2 op3) missing)
Figure 9: Generalization using structured background knowledge.
5. Computing maximal summarizations Each leaf cluster identi ed by the VAD is a set of interchangeable objects, Fig. 10. Summarizations are applied to the leaf clusters in order to reduce their size. A leaf cluster \contains" a collection of intervals such as those shown in Fig. 11 (left). Groups of intervals which do not overlap are identi ed and replaced by a unique summarized interval. {Resources}
{ Tasks + Resources} Time
Figure 10: Summarization of the tasks in the leaves reduces the size of sub-problems.
The rst step is to arrange the time tokens, by topological sorting, in a directed graph re ecting their temporal succession (Fig. 11 center). Nodes denote time tokens, links are built between nodes that do not overlap, and arrows re ect their precedence order. Every path in this graph represents a possible sequence of tasks that can be carried out by one and the same resource, and thus represents a possibly useful summarization. 6
(Task A) (Task B)
(Task C) Task A (Task D)
Task D (Task F)
Task B
Task C
Task F
Task G
Task B
Task C
Task D
Task E
(Task E) Task E
(Task G)
Time
Task F
Task G
Task A
Figure 11: Left: Token representation of tasks located in one cluster. Center: Comparability graph. Right: Same graph, after elimination of arcs of transitive precedence.
Obviously, accepting large gaps between successive tasks may be wasting resource capacity. The user may control the maximum interval between two successive tasks and 6
This is a directed acyclic graph and is called a comparability graph.
11
Algorithm
Greedy-1 [22] Greedy-min Greedy-max Minimal-change
Optimization Criterion Complexity Description None Min. minimal distance Max. minimal distance Min. change of location
( ) ( ) ( )
O n O n
O n
(
O k:n
2
)
Appendix A.1 Appendix A.2 Appendix A.3 Appendix A.4
Table 1: Four algorithms for building compact summarizations. The complexity measure assumes that
intervals are given by the interval orders and sorted in ascending value order of their endpoints. the number of tasks and k is the number of geographical locations of these tasks (k n).
n
is
the arcs that exceed this interval are omitted from the graph. Competing summarizations are apparent in the graph as di erent paths between the same nodes. Those that arise as a result of transitive precedences are eliminated to simplify the graph . If the users have some unquanti able criteria (subjective preferences) for grouping tasks, they can be presented with the graph of Fig. 11 (right) so they can interactively select the most suitable summarizations. In the appendices, we describe four algorithms for selecting the most compact summarizations, see Table 1. The goal is to replace the set of intervals by the set of summarized intervals of minimum cardinality. This can be done by coloring the intervals with the minimum number of colors then summarizing those intervals with the same color. Each of these algorithms, in addition to selecting the most compact summarizations, optimize some criterion. 7
6. Using abstractions for concept formation Fig. 12 shows an example of applying the summarization and generalization operators to a collection of intervals found in a leaf cluster. The atomic data at the bottom of this gure indicate that Tech-nicians were not found for a collection of surgical operations, denoted by Op , Op , Op , etc.. At the top of the same gure, the user is informed that technicians are missing for the traumatology and pediatrics units. Each level in the generated hierarchy de nes new concepts useful for solving this particular problem. 4
8
9
In Fig. 3, we showed how con ict sets are identi ed in a resource allocation problem. The concept formation procedure is applied to the tasks and resources contained in a con ict set like the one shown in Fig. 13. This procedure builds a general and concise description of a con ict from speci c information found in its raw representation. In this way, the abstraction process provides compact information useful for assessing problem tightness and for identifying types of underutilized resources. A similar process can be carried out at any level of the planning framework in Fig. 1. The new information generated is valuable for justifying and supporting strategic decisions such as hiring new personnel and making important investments in equipment. Eliminating transitive links is the problem of nding the minimum equivalent graph (MEG). For directed acyclic graphs, MEG is equivalent to transitive reduction. This operation can be done in less than ( 3 ) [1]. 7
O n
12
(Tech (traumatologie pediatrie) missing) Summarization (Tech traumatologie missing) (Tech pediatrie missing) Summarization (Tech traumatologie missing) (Tech traumatologie missing)
(Tech pediatrie missing)
(Tech pediatrie missing)
(Tech traumatologie missing)
(Tech pediatrie missing)
Generalization (Tech osteo missing) (Tech AMO missing)
(Tech strabisme missing)
(Tech hypospadias missing)
(Tech cicatrice missing)
(Tech orchidopexie missing)
Generalization (Tech op4 missing) (Tech op8 missing)
(Tech op34 missing)
Tasks in a leaf cluster
(Tech op37 missing)
(Tech op9 missing)
(Tech op38 missing)
Figure 12: Building abstractions by application of summarization and generalization operators.
We are currently working on an algorithm for interleaving these abstraction operators. It is clear that in order to generate useful abstractions, such a process should be user initiated and goal oriented . It can be con ned to leaf clusters delimited by the VAD or may encompass a given time window. We are currently working on the following unresolved issues: (1) integration of quantitative information within the generated abstractions, (2) manipulation of multiple concept hierarchies, and (3) use of constructive abstraction operators.
7. Related work Classical graph reduction methods by contraction used in graph coloring [4] can be understood as clustering of variables according to some similarity measures [33, 6, 29, 30, 5, 8]. These methods simplify the problem, but are not concerned with isolating and structuring con icts. In many real-world applications, the identi cation and analysis of con icts is the real issue. In [15, 16], Ellman proposes two heuristics to cluster either variables or values as an abstraction approach to solve CSPs. The user has to select, at compile time, the number of partitions (size of the abstract space) as well as the one feature function which guides the clustering. Ellman shows that the performance of the proposed method depends signi cantly on the prede ned number of partitions, a crucial parameter. The VAD heuristic does not su er from this shortcoming since the number of task clusters (variables) and resource pools (values) is determined dynamically during the clustering process. The feature function in our case is whether or not a resource is suitable for a 13
Conflict
{Resources}
{Resources} {Resources}
Figure 13: Concept formation is applied to con ict sets.
given task independently of the various attributes of the resource. In [27], Lansky discusses how activities and events in planning problems can be \clustered into groups|regions whose boundaries impose barriers on various forms of domain interaction", and how such a partitioning localizes domain properties. We think that this idea of localization is very close to the spirit of our approach. Our method determines these localizations and interactions dynamically at run time whereas in [27] they depend on the skill of the domain expert. The use of the temporal abstraction methods proposed in Section 4 can be extended to general reasoning about physical systems, such as diagnosis, as we suggested in [9]. In an independently developed work [31], Shahar et al. propose three di erent kinds of temporal abstractions, namely point temporal abstraction, temporal inference, and temporal interpolation, which are very close to the essence of temporal summarization and generalization. This fact shows the adequacy of the use of these methods to reason about applications as di erent as patient monitoring and diagnosis.
8. Future research and conclusions Abstractions have been formalized and thoroughly studied in the literature. We have identi ed resource allocation as a real-world application where abstractions are useful. In this paper, we showed how a decomposition heuristic for solving resource allocation problems discovers sets of interchangeable objects. We then argued that these sets can be used to support the user in visualizing opportunities and alternative solutions, and can be exploited to eciently reduce search e ort and backtracking. We also introduced two abstraction operators which can be applied to those sets to yield the formation of pertinent and new information. Importantly, for many practical applications, the abstractions can explain con icts and propose resources to be added to solve them. There are, of course, many open questions to be investigated further: (1) studying the criterion for the selection of the delayed value set and characterizing it with respect to discovering the maximal sets of interchangeable objects, (2) formalizing the use of interchangeable sets for solving con icts in a resource allocation problem, (3) proposing 14
and evaluating algorithms for building useful abstractions, and (4) generalizing these methods to other types of CSPs. Acknowledgments This research is supported by the Swiss National Science Foundation Project#5003034284. The authors are indebted to the following persons for insightful discussions: Dean Allemang, Sailesh Chutani, Ronen Feldman, Erika Foelling, Martin C. Golumbic, Alain Hertz, Simon Leinen, and Beat Liver.
Appendices A. Optimization criteria The following algorithms color a collection of interval orders with the minimum number of colors and thus nd maximal summarizations. Many such summarizations exist and we have identi ed two criteria which are of practical interest to discriminate among them: the distance (Appendices A.2 and A.3) and the change of an attribute value (Appendix A.4) between two tasks in a summarization.
De nition 6 The minimal distance is de ned as the minimum distance between a
begin point bj of an interval Ij and end point ei of another interval Ii such that Ii precedes Ij , Ii and Ij do not overlap, and are given the same color C . d = minf bj ? ei j C (Ii) = C (Ij )g (1) A.1. Interval orders coloring in linear time
Optimization criterion: none. Algorithm Greedy-1: [22] Create a list P of points pi by taking the two endpoints of 8
each interval. I (pi) is the interval for which pi is an endpoint. Sort the list P of points in ascending order of their values. Use a greedy algorithm to color the intervals. Algorithm Greedy-1 ( ) Let Heap := nil Repeat for every i in If i is a begin point Then If Heap is empty Then generate a new color and allocate it to ( i ) Else remove any color from the Heap and allocate it to ( i) Else put the color allocated to ( i ) in the Heap. 9
P
p
P
p
I p
I p
I p
Proof: See [22]. Complexity: The sorting can be done in O(n log n) but it is often considered as a
preprocessing step. Greedy-1 is clearly linear in time in the number of points and thus in the number of intervals.
The start time of an interval is called left endpoint and the end time is called the right endpoint. Sorting intervals according to their endpoints can obviously be done in ( log ). 9 The greedy algorithm is also called on-line or rst- t algorithm. 8
O n
15
n
A.2. Minimizing minimal distance
Optimization criterion: In some applications, it is preferable to choose the maxi-
mal summarizations which minimize the minimum distance between two intervals in a summarization. For instance, in a manufacturing environment, it is desirable to leave machines idle as little as possible. In an airline company, it is desirable to keep the aircrafts stationary as little as possible. Algorithm Greedy-min: In order to achieve this goal, it is sucient to manage the heap of released colors in a LIFO manner [20]: when a color is released, push it on top of the stack; when a color is needed, pop the one from the top of the stack. Proof: Let doptimal = minfbj ? ei j bj eig = bj ? ei (2) where bk and ek are the left and right endpoints of an interval Ik . We are going to show that the algorithm Greedy-min allocates the same color to Ij and Ii . Note that: 0
0
10
0
0
Lemma 6 For any coloring algorithm A, dA dopt. The rst interval colored after Ij must Ii . Suppose this is not the case. This implies the existence of an interval Ik such that ei < bk < bj . This implies that bk ? ei < bj ? ei . This is impossible because of Equation 2. Thus, the rst interval considered after Ii is Ij . In the same way, we can prove that no interval can terminate after Ii does and before Ij starts. After coloring Ii , Greedy-min considers Ij and gives it the color just released by Ii . Ii and Ij are allocated the same color and the resulting summarizations minimize the minimal distance.Thus, dgreedy?min = dopt. 2 Complexity: It is obvious that the algorithm remains linear in time in the number of intervals if the access to the stack is assumed constant. 0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
A.3. Maximizing minimal distance
Optimization criterion: In some applications, it is preferable to choose summariza-
tions that leave as much time as possible between two jobs to be given the same resource. For instance, this would give some time for the resource to rest (human), to be cleaned (machine), to recover lower temperature (tool), etc. Algorithm Greedy-max: In order to achieve this goal, manage the heap of colors in a FIFO-manner [20]: when a color is released, put it at the bottom of the heap; when a color is needed get the one on the top of the heap. Note that the algorithm starts with all the necessary colors already in the heap. Proof: We claim that the algorithm Greedy-max maximizes the minimum distance as de ned in Equation 1. Let Ii and Ij be two intervals satisfying Equation 1 and dIi0 ;Ij0 = ej ? bi . We claim that the algorithm Greedy-max is such that dIi0 ;Ij0 is maximal. We prove that no other method A does better: dA dgreedy?max . Let Cd 0
0
10
0
0
In case many such interval j 0 and i0 exist, pick anyone. I
I
16
be the number of colors in the heap just before the C (Ii ) is \released" and placed at the bottom of the heap, Cd = Maximum contention ? contention between ej and bi . Let S = fIk j ei bk bj g. Since all Ik 2 S will be allocated colors from the top of the heap, we have jS j = Cd: jS j cannot be less than Cd because that would imply that Ii and Ij are assigned di erent colors, and jS j cannot be more than Cd because that would mean two intervals Im; In 2 S are assigned the same color, which would imply that dIm ;In satis es Equation 1 and dIm ;In < dIi0 ;Ij0 , which is impossible. Any other algorithm with dIi1 ;Ij1 > dIi0 ;Ij0 would need to color S [ fIj g with the Cd colors initially available (without using any of the colors \released" between ei and bj , otherwise dIm ;In < dIi0 ;Ij0 ). This is impossible because jS j = Cd. 2 Complexity: as in Appendix A.2. 0
0
0
11
0
0
0
0
0
0
0
A.4. Minimizing change according to geographical location
Optimization criterion: In some applications, summarizations that minimize a criterion of change are preferred. In a hospital's operating unit, once operations have been assigned an operating room, personnel must be chosen to support surgeons during the operations. In this situation, it is preferable to keep the personnel as much as possible in the same operating room for psychological reasons (people do not like to be moved around constantly). The goal here is to choose those maximal summarizations that minimize the change of the geographical location. We propose an algorithm that nds the optimal solution in polynomial time. This algorithm is described for the case of minimizing the change of geographical location but can be extended to accommodate other criteria of change in a straightforward manner. Algorithm Minimal-change: In order to obtain maximal summarizations with minimal change in the geographical location between two jobs in the same summarization, we propose again to manipulate the heap. Any color returned to the heap is labeled by the name of the room in which the interval it served last takes place. When a new interval is considered for coloring, it is assigned the color from the heap which was last allocated to the same location. In case no such color exists, the algorithm tries to assign a color that will not be claimed by any room later in the problem. The details can be found in procedure Find a color . Algorithm Minimal-change ( ) Let Heap := nil and Contention := Maximum contention ( ) Repeat for every i in If i is a begin point Then If There is color in the Heap labeled with the operating room of ( i ) Then P
P
p
P
p
Select this color for ( i ) and remove it from the Heap ElseIf Number of generated colors Contention Then Generate a new color and assign it to ( i ) Else Find a color for ( i ) from the Heap and assign it to ( i ) Else put the color assigned to ( i ) in the Heap and label it with the name of the operating room of ( i ).
I p
I p
b
R
R
R
>
I
R
I
I
R
Procedure Maximum contention (P) Let Max := 0 and Contention := 0 Repeat for every k in f j g If k is a begin point Then p
p
p
increment Contention by one and
When Contention Max Then Max := Contention Else decrement Contention by one Return Max. >
Proof: We prove that Minimal-change is at least as good as any other algorithm, Better-method, for minimizing the number of changes. Let B = f bk j bk is the left endpoint of Ik g. B is sorted in ascending value order. Let Rk be the operating room where Ik takes place. Let Change(method; bm) be the total number of changes that a coloring algorithm method has done until time bm, including bm. Suppose that for a time point bi, Change(Better-method; bi? ) Change(Minimal-change; bi? ). At time bi, three situations are possible: 1
1
1. No color was ever assigned to Ri ) Change(Better-method; bi) Change(Better-method; b i? ) + 1 and Change(Minimal-change; bi) Change(Minimal-change; b i? ) + 1 ) Change(Better-method; bi) Change(Minimal-change; bi) (
1) (
1)
2. Ri had a color associated with it which was conceded to an interval Ic such that bc < bi. Thus, at bc, we selected for Ic the color associated with Ri because it is the color associated with the room of the interval that appeared last in the chronological order, see procedure Find a color. This means that Better-method has given to Ic a color conceded from another interval which starts between bc and bi. Thus, if Change(Minimal-change; bi) increases by one at bi, Change(Better-method; bc < bk < bi) must have increased at least once before bi. Thus, Change(Better-method; bj ) Change(Minimal-change; bj ). 3. At bi there is a color in the Heap associated with Ri. The change function need not be increased and Change(Better-method; bj ) Change(Minimal-change; bj ).
2
Complexity: The complexity of Minimal-change is in O(k:n ), where k is the number of rooms (k n). Sorting the interval endpoints can be done in O(n log n). The 2
complexity of the computation of the maximal contention can be done in O(n). The complexity of the procedure Find a color is O(k:n). 18
B. Other optimization criteria In those cases where the number of resources is limited (inferior to the maximal contention), some intervals should be dropped and excluded from summarizations. The problem becomes that of k-coloring in interval graphs addressed by [2, 7, 23]. The goal is to maximize the value of intervals taken in summarizations when each interval has a weight (for instance, in practical applications, this weight is the importance of a job). Hartman, in [23] shows that this can be solved in polynomial time, Arkin et al. in [2] show it is in O(n log n) for all k < maximal contention and Carlisle et al. in [7] show it is in O(kn log n). 2
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