Third ILOG International Users Meeting, 9 & 10 July 1997, Paris, France 1. Michel Lema^tre, Gerard Verfaillie ONERA/CERT

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1 Third ILOG International Users Meeting, 9 & 10 July 1997, Paris, France 1 Daily management of an earth observation satellite : comparison of ILOG Solver with dedicated algorithms for Valued Constraint Satisfaction Problems Michel Lema^tre, Gerard Verfaillie ONERA/CERT 2, avenue Edouard Belin { BP 4025 { Toulouse cedex 4 { France fmichel.lemaitre,gerard.verfaillieg@cert.fr Abstract The daily management of an earth observation satellite like Spot is a dicult combinatorial optimization problem. A simplied form of this problem can be stated as follows : given (1) a set of candidate photographs for the next day, each one associated with a weight reecting its importance, (2) a set of imperative constraints expressing physical limitations (non overlapping of photographs, respect of camera transition times, limitation of the instantaneous data ow), select a subset of candidates which meets all the constraints and which maximizes the sum of the weights of the selected candidates. This problem can be stated as a Valued Constraint Satisfaction Problem, a general framework for which our team has developed a powerful set of algorithms. This paper reports an experiment the aim of which was (1) to evaluate the ease with which this problem can be expressed with ILOG Solver, and (2) to compare the resolution performances obtained with ILOG Solver and with dedicated VCSP complete algorithms. The main conclusions are (1) a relative ease of expression of this problem with ILOG classes, in spite of a lack for expressing constraints given in extension, and (2) a relatively good performance of ILOG Solver when compared to classical VCSP algorithms. 1 Introduction Earth Observation Satellite Scheduling Problems (EOSSPs) consists in selecting, among a set of desired photographs, a maximal subset which satises imperative constraints expressing inescapable physical limitations. In the form that will be set more precisely below, this problem is NP-complete, which means that probably no algorithm can solve all EOSSP instances in polynomial time. In actual practice, computation times tend to grow exponentially with the size of the instances to be solved. ILOG Solver is a powerful tool for constraint programming, packaged as a C++ library. It provides a set of predened generic constraints and so-called

2 meta-constraints, widely open to extensions. It allows a clear separation of problem statement from problem resolution. Valued Constraint Satisfaction Problem formulation (VCSP) is a general framework which captures a large number of discrete constrained optimization problems, and for which we have developed a library of complete and approximate algorithms. EOSSPs can be casted in this general framework. The aim of the work reported in this paper is twofold : 1. to evaluate the facility of expressing EOSSPs with commercial available tools like ILOG Solver 2. to check the performance level of our algorithms, by comparison with a standard tool in discrete optimization. We choose ILOG Solver as a yardstick. This paper is organized as follows. The rst section sets the daily management EOSSP. The second one describes the VCSP framework, explains how EOSSPs can be casted into this framework, and succinctly describes our library of algorithms. The third section explains the modeling of EOSSPs using Solver. The next section presents the compared performances, and the last one concludes. 2 Earth Observation Satellite Scheduling Problems Earth Observation Satellite Scheduling Problems and, more precisely, the daily management problem for an earth observation satellite like the future satellite Spot5, is described as follows [1] : data we are given a set S of photographs which could be taken the next day from at least one of the instruments, with respect to the satellite trajectory for each photograph, a weight expressing its importance (aggregation of several criteria like the client importance, the demand urgency, the meteorological forecasts...) for each photograph, a set of dierent ways of taking it : up to three for a mono photograph (because of the three instruments on the satellite) and only one for a stereo (because such photographs require two trials : one with the front instrument and one with the rear one) a set of imperative constraints : non overlapping and minimal transition time between two successive photographs on the same instrument, limitation on the instantaneous data ow through the satellite telemetry for simultaneous photographs on dierent instruments, 2

3 question nd an admissible subset S 0 of S (imperative constraints met) which maximizes the sum of the weights of the photographs in S 0. 3 Valued Constraint Satisfaction Problems General Framework A Constraint Satisfaction Problem (CSP) instance is dened by a triple (X; D; C), where X = fx 1 ; : : : ; x n g is a set of variables, D = fd 1 ; : : :; d n g is a set of nite domains for the variables, and C is a set of constraints. A constraint c = (X c ; R c ) is dened by a subset of variables X c X on which it holds, and a subset R c Q x i 2X c d i of allowed tuples of values. A solution of an instance is an assignment of values to all the variables which satises all the constraints. Many CSP instances are so constrained that no solution exists. In this case, one can search for a solution minimizing the number of unsatised constraints. This is the partial CSP model, introduced by [2]. This model can be further generalized by giving a weight or a valuation to each constraint, mirroring the importance one gives to its satisfaction, and by considering aggregation operators other than the sum operator. We then search for a solution minimizing an aggregation of the valuations of the unsatised constraints. This extension of the CSP model is called the Valued Constraint Satisfaction Problem framework and was introduced by [5]. A VCSP instance is a quintuple (X; D; C; S; ') where (X; D; C) is a classical CSP instance, S = (E; ; ) is a valuation structure, and ' : C! E is a valuation function, giving the valuation of each constraint. E is the set of possible valuations; is a total order on E; > 2 E is the valuation corresponding to a maximal dissatisfaction, and?2 E is the valuation corresponding to a maximal satisfaction;, the aggregation operator, allows one to aggregate valuations. In order to have a sensible behavior, must satisfy some properties : commutativity, associativity, monotonicity w.r.t. ;? must be the identity element, and > an absorbing element. Let A be an assignment of values to all of the variables, namely a complete assignment. We dene the valuation of A for the constraint c by (? if c is satised by A '(A; c) = '(c) otherwise and the overall valuation of A by '(A) = c2c '(A; c): By instantiating the valuation structure S one can dene specic frameworks such as : { classical CSP, where E is ftrue; falseg, false true,? is true, > is false, 8c 2 C; '(c) = false and is ^ (boolean and) { additive VCSP, where E is N [ f+1g, is > (natural order),? is 0, > is +1 and is + 3

4 { max VCSP, or possibilistic VCSP, where E is [0; 1], is >,? is 0, > is 1 and is max { lexicographic CSP, providing a combination of additive and max CSP. It sometimes happens that a partial assignment of the variables is a meaningful solution of the problem instance. In this case, we will have to pay a penalty for the unassigned variables. This is exactly the case with EOSSPs, for which a solution is a collection of assignments of instruments to photographs, rejected photographs corresponding to unassigned variables. It is easy to express this situation into the general VCSP framework (in which we search for a complete assignment) : just add to each original domain a special value called the rejection value, and associate to each variable a soft (non imperative) unary constraint expressing the cost of assigning it the rejection value. We call this subset of the VCSP framework the Variable Valued CSPs (VVCSPs). The standard objective is to nd a complete assignment with minimal valuation. Two kinds of algorithm can be used for this purpose : { complete algorithms, based on Branch-and-Bound schemes, which explore systematically and completely the solution space; they nd an optimal solution, provided they are given enough time { incomplete or approximate algorithms which, leaving out completeness, try to nd quickly good solutions in an opportunistic way; these methods are generally based on greedy repair schemes and randomization techniques [7, 6]; although they often produce very good quality results, they cannot guarantee a distance to the optimal valuation. Earth Observation Satellite Scheduling Problems as VCSPs These problems can be casted as additive VVCSPs by : associating a variable v with each photograph p; associating with v a domain d to express the dierent ways of achieving p and adding to d a rejection value, to express the possibility of not selecting p; then associating with v a non imperative unary constraint, setting a penalty for using the rejection value, with a valuation equal to the weight of p translating as imperative constraints the constraints of non overlapping and minimal transition time between two photographs on the same instrument, and of limitation on the instantaneous data ow; in our EOSSP instances, imperative constraints are binary and ternary looking for a complete assignment of values to variables with minimum valuation, which is equivalent to maximizing the sum of the weights of the selected photographs. As a matter of fact, the data encoded as VCSP instances is the result of a preprocessing, which computes all the variables with their associated domain 4

5 and all the binary and ternary imperative constraints with their explicitly de- ned associated relation, expressed as lists of forbidden tuples of values. It should be noticed that the rejection value cannot appear in any forbidden tuple. A xed format has been dened for this data, and a le is produced for each EOSSP instance, using this format 1. A Library of VCSP Algorithms We have developed a library 2 of complete and approximate algorithms for solving VCSP instances. This library includes : complete algorithms based upon variations of Best First Branch-and- Bound search, with well known features such as Forward-Checking, and new ones such as Directed Arc Consistency [3] and Russian Doll Search [8] approximate algorithms based on local search techniques, especially Simulated Annealing. Complete algorithms of the library may be easily associated with variable and value ordering heuristics. A careful compromise between eciency and generality has been searched for. As far as generality is concerned, the implementation closely follows the mathematical (V)CSP model stated above. In particular, algorithms are fully parameterized by the valuation structure. Constraints can be dened either explicitly (by enumeration of allowed or rejected tuples) or implicitly (by mean of boolean functions). Special variants of complete algorithms have been built for the VVCSP sub-framework, in order to improve eciency. This software in written in Common Lisp. 4 Using ILOG Solver to solve EOSSPs The modeling of Spot5 daily management problem with ILOG Solver was entrusted to one of our student, as a last term engineer project, under our supervision [4]. When he started his work, he knew nothing about ILOG Solver. He took four months to achieve his work, including study of ILOG, trying several Solver models and heuristics, and conducting experiments. Input data for ILOG Solver were the data les resulting from the aforementioned preprocessing phase. Recall that a le denes for each EOSSP instance : { a list of variables with their domain and weight { a list of imperative constraints, each one dened as a list of explicitly forbidden tuples of values. ILOG Solver does not provide a predened class for a straightforward expression of explicit constraints (given by a list of forbidden or allowed tuples). 1 Some of these Spot5 instances can be down-loaded from ftp://ftp/cert.fr/pub/ lemaitre/lvcsp/pbs/spot5/. 2 Accessible from ftp://ftp/cert.fr/pub/lemaitre/lvcsp. 5

6 We have tried several models to circumvent this diculty. The most ecient seems to use the predened IlcIfThen constraint class. Suppose we want to forbid the tuple of values (2,3) for variables x1 and x2. This can be expressed by the instruction IlcIfThen(x1 = 2, x2!= 3). We must generate one such instruction for each forbidden tuple. This works also for ternary constraints. For example, IlcIfThen(x1 = 2, (x2!= 3 x3!= 1)) expresses that the tuple of values (2,3,1) is forbidden for the variables x1, x2, x3. The goal to be reached is just to minimize the expression P i w ir i over allowed complete assignments (that is : including no forbidden tuple), with w i being the weight of the variable number i and r i = 1 if this variable is assigned to the rejection value, r i = 0 elsewhere. This criterion is expressed using the \scalar product" IlcScalProd generic constraint. As said before, several trials have been done to express EOSSPs with Solver, and no particular diculty did arise during this modeling task. The report [4] gives more details about modeling EOSSPs using ILOG Solver. 5 Comparison of Performances Three complete algorithms, searching for optimal solutions, have been tested on some Spot5 daily management EOSSP instances : IS is our ILOG Solver version with the most ecient ordering heuristics we experimented. Here are some details about these heuristics : { trying the rejection value in the last place always improves computation time signicantly (typically divided by a factor of 4) { the best static ordering of variables seems to be the following : among the variables of minimum domain size, choose one involved in the greatest number of imperative constraints, and in case of a tie, choose one with maximum weight (the computation time is typically divided by 2 with respect to the initial ordering of variables) { choosing a variable having the smallest remaining domain size (dynamic order) surprisingly does not result in a signicant reduction of computation time. BB, from our VCSP library, is a classical Branch-and-Bound with Forward-Checking, associated with the following heuristics : { as with IS, try last the rejection value { dynamically choose a variable having the smallest remaining domain size; in case of a tie, choose one involved in the greatest number of imperative constraints. RDS (for Russian Doll Search) is an original algorithm [8] included in our library, which seems particularly well suited for small bandwidth constraint graphs like those of Spot5 daily management EOSSP instances; 6

7 instance IS BB RDS mean Table 1: Performance comparison on some Spot5 EOSSP instances. this algorithm replaces one backtrack search by n successive searches on nested subproblems (n being the number of variables), records the results of each search and uses them later, when solving larger subproblems, in order to improve the lower bound on the global valuation of any partial assignment. In this comparison, we use specialized versions of BB and RDS for VVC- SPs (soft unary constraints on variables, all other constraints being imperative ones). Table 1 shows the results obtained on 15 relatively small EOSSP instances, with the three algorithms. IS and BB are within the same order of performance, with a relatively small advantage for BB. Recall that IS is built from a very rich framework for Discrete Optimization Problems, whereas BB is a classical Branch-and-Bound algorithm, specialized for VVCSPs. RDS surpasses clearly the two other competing algorithms, probably because it exploits the small bandwidth structure of the constraint graphs. The results reported above concern comparatively small instances, ranging from 25 to 30 variables. Instances to be solved in practice may have more than 900 variables. When the size of the instances increases, it can be observed that computation time increases very rapidly. The BB algorithm cannot be reasonably used on instances having more than 60 variables, and RDS is generally overwhelmed by instances having more than 300 variables. For greater instances, we must abandon the search for optimality and turn towards approximate algorithms. 7

8 6 Conclusion We have presented the daily management scheduling problem of an earth observation satellite, a real-life combinatorial optimization problem of great economical importance. In this context, two opposite approaches for combinatorial optimization problem solving have been compared. The rst one, call it the language approach, is represented by ILOG Solver. It rests on a rich set of predened constructs for the expression of problems (generic constraints and search procedures) harmoniously included in the same ground language, providing the user with possibilities of extensions. The second approach, name it the model approach, is illustrated by our library of algorithms. It is built on a sound and general mathematical model, oering only basic and rather \at" constructs, but able to capture most (if not theoretically all) discrete combinatorial optimization problems. In this framework, a range of algorithms is proposed, some of them fully general, others being able to exploit particular features of problem instances. Undoubtedly, the language approach provides the user with easier ways of tackling problems : often, the user will nd easily a construct or a combination of constructs directly expressing his/her problem. On the other hand, the model approach will force the user towards a deeper understanding of his/her problem, possibly discovering the particular features of it. Once the modeling work is done, better performances may be expected. The performance comparison reported in this paper shows that the language approach is not much penalized by the rich set of constructs it provides, as far as general algorithms are used, and instances are small. Nevertheless, sophisticated algorithms exploiting the particular structure of problem instances are probably easier to bring into play within the model approach. We conclude by suggesting the following extensions to Solver : { the encapsulated expression of explicit constraints { the ability to use approximate algorithms, replacing systematic search by opportunistic local search. Acknowledgments This work was supported in part by the French space agency (Centre National d' Etudes Spatiales). We are indebted to Mustapha El Masbahi who conducted the ILOG experimentation, and to Denis Blumstein and Jean-Claude Agnese, from CNES, who rst conceived and experimented the Russian Doll Search method for EOSSPs. References [1] J.C. Agnese, N. Bataille, E. Bensana, D. Blumstein, and G. Verfaillie. Exact and Approximate Methods for the Daily Management of an Earth Observation Satellite. In Proc. of the 5th ESA Workshop on Articial Intelli- 8

9 gence and Knowledge Based Systems for Space, Noordwijk, The Netherlands, [2] E. Freuder and R. Wallace. Partial Constraint Satisfaction. Articial Intelligence, 58:21{70, [3] J. Larrosa and P. Meseguer. Expoiting the Use of DAC in MAX-CSP. In Proc. of the 2nd International Conference on Principles and Practice of Constraint Programming (CP-96, LNCS 1118), pages 308{322, Cambridge, MA, USA, [4] M. El Masbahi. Optimisation de la programmation journaliere des prises de vue du satellite SPOT5 avec Ilog Solver. Technical Report Projet de n d'etudes, INSA, Toulouse, France, [5] T. Schiex, H. Fargier, and G. Verfaillie. Valued Constraint Satisfaction Problems : Hard and Easy Problems. In Proc. of the 14th International Joint Conference on Articial Intelligence (IJCAI-95), pages 631{637, Montreal, Canada, [6] B. Selman, H. Levesque, and D. Mitchell. A New Method for Solving Hard Satisability Problems. In Proc. of the 10th National Conference on Articial Intelligence (AAAI-92), pages 440{446, San Jose, CA, USA, [7] P. J. M. van Laarhoven and E. H. L. Aarts. Simulated Annealing : Theory and Applications. D. Reidel Publishing Company, [8] G. Verfaillie, M. Lema^tre, and T. Schiex. Russian Doll Search for Solving Constraint Optimization Problems. In Proc. of the 13th National Conference on Articial Intelligence (AAAI-96), pages 181{187, Portland, OR, USA,