Iterative deepening multiobjective A*

Transcription

1 Iterative deepening multiobjective A* S. Harikumar, Shashi Kumar * Depurtment of Computer Science and Engineering, Indiun Institute of Technology, Delhi, New Delhi, India Received 19 January 1995; revised 17 January 1996 Communicated by M.J. Attalah Keywords: Algorithms; Combinatorial problems; Multiobjective search; Optimization; Analysis of algorithms 1. Introduction Many real-world optimization problems involve multiple objectives which are often conflicting. When conventional heuristic search algorithms such as A* and IDA* are used for solving such problems, then these problems have to be modeled as simple cost minimization or maximization problems. The task of modeling such problems using a single valued criterion has often proved difficult [6]. The problems involved in accurately and confidently determining a scalar valued criterion on which to base the selection of a most preferred alternative have led to the development of the multiobjective approach to alternative selection [7]. In [7], Stewart and White have presented a multiobjective generalization of the popular A* algorithm, the MOA*, which uses heuristic based best first search techniques to generate all nondominated solutions. Like A*, MOA* also has exponential space complexity. Depth first search techniques use linear space, but take too much time and do not lead to an admissible algorithm. A depth first version of A* called iterative deepening A* (IDA* ) [2] takes linear space and is shown * Corresponding author. to be admissible. Under certain conditions it has been shown that IDA* is optimal [4]. That is, the number of nodes expanded by IDA* is of the same order as A*. In this paper, we present a linear space multiobjective generalization of the iterative deepening algorithm called IDMOA* and prove its correctness, completeness and optimality with respect to MOA*. We have assumed that all objectives are formulated in terms of some quantity to be minimized. Maximization problems can be handled with simple modifications. We define dominance in the same way as in [7]. Definition (Dominance). Given a set Y of m-vectors, we define a relation R C Y X Y, as follows: (Y. Y ) ER * y,<yiviel,..., mand y#y. We say that y dominates y (in Y) if ( y, y ) E R. An element y E Y is said to be non-dominated (in Y) if there does not exist another element y E Y such that y dominates y (in Y ). We illustrate the concept of dominance in the context of the multiobjective traveling salesperson (TSP) problem in which we have to find the tours having non-dominated costs, given a city graph with each edge having two weights as shown in Fig. 1.

2 12 S. Harikumar, S. Kumar/Injbrmation Processing Letters 58 (1996) I I-15 Fig. I. Multiobjective TSP. The non-dominated tours are: (1) A + G + B -+ C --) D + E --) F: Cost = (15, IO). (2) A + G + B + C + F + E + D: Cost = (18, 9). (3) A + G --) F --) E + D --) C + B: Cost = (14, 12). The following notation will be used to describe the algorithm: m: The number of objectives. SOLUTION: The current set of all non-dominated solutions. Threshold,: Maximum value of threshold used for the ith objective. Si: The set of non-dominated solutions obtained after fixing threshold for the ith objective. s(w,, wz,..., w,): Cost vector associated with node s in the search graph or solution s. s(i): Cost component of node s corresponding to the ith objective. The basic idea As given in [2], IDA* proceeds by adjusting a threshold value and doing a depth first search till the evaluation function of the nodes exceed the threshold. The threshold is modified (increased) till we get a solution. In the multiobjective scenario, since we have multiple objectives, we need to have a threshold vector instead of a scalar value. If we have m objectives, the threshold vector may be represented by (t,, r,,... f,,,), where ci is the threshold for the ith objective during an iteration. We illustrate the basic idea of the algorithm by considering a problem with two objectives. In the IDMOA* algorithm presented here, we first apply a threshold for the first objectives, and using irerutive deepening [2], get a non-dominated solution which has the least value for the first objective. We use the value of the second objective for this solution as the maximum threshold for that objective and find more non-dominated solutions using iterative deepening search. This can be generalized for problems with more than two objectives. After applying a threshold for the ith objective (i > 1) we use max(s(i + 1); Vs E SOLUTION) as the maximum threshold for the (i + 1)t.h objective. We use this new maximum threshold to do iterative deepening search and generate more solutions. Like IDA* [2], IDMOA* also requires the evaluation functions to be monotone. The algorithm terminates when all the objectives have been considered. SO- LUTION now contains all the non-dominated solutions. 3. The IDMOA* algorithm Input: A graph G = (V, E) with vertices u E V and edge weights {(wl,, wzl...., w,,>, (w,~, w22..., w,,,~),...i. For each edge e E E, there corresponds an m-tuple ( w, e, wze,..., w,,) representing the selection values for the m objectives. Output: The set of non-dominated solutions namely SOLUTION. Global ObjIndex, MaxThreshold, Threshold, SOLU- TION. 0. Initialize by setting SOLUTION equal to the empty set and SNODE to the start node. 1. Find all solutions with the minimum value for the first objective Set Objlndex equal to 1 and Threshold equal to the heuristic estimate of the minimum threshold for the first objective If SOLUTION is empty, do the following: Perform DFSEARCH (SNODE) to find SO~Utions Heuristically increment Threshold by the minimum step considering the first objective.

3 S. Harikwnnr, S. Kumar/lnformation Processing Letters ) II Go to Step (1.2) Otherwise, continue. 2. Fix thresholds for other objectives and find all solutions using iterative deepening For ObjIndex := 2 to m do the following: Set MaxThreshold equal to max(s(objlndex); Vs E SOLUTION) Find solutions using iterative deepening search Set Threshold equal to the heuristic estimate of the minimum threshold for the current objective (Objlndex) If Threshold < MaxThreshold then do the following: Perform DFSEARCH (SNODE) to find solutions. Heuristically increment Threshold by the minimum step considering the current objective Go to Step ( ) Otherwise, continue. 3. stop. The DFSEARCHO routine does an exhaustive search to find out all the non-dominated solutions which satisfy the threshold criterion. Each solution is added to SOLUTION and dominated solutions are removed from SOLUTION. DFSEARCHC n) 1. Check for valid node If n(objzndex) is greater than Threshold go to Step (4) Otherwise, continue. 2. Identify solutions If n is a goal node and n(objindex> is not greater than Threshold, then do the following: Add it to SOLUTION Remove any dominated members of SOLU- TION Go to Step (4) 2.2. Otherwise, continue. 3. Expand n and examine its successors If n has no successors, go to Step (4) For all successors n of n, do the following: If n (ObjIndex) is not greater than Threshold and n is not dominated by any solution in SOLUTION, then do the following: l. Recursively do DFSEARCH (n ). 4. Return. 4. Properties of IDMOA* In the following we show that IDMOA* inherits the properties of IDA* and MOA*. We discuss the completeness of IDMOA* and its optimality with respect to MOA. Lemma 1. Si+, a Si; Vi: 1 d i < m. Proof. We have fixed Threshold,,, = max(s(i + 1); Vs E S,), i.e. Vs E Si, s(i + 1) Q Threshold,. 1. Also, for any new solution t found in the (i + 1)th step, t(i) > Threshold,. But, Vs E Si, s(i) Q Threshold,. We observe that t(i) > s(i) and so t cannot dominate s. Therefore, all solutions in Si continue to be non-dominated when new solutions are added in the (i + 1)th step. From the algorithm, Si+ 1 = (s i s(i + 1) < Threshold,,, and s is non-dominated}. Therefore, Vs, SESi * s=si+,. (1) We now prove that there can exist a non-dominated solution which is not in Si but in Si+,. Consider the case when there exists a non-dominated solution s and an integer k such that s (k) > s(k) Vs E Si, 1 < kg i and s (i + 1) < min(s(i + 1); Vs E Si>. Clearly, s 4Si, but S ES~+,, since s ( i + 1) d Threshold,,,. From (1) and (2), S,,, zsi; Vi: 1 gi<m. 0 Definition (Non-dominated node). A node in the search graph is said to be non-dominated if the cost vector associated with it is not dominated by that of any non-dominated solution. Lemma 2. All the non-dominated nodes will be expanded by IDMOA*. (2)

4 14 S. Harikumar. S. Kumar/lnformation Processing Letters ) II-15 Proof. We prove this by contradiction. Let us assume that IDMOA* does not expand a node n which is not dominated by any solution in S,,,. Since n was not expanded, it follows from the algorithm that n (i) > Threshold, Vi: 1 < i Q m. (3) Now consider the first solution detected by IDMOA*, say si, i.e. SEES,. Since s,es,, sf~sk Vk: 1 <k Q m according to Lemma 1. Therefore, sr( i) < Threshold, Vi: 1 < i Q m. (4) Due to facts (3) and (4), and by the definition of dominance, n is dominated by sr. This contradicts our assumption that n is a non-dominated node. So IDMOA expands all the non-dominated nodes. 0 Now we prove that IDMOA is complete, meaning that it finds all the non-dominated solutions. Theorem 3. IDMOA is complete. Proof. Let us assume that IDMOA* is incomplete, and it has missed a solution s which is not dominated by any solution in S,,,, S es,. But by Lemma 2, since IDMOA* expands all the non-dominated nodes, s will be considered for expansion by IDMOA*. Then it follows from the algorithm that s will be identified as a solution during DFSEARCHO. This contradicts our assumption that IDMOA* has missed the non-dominated solution s. Therefore IDMOA is complete. 0 Now we see the optimality of IDMOA with respect to MOA* by considering the nodes expanded (like in the case of IDA* [3]) by IDMOA* and MOA". The asymptotic optimality of IDMOA will still be subject to the optimality conditions for IDA* described in [4]. Theorem 4. IDMOA* does not expand a node which is not expanded by MOA*. Proof. By Lemma 17 in [7], we know that MOA* expands all the non-dominated nodes, and only the non-dominated nodes. Let us assume that IDMOA* expands a node n which is not expanded by MOA*. So n is dominated by some solution. If n(w,, w2,... ) w,> is a dominated node, then there should exist a solution s( w,, w;,..., wk> which dominates n. By the definition of dominance, Vi: l,<igm, w:,<wi and 3j: l<j<m, w;<wj. Since s is a non-dominated solution, according to Lemma 2, 3 i: 1 < i < m such that s(i) < Threshold,.. Let j be the first such objective (in the order in which objectives are considered by the algorithm) for which s( j> Q Thresholdj. Consider the iterative deepening search phase in which threshold for objective j is fixed. We have two possibilities with s dominating n. Case 1: n(j) > s(j). Clearly, s will be encountered before n in iterative deepening. Then s, being a solution, will get added to SOLUTION. Since n is dominated by s, n will not be expanded in further iterations. Case 2: n(j) = s(j). Consider the first iteration during iterative deepening in which s is encountered. In this case, both s and n are encountered in the same invocation of DFSEARCHO, though either of them can be encountered first. n will not be expanded in the same invocation of DFSEARCHO since any child of n will have evaluation function greater than the threshold used in that iteration, as the evaluation function is monotone and threshold is incremented by the minimum steps. s gets added to SOLUTION since it is a non-dominated solution. Since n is dominated by s, it is not considered for expansion in the next iteration. Therefore, we contradict our assumption that a dominated node n is expanded by IDMOA. 0 Because of the above result, the following corollary can be stated. Corollary 5. The order in which IDMOA* considers the objectives does not affect the set of nodes expanded. Proof. From Lemma 2, Theorem 4 and Lemma 17 of [7] we see that both IDMOA and MOA expand the same set of nodes. Since Lemma 2 and Theorem

5 S. Harikumar, S. Kumar / Information Processing Leners 58 (1996) I I hold irrespective of the order in which the objectives are considered, we conclude that the set of nodes expanded by IDMOA* is not affected by the order in which it considers the objectives Implementation and performance The IDMOA algorithm has been tested on search problems like multiobjective TSP. In practice IDMOA is found to be more efficient than MOA* in terms of actual time. This is basically due to two reasons: (1) MOA finds the non-dominated set from OPEN [7], in each iteration. Since OPEN is a large set (growing exponentially), finding the nondominated set is time consuming. IDMOA* tests whether a node is dominated by any of the solutions in SOLUTION before expanding it. Since SOLU- TION is much smaller than OPEN, IDMOA* proceeds faster. (2) The overheads associated with IDMOA* are much less compared to MOA* where we have to maintain many sets. 6. Conclusion Multiobjective search is a general problem solving technique for problems in which there are multi- ple objectives. We have presented an extension of MOA called iterative deepening multiobjective A* (IDMOA* > to tackle the same class of problems. We have shown the correctness and completeness of the algorithm. Since IDMOA* uses only linear space, it can be a good candidate for the general class of multiobjective search problems. Finding out general approximate versions of IDMOA* will be an interesting future work. References t11 A.V. Aho, J.E. Hopcroft and J.D. Ullman. Data Structures and Algorithms (Addison-Wesley, Reading, MA, 1983). El R.E. Korf, Depth first iterative deepening: An optimal admissible tree search, Artificial Intelligence 27 (1985) [31 R.E. Korf, Optimal path-finding algorithms, in: L. Kanal and V. Kumar, eds., Search in Artificial Intelligence (Springer, Berlin, 1988) t41 A. Mahanti, S. Ghosh, D.S. Nau, A.K. Pal and L. Kanal, Performance of IDA on trees and graphs, in: Proc. AAAI-92 (1992) [51 N.J. Nilsson, Principles of Artijicial Intelligence (Tioga, Palo Alto, CA, 1980). l61 R.K. Singh, S. Kumar and V. Arvind, A heuristic search strategy for optimization of trade-off cost measures, in: Proc IEEE Conf on Tools Internat. for AI, San Jose, CA (1991). 171 B.S. Stewart and Chelsea C. White, Multiobjective A, J. ACM 38 (4) (1991)