Flexbile-Attribute Problems - Semantic Scholar

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Flexbile-Attribute Problems Jurij Miheliˇ c, Borut Robiˇ c Faculty of Computer and Information Science, University of Ljubljana, Slovenia e-mail: [email protected], e-mail: [email protected] Received: date / Revised version: date

Abstract Problems with significant input-data uncertainty are very common in practical situations. One approach to dealing with this uncertainty is called scenario planning, where the data uncertainty is represented with scenarios. A scenario represents a potential realization of the important parameters of the problem. In this paper we present a new approach to coping with data uncertainty, called the flexibility approach. Here a problem is described as a set of interconnected simple scenarios. The idea is to find a solution for each scenario such that, after a change in scenario, transforming from one solution to the other one is not expensive. We define two versions of flexibility and hence two versions of the problem, which are called the sum-flexible-attribute problem and the max-flexibleattribute problem. For both problems we prove the N P-hardness as well as the non-approximability. We present polynomial time algorithms for solving the two problems to optimality on trees. Finally, we discuss the possible applications and generalizations of the new approach. Key words gorithms

Uncertainty – flexibility – NP-hard – approximability – al-

1 Introduction In reality we often run into problems that have significant input-data uncertainty. For example, the uncertainty can be in price, demand, production costs, growth of the market as well as potential future changes in consumer tastes, competitors reactions, the emergence of new technologies, etc. There are several well-known ways to represent data uncertainty [10]. In this paper we use the scenario-based representation of data uncertainty. Here, a

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Jurij Miheliˇc, Borut Robiˇc

scenario represents a potential realization of the parameters of the problem. Due to the unpredictability of the future, easy transitions among scenarios (as well as their solutions) are desired. The area of research dealing with scenarios is called scenario optimization. In the simplest approach to scenario optimization, each scenario can be treated as an independent optimization problem. However, the solutions found may be so dissimilar that the transformation from one solution to another (i.e., adapting to scenarios) becomes prohibitively expensive. Another approach, called the robustness approach, is to find only one solution that is suitable for each and every (possible) scenario. This approach has already received considerable attention [1,2,5,9,10,13,14]. Unfortunately, however, the solution might not be achievable due to scenario diversity, or it might be suitable for some scenarios but very unsuitable for others. Yet another recent approach, called the flexibility approach, is to find many solutions, one for each scenario, while ensuring that these solutions are “similar” [11,12]. Such solutions are flexible in the sense that they allow a cheap transformation from one to another as a response to the change of the actual scenario. Let us now present an application of the flexibility approach. (For broader discussion on the applicability see below.) Consider the following problem of supply-chain management type. An organization or a company is considering the possibility of starting the distribution of goods, e.g., food or medicine supplies, in several different regions worldwide. However, the changing of local and non-local conditions, e.g., poverty level, natural disasters, wars, political decisions, may influence the decision whether goods are to be supplied in a region. In order to support the potentially changing distribution the organization intends to set up several new facilities, i.e., distribution centers, in different regions. In each region there are several candidate locations (constituting the scenario) for setting up a facility, but only one of them is to be chosen. Additionally, there are known transportation routes between the regions. Clearly, the transportation cost between two regions depends on the locations that were chosen in the two regions. Thus, finding a good location in every region can greatly reduce the distribution costs. The aim of the organization is, therefore, to determine the set of locations, one in each region, that will minimize the distribution costs. In this paper we present the flexibility approach with many interconnected and simple scenarios. In Section 2, we formalize two versions of the flexibility and hence introduce two important types of flexibility problem, i.e., the sum-flexible-attribute problem and the max-flexible-attribute problem. In Section 3, we describe a numerical example of the flexibility problems. In Section 4, we discuss the computational complexity of the two problems. In particular, we prove the N P-hardness of both problems as well as presenting several approximability results. In Section 5, assuming simplified input data, we present polynomial time algorithms for solving the problems to optimality. Finally, in Section 6 we discuss the generalizations

Flexbile-Attribute Problems

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and the applicability of the flexibility approach. Section 7 concludes the paper. 2 Flexbile-attribute problems (FAPs) In flexible-attribute problems we deal with interconnected and simple scenarios. Each scenario is described with a set of elements, called attributes. The possible transitions among scenarios are represented by an undirected graph whose vertices represent scenarios. Our aim is to choose one attribute for each vertex (scenario) in such a way that the total cost of transforming the chosen attributes is optimized. Let G = (V, E) be a graph where V = {1, 2, . . . , n} is a set of vertices and E is a set of edges (i, j) ∈ V × V . To each vertex i ∈ V is assigned a set of attributes Ai = {ai1 , ai2 , . . . , ai|Ai | }. Notice that Ai describes the i-th scenario si . Let i and j be two connected vertices, i.e., (i, j) ∈ E. The cost of transforming attribute aik ∈ Ai to ajl ∈ Aj is denoted by cij (aik , ajl ). The solution of a flexible-attribute problem will be a list (a1 , a2 , . . . , an ), where ai ∈ Ai is an attribute chosen for scenario si . Thus, the cost of the transition between si and sj , where (i, j) ∈ E, is simply the cost of transforming the attribute ai ∈ Ai to aj ∈ Aj , i.e., cij (ai , aj ) There are several ways to define the total cost of transforming the attributes. We are interested in two of them and thus define two different optimization problems. The first problem is to minimize the function fsum , where X cij (ai , aj ). fsum = (i,j)∈E

We call this problem the minimum sum-flexible-attribute problem (denoted by SumFAP). The approach of calculating such a total cost will be termed the sum-flexibility. The second problem is to minimize the objective function fmax , where fmax = max cij (ai , aj ). (i,j)∈E

This problem is called the minimum max-flexible-attribute problem (denoted by MaxFAP) and the corresponding approach, the max-flexibility. Notice that in the former approach the optimization is focused on the average case (the equality of all transitions) and in the latter on the worst case. We discuss several generalizations of these problems in Section 6. 3 Example Let us now give a small numerical example. Consider also the application of the supply-chain management type mentioned in Section 1. Let there

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be four regions represented by scenarios s1 , s2 , s3 , and s4 . Candidate locations for placing a facility in the region si are represented by attributes. Let there be four candidate locations in the first and the third region and three locations in the second and the fourth region. Thus, the corresponding sets of attributes are A1 = {a11 , a12 , a13 , a14 }, A2 = {a21 , a22 , a23 }, A3 = {a31 , a32 , a33 , a34 }, and A4 = {a41 , a42 , a43 }. Let there also be distribution routes between all regions except between the second and the third one. The scenario graph representing distribution routes and attributes is shown in Figure 1.

Fig. 1 Scenario graph and optimal solution

For each pair of locations (represented by attributes aik and ajl ) in two connected regions (represented by scenarios si and sj ) the transportation cost cij (aik , ajl ) is also known. For simplicity of our example, let this cost be cij (aik , ajl ) = 1 + |i + k − j − l|. For both problems, i.e. SumFAP and MaxFAP, the optimal solution is (a14 , a23 , a32 , a41 ) with cost 4 and 1, respectively.

4 Time complexity of the FAP 4.1 Size of the solution space Let us first discuss the size of the solution space. For both flexible-attribute problems the feasible solutions are sequences (a1 , a2 , . . . , an ) of attributes where ai ∈ Ai is the attribute chosen for scenario si . The number of feasible solutions equals the number of such possible sequences. For each si we can choose Q one of |Ai | possible attributes. Thus, the size of the solution space is i∈V |Ai |, which is exponential in n when the input is non-trivial. Thus, for a fixed n it is, in principle, possible to solve both problems to optimality in polynomial time by a simple, exhaustive search of the solution space. In practice, however, even when n is a small constant the size of the solution space is often too large for an efficient, exhaustive search. Additionally, in the following we also show that both problems are N P-hard when n is variable.


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4.2 NP-hardness Our first result relating to the complexity of solving the flexible-attribute problems is that SumFAP and MaxFAP are both N P-hard optimization problems. To prove this we show that decision versions of both these problems are N P-complete. To show the latter we use the technique of reduction [3]. Theorem 1 Decision version of SumFAP is N P-complete. Proof Reduction of the traveling salesperson problem (TSP), a well-known N P-complete problem [8]. Given a set {c1 , c2 , . . . , cn } of cities and a matrix D = (dij )n×n , where dij ∈ N represents the distance between ci and cj , and a constant B > 0, the TSP problem asks whether there is a permutation T = (ci1 , ci2 , . . . , cin ) of cities such that the length m(T ) of the tour T is at Pn−1 most B, i.e., if m(T ) = k=1 dik ,ik+1 + din ,i1 ≤ B. Let P be the decision version of SumFAP. We can quickly show that P belongs to N P. For a given instance of P guess the attributes (a1 , . . . , an ) and check if fsum ≤ B. Now we describe the reduction of the TSP to P . Given an instance of the TSP, i.e., a set of cities {c1 , c2 , . . . , cn }, a matrix D, and a constant B, we construct an instance of P , i.e., a graph G = (V, E), sets of attributes A1 , A2 , . . . , An , and costs cij . The construction is shown in Figure 2.

Fig. 2 Reduction.

We construct a complete graph G on n vertices and for each i ∈ V we define Ai = {c1 , c2 , . . . , cn }. Let N = {(n, 0)} ∪ {(i, i + 1)|1 ≤ i < n} be the set of edges between “immediate” neighbors (vertices with consecutive indices). For each (i, j) ∈ E we define the cost of the transformation of the attributes ck ∈ Ai and cl ∈ Ai+1 as   ∞ k = l cij (ck , cl ) = dkl (i, j) ∈ N   0 otherwise. The construction produces n vertices (scenarios), n(n − 1)/2 edges and n2 attributes. Clearly, this is a polynomial time reduction. Next we show its

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correctness by proving that for each instance of the TSP there is a solution T = (cl1 , cl2 , . . . , cln ) with cost z if and only if T is the solution with the cost z for the corresponding instance of P . (⇒) Let T = (cl1 , cl2 , . . . , cln ) be a solution of an instance of the TSP with the Pcost z. Since T is a permutation, we have li 6= lj for all 1 ≤ i, j ≤ n. Thus, (i,j)∈E/N cij (cli ,lj ) = 0 and hence X cij (cli , clj ) = fsum (T ) = (i,j)∈E

=

X

cij (cli , clj ) =

X

dli ,lj =

(i,j)∈N

=

(i,j)∈N

=

n−1 X

dlk ,lk+1 + dln ,l1 =

k=1

= m(T ). (⇐) Let T = (cl1 , cl2 , . . . , cln ) be a solution for P with the cost z 6= ∞. Because z 6= ∞ we have cij (cli , clj ) 6= ∞ for all (i, j) ∈ E, and consequently li 6= lj . Thus, T is a permutation. We deduce that fsum (T ) = m(T ) in a similar fashion as in the (⇒) part. Recall that the optimization problem is N P-hard if its decision version is N P-complete [3]. Thus, we have the following corollary. Corollary 2 SumFAP is an N P-hard optimization problem. Let us now turn to the MaxFAP problem and prove the following theorem. Theorem 3 Decision version of MaxFAP is N P-complete. Proof Reduction of the Hamiltonian cycle problem (HCP). Given a graph H = (U, F ), is there a cycle T = (ui1 , ui2 , . . . , uin ) of vertices such that each vertex is in T exactly once? Let P denote the decision version of MaxFAP. It is clear that P ∈ N P, so we describe the reduction of the HCP to P . Given an instance H = (U, F ) of the HCP, we construct a complete graph G = (V, E) with |V | = |U | = n vertices and define sets of attributes A1 , A2 , . . . , An as Ai = {u1 , u2 , . . . , un } for each i ∈ V . In addition, we define, for each (i, j) ∈ E, the cost of the transformation of the attributes uk ∈ Ai and ul ∈ Ai+1 as  0 (i, j) ∈ N, (uk , ul ) ∈ F    ∞ (i, j) ∈ N, (u , u ) ∈ k l / F cij (uk , ul ) =  ∞ (i, j) ∈ / N, k = l    0 (i, j) ∈ / N, k 6= l


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where N = {(n, 0)} ∪ {(i, i + 1)|1 ≤ i < n}. This is clearly a polynomial reduction. Now we show that for each instance of the HCP there is a solution T = (ul1 , cu2 , . . . , cun ) if and only if T is the solution with cost 0 of the corresponding instance of P . (⇒) Let T = (ul1 , ul2 , . . . , uln ) be a solution of the HCP. We compute fmax on T , i.e., fmax (T ) = max cij (uli , ulj ) = (i.j)∈E

= max[ max

(i,j)∈E/N

cij (uli , ulj ), max cij (uli , ulj )]. (i,j)∈N

Since T is a permutation, we have li 6= lj for all (i, j) ∈ E/N . Thus, the first max(i,j)∈E/N cij (uli , ulj ) = 0. Since T is a cycle, we have (uli , ulj ) ∈ F for all (i, j) ∈ N . Consequently, the second term is zero, too. Hence, fmax = 0. (⇐) Let T = (ul1 , ul2 , . . . , uln ) be a solution of P with cost 0. Since fmax = 0, for all (i, j) ∈ E and for all uli and ulj holds, either (i, j) ∈ N / N and li 6= lj and (uli , ulj ) ∈ F (meaning that T is a cycle), or either (i, j) ∈ (meaning that T visits every vertex in U exactly once). Consequently, T is a Hamiltonian cycle in H. Corollary 4 MaxFAP is an N P-hard optimization problem. 4.3 Approximability We have just shown that both SumFAP and MaxFAP are N P-hard optimization problems. Searching for approximation algorithms is, therefore, a reasonable approach [3]. The following theorem is not surprising, and we omit the proof because of its simplicity (by contradiction). Theorem 5 There are no polynomial absolute approximation algorithms for SumFAP and MaxFAP, unless P=N P. A more interesting fact is that both problems are non-approximable, i.e., using the gap-technique [3] we prove the following theorem, which states that there are no polynomial approximation algorithms having a constant approximation factor for SumFAP and MaxFAP. Theorem 6 SumFAP∈ / AP X and MaxFAP∈ / AP X. Proof By contradiction. In the proof of Theorem 3 replace the function of the attribute transformation cost with the following  1 (i, j) ∈ N, (uk , ul ) ∈ F    M (i, j) ∈ N, (u , u ) ∈ k l / F cij (uk , ul ) =  M (i, j) ∈ / N, k = l    1 (i, j) ∈ / N, k 6= l

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Let x be an instance of HCP and x′ the corresponding instance of MaxFAP. Let A be the (M − ǫ)-approximation algorithm for MaxFAP, where ǫ > 0. If x has a Hamiltonian cycle then the algorithm A must return the optimal solution with the cost 1 on x′ . Otherwise, if x has no Hamiltonian cycle then the optimal solution on x′ has the cost M . Thus, it is possible to solve HCP to optimality with A. This is not possible unless P=N P. The proof for SumFAP is similar and is, therefore, omitted.

4.4 Some additional properties In this subsection we state a few additional properties of the two F AP problems. Let S be a solution of an instance of SumFAP respectively MaxFAP. It is easy to prove the following theorem. Theorem 7 fmax (S) ≤ fsum (S) ≤ |E|fmax (S). ∗ ∗ Let Ssum be an optimal solution for an instance of SumFAP and Smax be an optimal solution for the same instance of MaxFAP. ∗ ∗ ∗ ∗ Corollary 8 fmax (Smax ) ≤ fmax (Ssum ) ≤ fsum (Ssum ) ≤ fsum (Smax ) ≤ ∗ |E|fmax (Smax ).

Corollary 9 If A is an r-approximation algorithm for SumFAP, then A is an r|E|-approximation algorithm for MaxFAP. Proof Denote with x an instance of SumFAP resp. MaxFAP. Let A(x) be a solution returned by A on x. As by assumption A is an r-approxi∗ ∗ mation algorithm, we have fsum (Ssum ) ≤ fsum (A(x)) ≤ rfsum (Ssum ), and ∗ ∗ consequently fmax (Smax ) ≤ fmax (A(x)) ≤ fsum (A(x)) ≤ rfsum (Ssum ) ≤ ∗ ). r|E|fmax (Smax 5 Algorithms on trees Since both SumFAP and MaxFAP are N P-hard, it is very unlikely that they can be solved to optimality in polynomial time, unless P=N P. In addition, we have just presented some negative results relating to their approximability. In this situation it is, therefore, reasonable to search for simplifications of the two problems that enable the construction of efficient algorithms. In the following we describe algorithms for solving SumFAP and MaxFAP to optimality where the input graph G is a tree. For the simplicity of the description of the algorithms let s1 be a root in G and let us increasingly number the remaining scenarios with breadth-first search, as shown in Figure 3. Both algorithms are based on dynamic programming [6]. The heart of the algorithms is the assignment of the appropriate weights to the attributes. The algorithms consist of two parts. The first part is the initialization, where


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Fig. 3 Example of a tree for flexible-attribute problems.

the weights of all the attributes belonging to leaf scenarios (vertices) are set to zero. The second part is the iterative processing of all the remaining scenarios. During each step of the processing the weights of the attributes belonging to one scenario are computed. The difference between the algorithms SumFAP and MaxFAP is in the way the weights are calculated. Denote with w(aik ) the weight of the attribute aik ∈ Ai . Let A = {(i, j) ∈ E ∧ i < j}. Imagine A being arcs in the direction from the root s1 to the leaves of G. Denote with AlgSumFAP the polynomial algorithm for solving SumFAP to optimality on a tree. The weight w(aik ) of the attribute aik ∈ Ai is, in AlgSumFAP, calculated according to the following formula w(aik ) =

X

min [cij (aik , ajl ) + w(ajl )].

(i,j)∈A

ajl ∈Aj

(1)

Denote by AlgMaxFAP polynomial algorithm for solving MaxFAP to optimality on a tree. The calculation of weight w(aik ) of the attribute aik ∈ Ai is in AlgMaxFAP done according to the formula w(aik ) = max

min max[cij (aik , ajl ), w(ajl )].

(i,j)∈A ajl ∈Aj

(2)

The algorithms AlgSumFAP and AlgMaxFAP are shown in Figure 4. The initialization is from line 1 to line 5. First (line 1), the variable L is assigned to the set of leaf vertices. Then (lines 2 to 4), the weight of each attribute of any leaf scenario is set to 0. The processed scenarios are stored in the variable W . Since all the leaf scenarios are already processed, W is initialized to L (line 5). The main loop of the algorithm is from line 6 to line 11. The loop is executed until all the scenarios are processed (line 6). In line 7, a scenario i is selected such that it is not yet processed and that all its output arcs lead to already-processed scenarios. In lines 8 and 9, the weights of the attributes aik ∈ Ai belonging to the selected scenario i are calculated. The weight w(aik ) of the attribute aik is calculated (line 9) according to formula 1 or 2 for the algorithms AlgSumFAP or AlgMaxFAP, respectively. After the

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Jurij Miheliˇc, Borut Robiˇc Input: Instance of SumFAP resp. MaxFAP on a tree. Output: Value f ∗ . Algorithm: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

L = {i|¬∃j : (i, j) ∈ A}; for all i ∈ L do for all aik ∈ Ai do w(aik ) := 0; W := L; while W 6= V do Choose i ∈ V /W : ∀(i, j) ∈ A : j ∈ W ; for all aik ∈ Ai do Calculate w(aik ); W := W ∪ {i}; end; f ∗ = mina1k ∈Ai w(a1k ); return f ∗ .

Fig. 4 Algorithms AlgSumFAP and AlgMaxFAP.

calculation, the vertex i is added to W (line 10). At the end of the algorithm the minimum weight of any attribute belonging to the root scenario is assigned to f ∗ (line 12) and returned as a result (line 13). Time complexity. Let n = |V | and l = maxi∈V |Ai |. Since G is a tree, |E| = n − 1. Initialization takes O(nl) time. During each step of the main loop one scenario is processed. Notice that since the input graph is a tree, for each loop iteration exactly one different edge is used in the weight calculation. For each edge there are O(l) weights to calculate and each calculation uses O(l) weights. Thus, the total time complexity of the algorithm is O(l2 n), and we have proved the following lemma. Lemma 1 Algorithms AlgSumFAP and AlgMaxFAP have a polynomial time complexity O(l2 n). Correctness. To prove the correctness of the algorithm the following definition is needed. Denote with T (V ′ ) = (V ′ , E ′ ) an induced subgraph in G = (V, E) on the vertices V ′ ⊆ V and with N (i) a set of vertices V ′ ⊆ V consisting of i, and all vertices that are accessible from i following arcs in A. Now consider T (N (i)). Denote with (aik , . . . ) an optimal solution for T (i) where the selection in scenario i is fixed to the attribute aik . Denote with fsum (T (N (i)), (aik , . . . )) a value of fsum calculated on T (N (i)) and (aik , . . . ). Let waik be the weight calculated with the algorithm AlgSumFAP for the attribute aik . We have the following lemma. Lemma 2 w(aik ) = fsum (T (N (i)), (aik , . . . )) for all aik ∈ Ai , where i ∈ V .


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Proof By induction on the size of the sub-trees T (N (i)) in G. We begin with the trees T (N (i)) consisting of only one vertex, i.e., the leaves of G, and proceed by adding the remaining neighbor vertices. Basis. Let L be the leaves of G. Consider the attribute aik ∈ Ai , where i ∈ L. Clearly, T (N (i)) = ({i}, ∅). Hence, fsum (T (N (i)), (aik )) = 0. The weight w(aik ) is initialized to 0 in AlgSumFAP. Clearly, w(aik ) = fsum (T (N (i)), (aik )). Inductive step. Consider aik , where i ∈ V /L. With the induction hypothesis the weight w(ajl ) = fsum (T (N (j)), (ajl , . . . )) for all ajl ∈ Aj , where (i, j) ∈ A. Use the hypothesis to replace w(ajl ) in Formula 1. X min [cij (aik , ajl ) + fsum (T (N (j)), (ajl , . . . ))] = w(aik ) = (i,j)∈A

=

X

min fsum (T ({i} ∪ N (j)), (aik , ajl , . . . )) =

(i,j)∈A

=

X

ajl ∈Aj

ajl ∈Aj

fsum (T ({i} ∪ N (j)), (aik , . . . )) =

(i,j)∈A

= fsum (T (N (i)), (aik , . . . )) We have a similar lemma for AlgMaxFAP. Let fmax (T (N (i)), (aik , . . . )) be the value of fmax calculated for T (N (i)) and (aik , . . . ). Let waik be the weight calculated with the algorithm AlgMaxFAP for attribute aik . Lemma 3 w(aik ) = fmax (T (N (i)), (aik , . . . )) for all aik ∈ Ai , where i ∈ V . Proof By induction on the size of the sub-trees T (N (i)) in G. We begin with the leaves of G, and proceed by adding the remaining neighbor vertices. Basis. Let L be the leaves of G. Consider aik ∈ Ai , where i ∈ L. Clearly, T (N (i)) = ({i}, ∅). Hence, fsum (T (N (i)), (aik )) = 0. The weight w(aik ) is initialized to 0 in AlgSumFAP. Clearly, w(aik ) = fsum (T (N (i)), (aik )). Inductive step. Consider aik , where i ∈ V /L. With the induction hypothesis w(ajl ) = fsum (T (N (j)), (ajl , . . . )) for all ajl ∈ Aj , where (i, j) ∈ A. Use the hypothesis to replace w(ajl ) in Formula 2. w(aik ) = max

min max[cij (aik , ajl ), fmax (T (N (j)), (ajl , . . . ))] =

(i,j)∈A ajl ∈Aj

= max

min fmax (T ({i} ∪ N (j)), (aik , ajl , . . . )) =

(i,j)∈A ajl ∈Aj

= max fmax (T ({i} ∪ N (j)), (aik , . . . )) = (i,j)∈A

= fmax (T (N (i)), (aik , . . . )) By Lemma 2 resp. Lemma 3 we have the following corollary. Corollary 10 The value f ∗ returned by AlgSumFAP resp. AlgMaxFAP is the value of the optimal solution for SumFAP resp. MaxFAP, where G is a tree.

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Proof For all a1k ∈ A1 , where i ∈ V is the root vertex in G, the weight w(a1k ) is the value of the optimal solution if the selection in s1 is fixed to a1k . Since f ∗ = mina1k ∈Ai w(a1k ), f ∗ is optimal. From Lemma 1 and Corollary 10 we have the following theorem. Theorem 11 AlgSumFAP resp. AlgMaxFAP is a polynomial time algorithm for optimally solving SumFAP resp. MaxFAP on a tree.

6 Applicability and generalizations In previous sections we considered scenarios that are simple, i.e., each scenario is a set of attributes. There are many practical cases where such a description is appropriate. Such situations often arise when the attributes represent real-world objects that are, in a way, independent. For example, in economics or game theory, where one models strategic interactions among economic agents, attributes may represent actions that could be taken by a decision maker in an economic situation (scenario). The number of such actions, which are considered in practical situations, must be manageable. Therefore, the set describing the scenario consists of a moderate number of attributes. Another example where a flexible-attribute framework can be straightforwardly applied is when resolving lexical ambiguities in natural-language translation [4,7]. For each word in the sentence a set of its alternative translations is found. However, not all combinations of translations form a proper interpretation. To select the preferred interpretation a statistical model of the target language is used. To do this one might represent words with scenarios, translations with attributes, and the cost of transforming attributes with the statistical matching of the two translations. Additionally, a scenario graph might depend on the syntactic structure of the original sentence. There are, however, practical cases where scenarios are more appropriately described by complex structures, such as graphs, matrices etc., depending on the optimization problem with input uncertainty. One approach to dealing with such cases is that attributes represent possibly complex solutions of the instance of the optimization problem (scenario). For example, when solving the flexible 1-center problem one must enumerate, for each scenario, all the optimal solutions of the 1-center problem. The number of these is polynomial in the size of the problem instance. However, there are problems where enumerating all the solutions leads to an exponential explosion in the number of attributes. This can be mitigated by generating a subset of feasible solutions whose size depends on the available time for computation. If possible, the solutions should be generated in order of decreasing quality. Another approach is to consider a generalization of the two FAPs when several attributes have to be selected in each scenario. Although the total


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cost of transforming the solutions may seem to be difficult to calculate, it is tractable by solving the minimum assignment problem [11]. Yet another approach in cases where scenarios are described by complex structures is that each attribute is only a component of a solution of the problem instance. Hence, the attributes must be selected in such a way that they together represent the solution. For example, for the flexible minimum spanning tree problem, each attribute is an edge of the network. Thus, when selecting attributes (edges) one must be careful (1) to select those that constitute a minimum spanning tree, while (2) minimizing the cost of transforming the tree to another one. Hence, in this approach an exponential explosion in the number of attributes is avoided at the cost of a more complex attribute selection. A similar example is to introduce flexibility to the k-center problem, which is known to be N P-hard. Each attribute is a vertex of a network. Given a network instance, a feasible solution is any selection of k attributes. Clearly, one might consider such selections that form good (possibly 2approximate) solutions. In many cases it is easy to show the N P-hardness of flexible versions of optimization problems. For example, one can straightforwardly reduce the flexible minimum spanning tree or the flexible 1-center problem to corresponding FAP. Notice that the original, i.e., non-flexible, problems are not N P-hard. We have observed, however, that many other optimization problems, though polynomially solvable, become N P-hard when a flexibility requirement is added. An interesting problem would be to find out whether this holds for every problem in P, i.e., if the flexibility requirement is hard per se.

7 Conclusions In this article we proposed a new approach, called the flexibility approach, for dealing with uncertainty in optimization problems. To demonstrate the usefulness of the approach we presented a few applications and defined two optimization problems, SumFAP and MaxFAP. We proved that both these problems are N P-hard. Additionally, we showed the non-approximability of the two problems. Since it is very unlikely that any exact polynomial time algorithm for the problems will be found, we considered a simplification of the two problems where the input graph is a tree. For this case we described a polynomial time algorithm for solving the problems to optimality. Some questions about the problems SumFAP and MaxFAP are still open, such as (i) the solvability of the generalized problems where more than one attribute must be selected in each scenario, (ii) the exploitation of the other models for the transformation cost, and (iii) the solvability of the problems on various other input graphs, such as cycles, circulant graphs, grids, etc.

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