On the weak robustness of interval fuzzy matrices

Linear Algebra and its Applications 474 (2015) 243–259

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Linear Algebra and its Applications www.elsevier.com/locate/laa

On the weak robustness of interval fuzzy matrices Helena Myšková, Ján Plavka ∗,1 Department of Mathematics and Theoretical Informatics, Technical University, B. Němcovej 32, 04200 Košice, Slovakia

a r t i c l e

i n f o

Article history: Received 30 June 2014 Accepted 22 February 2015 Available online 6 March 2015 Submitted by P. Butkovic MSC: primary 08A72, 90B35 secondary 90C47 Keywords: Weak robustness Fuzzy matrices Interval matrix

a b s t r a c t A max–min (fuzzy) matrix A (operations max and min are denoted by ⊕ and ⊗, respectively) is called weakly robust if the only possibility to arrive at an eigenvector is to start the sequence (orbit) x, A ⊗ x, A2 ⊗ x, . . . by a vector that is itself an eigenvector. The weak robustness of a fuzzy matrix is extended to interval fuzzy matrices distinguishing two possibilities, that at least one matrix or all matrices from a given interval are weakly robust. Characterization of weak robustness of interval fuzzy matrices is presented and an O(n3 ) algorithm for checking the weak robustness of interval fuzzy matrices is described. © 2015 Elsevier Inc. All rights reserved.

1. Introduction In (max, min)-algebra (fuzzy algebra) the standard pair of operations plus and times is substituted by the pair of operations maximum and minimum, which are involved in many optimization problems. Fuzzy matrix operations are useful for expressing applications of fuzzy discrete dynamic systems, graph theory, scheduling, knowledge engineering,

* Corresponding author. 1

E-mail addresses: [email protected] (H. Myšková), [email protected] (J. Plavka). Supported by KEGA grant 032TUKE-4/2013, APVV grant 04-04-12.

http://dx.doi.org/10.1016/j.laa.2015.02.025 0024-3795/© 2015 Elsevier Inc. All rights reserved.

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cluster analysis, fuzzy systems and for describing diagnosis of technical devices [5,6,15] or medical diagnosis [12,13]. The problem studied in [12] and recently adapted in [9,10] has interesting applications in informatics and leads to the problem of finding the invariants and the greatest invariant of the fuzzy system. Fuzzy matrices with interval coefficients are also of practical importance, see [4,8,10]. The matrices of the fuzzy system for which an invariant is reached with any starting vector are called robust. Such matrices have been studied in [7]. The matrices for which an invariant is reached only if a starting vector is an eigenvector of the fuzzy matrix are called weakly robust. Efficient characterizations of such matrices are described in [11]. The values of vector or matrix inputs in practice are usually not exact numbers and they can be rather considered as values in some intervals. In this paper the properties of matrices and vectors with interval coefficients are studied and complete solution of both the possible weak robustness (at least one matrix from a given interval is weakly robust) and universal weak robustness (all matrices from a given interval are weakly robust) in fuzzy algebra are presented. Moreover polynomial algorithms for verifying the equivalent conditions and the corresponding properties of interval fuzzy matrices are suggested. The paper is organized as follows. In Section 2 we give definitions and basic results used in the paper. The known results describing the equivalent conditions on the weak robustness of fuzzy matrices are given in Section 3. Section 4 presents definitions and results on the weak interval robustness. Sections 5, 6 describe the criteria for the possible weak interval robustness and the universal weak interval robustness. 2. Background of the problem The fuzzy algebra B is a triple (B, ⊕, ⊗), where (B, ≤) is a bounded linearly ordered set with binary operations maximum and minimum, denoted by ⊕, ⊗. The least element in B will be denoted by O, the greatest one by I. By N we denote the set of all natural numbers and by N0 the set N0 = N ∪ {0}. For a given natural n ∈ N, we use the notations N and M for the set of all smaller or equal natural numbers, i.e., N = {1, 2, . . . , n} and M = {1, 2, . . . , m}, respectively. For any n ∈ N, B(n, n) denotes the set of all square matrices of order n and B(n) the set of all n-dimensional column vectors over B. If each entry of a matrix A ∈ B(n, n) is equal to O we shall denote it as A = O. The matrix operations over B are defined formally in the same manner (with respect to ⊕, ⊗) as matrix operations over any field. Let x = (x1 , . . . , xn ) ∈ B(n) and y = (y1 , . . . , yn ) ∈ B(n) be vectors. We write x ≤ y (x < y) if xi ≤ yi (xi < yi ) holds for each i ∈ N . For a given square matrix A ∈ B(n, n) let us denote sk (A) = A ⊕ A2 ⊕ · · · ⊕ Ak where Ai stands for the i-fold iterated product A ⊗ A ⊗ . . . ⊗ A. A square matrix is called diagonal if all its diagonal entries are arbitrary elements of B and off-diagonal entries are equal to O. A diagonal matrix with all diagonal entries equal to I is called a unit matrix and denoted by U . A matrix obtained from a diagonal matrix (unit matrix) by permuting the rows and/or columns is called a permutation

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245

matrix (unit permutation matrix) and denoted by P (PU ). If C = PUT ⊗ A ⊗ PU for some unit permutation matrix PU then we say that A and C are equivalent (denoted by A ∼ C), whereby the matrix PUT is the transpose of the matrix PU (if PU = (pij ) then PUT = (pji )) and PUT ⊗ PU = PU ⊗ PUT = U . For a matrix A ∈ B(n, n) the symbol G(A) = (N, E) stands for a complete, arcweighted digraph associated with A, i.e., the node set of G(A) is N , and the capacity of any arc (i, j) is aij . In addition, for given h ∈ B, the threshold digraph G(A, h) is the digraph with the node set N and with the arc set E = {(i, j); i, j ∈ N, aij ≥ h}. A path in the digraph G(A) = (N, E) is a sequence of nodes p = (i1 , . . . , ik+1 ) such that (ij , ij+1 ) ∈ E for j = 1, . . . , k. The number k is the length of the path p and it is denoted by (p). If i1 = ik+1 , then p is called a cycle and it is called an elementary cycle if moreover ij = im for j, m = 1, . . . , k. A digraph G = (N, E) without cycles is called acyclic. If G = (N, E) contains at least one cycle G is called cyclic. A matrix A is called generalized Hamiltonian permutation if all nonzero entries of A lie on a Hamiltonian cycle (the threshold digraph G(A, h), h = min {aij ; aij > O} is i,j∈N

elementary cycle containing all nodes). Let A ∈ B(n, n) and x ∈ B(n). The orbit O(A, x) of x = x(0) generated by A is the sequence x(0) , x(1) , x(2) , . . . , x(n) , . . . , where x(r) = Ar ⊗ x(0) for each r ∈ N. For a given matrix A ∈ B(n, n), the number λ ∈ B and the n-tuple x ∈ B(n) are the so-called eigenvalue of A and λ-eigenvector of A, respectively, if A ⊗ x = λ ⊗ x. The eigenspace V (A, λ) is defined as the set of all λ-eigenvectors of A with associated eigenvalue λ, i.e., V (A, λ) = {x ∈ B(n); A ⊗ x = λ ⊗ x}. There is a well-known connection between the entries in powers of matrices and paths in associated digraphs: (i, j)th entry akij in Ak is equal to the maximum capacity of a path k k from Pij , where Pij is the set of all paths of length k beginning at node i and ending at node j. If Pij denotes the set of all paths from i to j, then a∗ij = max{akij ; k = 1, 2, . . .} is the maximum capacity of a path from Pij and a∗jj is the maximum capacity of a cycle containing node j. Let a matrix A = (aij ) ∈ B(n, n) and λ ∈ B. Let us define the greatest λ-eigenvector x∗ (A, λ) corresponding to a matrix A and λ as x∗ (A, λ) =

 x∈V (A,λ)

x

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and denote mA =

 i,j∈N

aij ,

c(A) =

 

aij ,

c∗ (A) = (c(A), . . . , c(A))T ∈ B(n).

i∈N j∈N

It has been proved in [14] for a more general structure (distributive lattice) that the greatest λ-eigenvector x∗ (A, λ) of A exists for each λ ∈ B and a given matrix A ∈ B(n, n). A matrix A ∈ B(n, n) is ultimately periodic if there is a natural number p such that the following holds for some λ ∈ B and natural number R: Ak+p = λ ⊗ Ak for all k ≥ R. The smallest natural number p with the above property is called the period of A, denoted by per(A, λ). Let us denote the sets T (A, λ) and T ∗ (A, λ) as follows T (A, λ) = {x ∈ B(n); O(A, x) ∩ V (A, λ) = ∅}, T ∗ (A, λ) = {x ∈ B(n); x∗ (A, λ) ∈ O(A, x)}. The set T (A, λ) (T ∗ (A, λ)) allows us to describe matrices for which a λ-eigenvector (the greatest λ-eigenvector) is reached with any start vector. Let A ∈ B(n, n) be a matrix and λ ∈ B. The matrix A is called λ-robust and strongly λ-robust if T (A, λ) = B(n) and T ∗ (A, λ) = B(n) \ {x ∈ B(n); x < c∗ (A)}, respectively. Notice that the concept of λ-robustness and strongly λ-robustness have been studied in [9] and their interval versions in [7] and [10]. 3. Weak λ-robustness of fuzzy matrices It follows from the definitions of V (A, λ) and T (A, λ) that x ∈ V (A, λ) implies A ⊗x ∈ V (A, λ) and V (A, λ) ⊆ T (A, λ) ⊆ B(n) is fulfilled for every matrix A ∈ B(n, n) and λ ∈ B. Definition 3.1. Let A = (aij ) ∈ B(n, n), λ ∈ B. A is called weakly λ-robust if T (A, λ) = V (A, λ). Notice that a given matrix A is weakly λ-robust if Ak ⊗ x is not an eigenvector for any x and any k unless x is an eigenvector itself. The next lemma describes a universal criterion for weak λ-robustness in max–plus algebra and fuzzy algebra, see [1,9,11]. Lemma 3.1. Let A = (aij ) ∈ B(n, n), λ ∈ B. Then T (A, λ) = V (A, λ) if and only if (∀x ∈ B(n))[ A ⊗ x ∈ V (A, λ) ⇔ x ∈ V (A, λ)].

H. Myšková, J. Plavka / Linear Algebra and its Applications 474 (2015) 243–259

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Let us denote CA the square matrix which arose from the matrix A by deleting O columns and corresponding rows. Theorem 3.1. (See [11].) If A = (aij ) ∈ B(n, n), A = O and λ = O then A is weakly λ-robust if and only if CA contains no O columns. Theorem 3.2. (See [11].) Let A = O and λ > O. If A is weakly λ-robust then A contains no O column and no O row. Theorem 3.3. (See [11].) Let A = O and λ > O. If A is weakly λ-robust then A is a permutation matrix. Theorem 3.4. (See [11].) Let A = (aij ) ∈ B(n, n), A = O be a generalized Hamiltonian permutation matrix and λ > O. Then A is weakly λ-robust if and only if λ < c(A) or all entries on the Hamiltonian cycle are equal to λ (i.e. mA = c(A) = λ). Notice that any orbit of a non-diagonal matrix A with the period equal to 1 arrives at an eigenvector of A, so such matrices are λ-robust and never weakly λ-robust. Let us suppose now that A = (aij ) ∈ B(n, n) is a permutation matrix and λ ∈ B. Then the digraph G(A, c(A)) is the set of Hamiltonian cycles, say ci = (k1i , . . . , klii ) for i ∈ S = {1, . . . , s}. Without loss of generality the matrix A can be considered in block-diagonal form (denoted by A = (A1 , . . . , As )) ⎛

A1 ⎜ ⎜ O A=⎜ ⎜ .. ⎝ . O

O A2 O

⎞ O ⎟ O ⎟ ⎟, ⎟ ⎠ . . . As ... ...

(1)

where submatrices A1 , . . . , As are generalized Hamiltonian permutation matrices corresponding to the Hamiltonian cycles c1 , . . . , cs . Theorem 3.5. (See [11].) Let A ∈ B(n, n), A = O, A = (A1 , . . . , As ), s ≥ 2 be a block-diagonal permutation matrix and λ > O. Then A is weakly λ-robust if and only if (∀i ∈ S)[λ < c(Ai ) ∨ λ = c(Ai ) = mAi ]. Theorem 3.6. (See [11].) Let A ∈ B(n, n), λ ∈ B and A ∼ C. Then A is weakly λ-robust if and only if C is weakly λ-robust. 4. Weak λ-robustness of interval fuzzy matrices In this section we shall deal with matrices with interval elements. Sufficient and necessary conditions for an interval matrix to be weakly λ-robust will be proved. In addition

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we introduce a polynomial algorithm to check the weak λ-robustness of interval fuzzy matrices. Similarly to [3,4,8] we define an interval matrix A. Definition 4.1. Let A, A ∈ B(n, n). An interval matrix A with bounds A and A is defined as follows A = [A, A] =



 A ∈ B(n, n); A ≤ A ≤ A .

Investigating interval weak λ-robustness for an interval matrix A following questions can arise. Is A weakly λ-robust for some A ∈ A or for all A ∈ A? Definition 4.2. Let A be an interval matrix and λ ∈ B. A is called (i) possibly weakly λ-robust if there exists a matrix A ∈ A such that A is weakly λ-robust, (ii) universally weakly λ-robust if each matrix A ∈ A is weakly λ-robust. The notion of equivalence of fuzzy matrices can be generalized into interval forms of fuzzy matrices as follows. For a given interval matrix A and a unit permutation matrix PU define the interval matrix C such that

 C = PUT ⊗ A ⊗ PU = PUT ⊗ A ⊗ PU ; A ∈ A and we say that A and C are equivalent (denoted by A ∼ C). By Theorem 3.6 simultaneous permutations of rows and columns of the matrix A have no influence on conditions of weak λ-robustness describing in Theorem 3.1 and Theorem 3.4. Thus we can formulate the generalization of Theorem 3.6. Theorem 4.1. Let A, C be interval matrices, A ∼ C and λ ∈ B. Then A is possibly (universally) weakly λ-robust if and only if C is possibly (universally) weakly λ-robust. 5. Possible weak λ-robustness of interval fuzzy matrices Sufficient and necessary conditions for an interval matrix to be possibly weakly λ-robust will be proved in this section. Let an interval matrix A = [A, A], A ∈ A and k ∈ N be given. Denote the k × k matrix consisting of i1 st, . . . , ik th columns and corresponding rows of A by

A

i1 i1

i2 i2

. . . ik . . . ik

⎛

ai1 i1 ⎜ .. =⎝ . aik i1

⎞ ai1 ik .. ⎟ . . ⎠ . . . aik ik ...

H. Myšková, J. Plavka / Linear Algebra and its Applications 474 (2015) 243–259



i1 i1

Definition 5.1. The column i of A

249

i2 i2

... ...

ik ik

is called removable if max ais i = 1≤s≤k

i1 i2 . . . ik O ∧ max a si = O, or equivalently, the column i of matrices A 1≤s≤n i1 i2 . . . ik and A is O column. Notice that each O column of A is removable one. (0) Let us denote A = A and for j = 1, . . . , r, r ≤ n recurrently define the kj × kj matrix (j)

A

(j−1)

=A

(j−1)

which arose from the matrix A rows.

ij−1 1

ij−1 2

...

ij−1 kj−1

ij−1 1

ij−1 2

...

ij−1 kj−1

(2)

by deleting all removable columns and corresponding

Example 5.1. Let B = [0, 10] and A, A have the form ⎛

0 ⎜0 ⎜ ⎜ A = ⎜0 ⎜ ⎝1 0

1 1 1 1 1

0 0 0 0 0

1 1 1 1 1

⎛

⎞ 0 0⎟ ⎟ ⎟ 0⎟, ⎟ 0⎠ 0

0 ⎜0 ⎜ ⎜ A = ⎜1 ⎜ ⎝1 1

1 1 1 1 1

0 0 0 0 1

1 1 1 1 1

⎞ 0 0⎟ ⎟ ⎟ 0⎟. ⎟ 0⎠ 0

(0)

(1)

Put A = A. We shall recurrently construct the sequence of matrices A column 5 of A is removable then we get (1)

1 2 3 4 1 2 3 4

(0)

A

=A

(1)

Here the column 3 of A

(2)

0 ⎜0 ⎜ =⎜ ⎝1 1

1 1 1 1

0 0 0 0

(2)

. The

⎞ 1 1⎟ ⎟ ⎟. 1⎠ 1

is removable as well and then we get

A

⎛

,A

(1)

=A

1 1

2 2

4 4

⎛

⎞ 0 1 1 ⎜ ⎟ = ⎝0 1 1⎠. 1 1 1

Theorem 5.1. Let A be an interval matrix and A = O. Then A is possibly weakly O-robust (r) if and only if there exists r ∈ N such that A contains no O columns.

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(r)

Proof. Suppose that there exists r ∈ N such that A contains no O columns. Since (r) A arose as results of recurrent process (2), i.e., by deleting all removable columns and (0)

corresponding rows in A

,A

(1)

(r−1)

,...,A

then we obtain that max a s = O for each ∈N

{ir1 , . . . , irkr }

s∈ / (it follows from the definition of removable column, i.e., if column s is removable then column s is O column of A). Let us construct the matrix A˜ = (˜ auv ), where  auv , for u ∈ N ∧ v ∈ {ir1 , . . . , irkr } a ˜uv = a uv , otherwise. (r) It is clear that A˜ ∈ A, CA˜ = A contains no O columns and A˜ is weakly O-robust by Theorem 3.1. Let us assume that A is possibly weakly O-robust, i.e., there exists a matrix A ∈ A which is weakly O-robust. By Theorem 3.1 the O-robustness of A is equivalent to the condition that the submatrix CA contains no O columns. For the sake of a contradiction (r) let us suppose that for each r ∈ N the matrix A contains at least one O column. Put

CA = A

j1 j1

j2 j2

. . . j . . . j

(3)

and (r)

A

=A

(0)

Since the matrix A inclusion

(r−1)

ir−1 1

ir−1 2

. . . ir−1 kr−1

ir−1 1

ir−1 2

. . . ir−1 kr−1

, 1 ≤ r ≤ n.

(4)

= A contains at least one O column (removable column) then the

N \ {i01 , i02 , . . . , i0k0 } ⊆ N \ {j1 , j2 , . . . , j } (1)

holds true. If N \ {i01 , i02 , . . . , i0k0 } = N \ {j1 , j2 , . . . , j } then A at least one O column. This is a contradiction. In the second case if

and hence CA contains

N \ {i01 , i02 , . . . , i0k0 } ⊂ N \ {j1 , j2 , . . . , j } (1)

then there is at least one index s1 ∈ {i01 , i02 , . . . , i0k0 } such that the column s1 of A is removable. Since CA contains no O column then s1 is element of N \ {j1 , j2 , . . . , j }. (2)

Similarly, the matrix A

contains at least one O column, say s2 ∈ {i11 , i12 , . . . , i1k1 } \{s1 }, (2)

such that the column s2 of A

is removable and s2 ∈ N \{j1 , j2 , . . . , j }. By repeating of (r)

the above process at most r times, r ≤ n we obtain the matrix A with at least one O column which is also O column in CA . This is a contradiction with the assumption. 2

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251

Let A = (aij ) ∈ B(n, n) be a given matrix and Pn be the set of all permutations of N . The bottleneck assignment problem is defined as follows: for a given matrix A to find a permutation π ∈ Pn which maximizes the objective function min aiπ(i) . i∈N

The bottleneck assignment problem has been studied by Gabow and Tarjan [2], who gave an algorithm for solving the bottleneck assignment problem with worst case complexity √ O(n2 n log n ). Denote ap(A) = max min aiπ(i) . π∈Pn i∈N

The next assertion describes the necessary condition of possible weak λ-robustness for λ > O. Notice that if A is a permutation matrix then c(A) = ap(A). Lemma 5.1. Let A be an interval matrix and λ > O. If A is possibly weakly λ-robust then there is a permutation π ∈ Pn such that (i) a kl = O for (k, l) ∈ / {(1, π(1)), . . . , (n, π(n))}, (ii) aiπ(i) > O for i = 1, . . . , n. Proof. Suppose that A is possibly weakly λ-robust and λ > O, i.e., there is a weakly λ-robust matrix A ∈ A. By Theorem 3.2 A contains no O column and no O row and by Theorem 3.3 A is a permutation one. Hence there is a permutation π ∈ Pn such that (aiπ(i) ≥) aiπ(i) > O for i = 1, . . . , n and (a kl ≤) akl = O for (k, l) ∈ / {(1, π(1), . . . , (n, π(n)} and the assertion follows. 2 Let A be an interval matrix and λ > O. Suppose that A is possibly weakly λ-robust. By Lemma 5.1 there is a permutation π ∈ Pn such that (i) a kl = O for (k, l) ∈ / {(1, π(1)), . . . , (n, π(n))}, (ii) aiπ(i) > O for i = 1, . . . , n. Denote S = {(ir , jr ); a ir ,jr > O} = {(i1 , j1 ), . . . , (ik , jk )} (according to (i) we get that iu = iv for u = v and ja = jb for a = b), a matrix DA = (duv ), where  duv =

O, auv ,

if (∃q)[u = iq ∧ v = jq ] ∨ (∃q)[v = jq ∧ u = iq ], otherwise

(5)

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and sets Sn = {π ∈ Pn ; {(i1 , j1 ), . . . , (ik , jk )} ⊆ {(1, π(1)), . . . , (n, π(n))}}, Snopt = {π ∈ Sn ; ap(DA ) = min aiπ(i) }. i∈N

In the next part of the section we are looking for a weakly λ-robust matrix A ∈ A, A ≤ DA with c(A) ≤ ap(DA ) whereby entries of A are as large as possible. For a given A ∈ A and π ∈ Pn define auxiliary permutation matrices Aπ = (aπuv ) as follows  auv , if (u, v) ∈ {(1, π(1)), . . . , (n, π(n))}, aπuv = O, otherwise. By Theorem 3.6 suppose that for π ∈ Sn the matrix Aπ = (Aπ,1 , . . . , Aπ,p ) (A π = π,i (Aπ,1 , . . . , A π,p )) is block-diagonal permutation with Aπ,i = (aπ,i uv ) (A π,i = (a uv )), Aπ,i are generalized Hamiltonian permutation matrices with c(Aπ,i ) ≥ c(Aπ ) for i ∈ {1, . . . , p} π ) as follows and define the block-diagonal permutation matrix Fπ = (fuv

π fuv

⎧ c(Aπ ), ⎪ ⎪ ⎪ ⎪ ⎨ a π,i , uv = ⎪ aπ,i , ⎪ uv ⎪ ⎪ ⎩ O,

π,i if aπ,i uv ≥ c(Aπ,i ) = c(Aπ ) ≥ a uv π,i if aπ,i uv ≥ a uv > c(Aπ,i ) = c(Aπ )

(6)

if aπ,i uv ≥ c(Aπ,i ) > c(Aπ ) otherwise.

Since the matrix Fπ plays crucial role for the next assertion which describes the equivalent conditions for possibly weakly λ-robustness we present the construction of Fπ in the following example. Example 5.2. Let B = [0, 10], A, A have the forms ⎛

0 ⎜0 ⎜ ⎜ A = ⎜0 ⎜ ⎝0 0 and π1 = (1, 3, 2, 5, 4), π2 follows ⎛ 3 ⎜0 ⎜ ⎜ Aπ1 = ⎜ 0 ⎜ ⎝0 0

0 0 0 0 0

0 3 0 0 0

0 0 0 0 2

⎞ 0 0⎟ ⎟ ⎟ 0⎟, ⎟ 4⎠ 0

⎛

3 ⎜9 ⎜ ⎜ A = ⎜2 ⎜ ⎝7 9

2 0 3 1 1

1 4 1 8 1

⎞ 1 1⎟ ⎟ ⎟ 1⎟ ⎟ 5⎠ 7

2 1 1 1 6

= (2, 3, 1, 5, 4) be given permutations. Then Aπ1 , Aπ1 look as

0 0 3 0 0

0 4 0 0 0

0 0 0 0 6

⎞ 0 0⎟ ⎟ ⎟ 0⎟, ⎟ 5⎠ 0

⎛

Aπ2

0 ⎜0 ⎜ ⎜ = ⎜2 ⎜ ⎝0 0

2 0 0 0 0

0 4 0 0 0

0 0 0 0 6

⎞ 0 0⎟ ⎟ ⎟ 0⎟ ⎟ 5⎠ 0

H. Myšková, J. Plavka / Linear Algebra and its Applications 474 (2015) 243–259

253

and c(Aπ1 ) = 3, c(Aπ2 ) = 2. By (6) we can construct Aπ1 ,1 , Aπ1 ,2 , Aπ1 ,3 , Fπ1 and Aπ2 ,1 , Aπ2 ,2 , Fπ2 : ⎛

Aπ1 ,1

  = 3 , Aπ1 ,2 =

⎛

0 ⎜ Aπ2 ,1 = ⎝ 0 2

2 0 0





0 3

4 0

⎞

0 ⎟ 4⎠, 0

0 6

, Aπ1 ,3 =

Aπ2 ,2 =

0 6

5 0

,

3 ⎜0

⎜ 5 ⎜ , Fπ1 = ⎜ 0 ⎜ 0 ⎝0 0 ⎛ 0 2 ⎜0 0 ⎜ ⎜ Fπ2 = ⎜ 2 0 ⎜ ⎝0 0 0 0

0 0 3 0 0

0 3 0 0 0

0 3 0 0 0

0 0 0 0 6

⎞ 0 0 0 0⎟ ⎟ ⎟ 0 0⎟, ⎟ 0 5⎠ 6 0 ⎞ 0 0⎟ ⎟ ⎟ 0⎟. ⎟ 5⎠ 0

It is easily to check that Fπ1 is weakly 3-robust by Theorem 3.5 and hence the interval matrix A is possibly weakly 3-robust. π1 and By Theorem 3.5 the matrix Fπ2 is not weakly 2-robust because 2 = c(Fπ2 ) < f23 Fπ2

1 2 3 1 2 3

⎛

0 ⎜ = ⎝0 2

2 0 0

⎞ 0 ⎟ 3⎠. 0

Theorem 5.2. Let A be an interval matrix and λ > O. Then A is possibly weakly λ-robust if and only if there is a permutation π ∈ Pn such that (i) a kl = O for (k, l) ∈ / {(1, π(1)), . . . , (n, π(n))}, (ii) aiπ(i) > O for i = 1, . . . , n, (iii) λ < ap(DA ) ∨ [λ = ap(DA ) ∧ (∃σ ∈ Snopt )[Fσ is weakly λ-robust]]. Proof. “⇒” We shall prove the implication by a contradiction. Suppose that A is an interval matrix, λ > O and A is possibly weakly λ-robust, i.e., there is a weakly λ-robust matrix A˜ ∈ A and by Lemma 5.1 there is a permutation π ∈ Pn such that (i) a kl = O for (k, l) ∈ / {(1, π(1)), . . . , (n, π(n))}, (ii) aiπ(i) > O for i = 1, . . . , n. Since A˜ is weakly λ-robust then by Theorem 3.5 and Lemma 5.1 there is a permutation τ ∈ Sn such that A˜ = A˜τ = (A˜τ,1 , . . . , A˜τ,p ) is a block-diagonal permutation matrix with A˜τ,i = (aτ,i uv ), Aτ,i are generalized Hamiltonian permutation matrices and (∀i, 1 ≤ i ≤ p)[λ < c(A˜τ,i ) ∨ λ = c(A˜τ,i ) = mA˜τ,i ].

(7)

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For the contrary suppose that λ ≥ ap(DA ) ∧ [λ = ap(DA ) ∨ (∀σ ∈ Snopt )[Fσ is not weakly λ-robust]]. Consider two cases. Case 1. λ > ap(DA ). In this case it follows from the definition of DA that (∀α ∈ Sn )[λ > ap(DA ) ≥ ap(Aα ) = c(Aα )]. Particularly for α = τ we get λ > ap(DA ) ≥ ap(Aτ ) ≥ c(A˜τ ) and by Theorem 3.5 we obtain a contradiction with the weak λ-robustness of A˜τ . Case 2. λ = ap(DA ) and (∀σ ∈ Snopt )[Fσ is not weakly λ-robust]. The inequalities λ ≤ c(A˜τ ) ≤ c(A˜τ,i )) follows from (7) and hence we obtain λ ≤ c(A˜τ ) = ap(A˜τ ) ≤ ap(Aτ ) ≤ ap(DA ) = λ ⇒ c(A˜τ ) = c(Aτ ) = λ.

(8)

˜τ implies the weak λ-robustness of We shall show that the weak λ-robustness of A τ Fτ = (fuv ), where

τ fuv

⎧ ⎪ ⎨ c(Aτ ) = λ, = aτ,i uv , ⎪ ⎩ O,

τ,i if aτ,i uv ≥ c(Aτ,i ) = c(Aτ ) = λ ≥ a uv

if aτ,i uv ≥ c(Aτ,i ) > c(Aτ ) = λ

(9)

otherwise.

Notice that if λ = c(A˜τ,i ) = mA˜τ,i then the second possibility of (6) τ,i aτ,i ˜τ,i uv ≥ a uv ≥ a uv > c(Aτ,i ) = λ

(10)

τ,i does not occur (if there is u, v such that aτ,i ˜τ,i uv ≥ a uv > c(Aτ,i ) = λ then λ = uv ≥ a c(A˜τ,i ) < mA˜τ,i and by Theorem 3.5 A˜ is not weakly λ-robust). For a given index i ∈ {1, . . . , p} we shall prove two implications following from (7):

(i) λ = c(A˜τ,i ) = mA˜τ,i ⇒ λ = c(Fτ,i ) = mFτ,i . ˜ If λ = c(A˜τ,i ) = mA˜τ,i then each non-O element a ˜τ,i uv of Aτ,i is equal to λ and by (8) we have that τ,i ˜ ˜ aτ,i ˜τ,i uv ≥ c(Aτ,i ) ≥ c(Aτ,i ) ≥ c(Aτ ) = c(Aτ ) = λ = a uv ≥ a uv

(11)

τ,i and hence by (9) we get fuv = λ. ˜ (ii) λ < c(Aτ,i ) ⇒ λ < c(Fτ,i ). ˜ If λ < c(A˜τ,i ) then each non-O element a ˜τ,i uv of Aτ,i is greater than λ and by (8) we obtain

˜ ˜ aτ,i uv ≥ c(Aτ,i ) ≥ c(Aτ,i ) > λ = c(Aτ ) = c(Aτ ) τ,i and hence by (9) we get fuv = aτ,i uv > λ.

(12)

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255

τ As a consequence of (10), (11) and (12) matrix Fτ = (fuv ) has the form (9) and the weak λ-robustness of Fτ follows from the fulfilled conditions of Theorem 3.5. This is a contradiction with the assumption that Fτ is not weakly λ-robust. “⇐” The converse implication trivially follows for the case that λ = ap(DA ) ∧ (∃σ ∈ opt Sn )[Fσ is weakly λ-robust]. Thus Fσ ∈ A and A is possibly weakly λ-robust. Suppose that λ < ap(DA ) = min aiπ(i) for π ∈ Snopt . Then by Theorem 3.5 matrix i∈N

aπij ) with entries A˜π = (˜  a ˜πij =

aij ,

if (i, j) ∈ {(1, π(1)), . . . , (n, π(n))}

O,

otherwise

is weakly λ-robust and hence A is possibly weakly λ-robust.

2

We can use the obtained results to derive a simple procedure for checking the possible weak λ-robustness of a given interval matrix A = [A, A] and O ≤ λ < ap(DA ). Algorithm (Possible Weak Robustness). Input. A = [A, A] and O ≤ λ < ap(DA ). Output. ‘yes’ in variable pwr if A is weakly λ-robust; ‘no’ in pwr otherwise. begin (i) If λ = O and A = O then pwr = ‘yes’; (j)

(ii) For j = 1, . . . , r, r ≤ n compute A

;

(r)

(iii) If λ = O ∧ A = O ∧ A contains no O columns then pwr = ‘yes’; (iv) If there is π ∈ Pn such that (a) a kl = O for (k, l) ∈ / {(1, π(1)), . . . , (n, π(n))}, (b) aiπ(i) > O for i = 1, . . . , n then compute DA else pwr = ‘no’; (v) If O < λ < ap(DA ) then pwr = ‘yes’ else pwr = ‘no’; end Theorem 5.3. Let A be an interval matrix and O ≤ λ < ap(DA ). The algorithm Possible Weak Robustness correctly decides whether a matrix A is possibly weakly λ-robust in O(n3 ) arithmetic operations. (j)

Proof. The number of operations for constructing A , j = 1, . . . , r in step (ii) and checking the condition of step (iii) is rO(n2 )+O(n2 ) ≤ O(n3 ). Step (iv) needs O(n2 ) operations √ for looking for a permutation π and computing the value ap(DA ) needs O(n2 n log n ) operations [2]. The others steps of the algorithm need at most O(n2 ) operations, thus the √ complexity of all steps of the algorithm is O(n2 ) + O(n3 ) + O(n2 n log n ) = O(n3 ). 2

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Notice that Theorem 5.2 implies that the computational complexity of a procedure based on checking all matrices Fπ for π ∈ Snopt and which decides whether A is possibly weakly ap(DA )-robust can be exponentially large. Moreover, we are able neither to suggest polynomial algorithm nor to prove NP-completeness of the above problem. We illustrate the hardness of the conditions of Theorem 5.2 for λ = ap(DA ) in the following example. Example 5.3. Let B = [0, 10] and A, A, DA have the form ⎛

0 ⎜0 ⎜ ⎜ A = ⎜0 ⎜ ⎝0 0

0 0 3 0 0

0 0 0 0 0

1 0 0 0 0

⎞ ⎛ 0 1 ⎟ ⎜ 0⎟ ⎜9 ⎟ ⎜ 0⎟,A = ⎜1 ⎟ ⎜ ⎝7 0⎠ 0 9

1 0 5 1 1

1 9 1 8 1

2 1 1 1 1

⎞ ⎛ 1 0 ⎟ ⎜ 1⎟ ⎜9 ⎟ ⎜ 1 ⎟ , DA = ⎜ 0 ⎟ ⎜ ⎝7 9⎠ 7 9

0 0 5 0 0

0 9 0 8 1

2 0 0 0 0

⎞ 0 1⎟ ⎟ ⎟ 0⎟. ⎟ 9⎠ 7

Then we get S = {(1, 4), (3, 2)}, ap(DA ) = 2 (= a14 ⊗ a21 ⊗ a32 ⊗ a43 ⊗ a55 ), S5opt = {π1 = (4, 1, 2, 3, 5), π2 = (4, 3, 2, 1, 5), π3 = (4, 3, 2, 5, 1)}. By Theorem 5.2 it follows that A is possibly weakly λ-robust for λ < 2. In the case when λ = 2 we shall show that each permutation from the set S5opt has to be considered. ⎛

0 ⎜9 ⎜ ⎜ A π1 = ⎜ 0 ⎜ ⎝0 0 ⎛ 0 ⎜0 ⎜ ⎜ Aπ2 = ⎜ 0 ⎜ ⎝7 0 ⎛ 0 ⎜0 ⎜ ⎜ Aπ3 = ⎜ 0 ⎜ ⎝0 9

0 0 5 0 0

0 0 0 8 0

2 0 0 0 0

0 0 5 0 0

0 9 0 0 0

2 0 0 0 0

0 0 5 0 0

0 9 0 0 0

2 0 0 0 0

⎞ 0 0⎟ ⎟ ⎟ 0⎟, ⎟ 0⎠ 7 ⎞ 0 0⎟ ⎟ ⎟ 0⎟, ⎟ 0⎠ 7 ⎞ 0 0⎟ ⎟ ⎟ 0⎟, ⎟ 9⎠ 0

⎛

0 ⎜2 ⎜ ⎜ Fπ1 = ⎜ 0 ⎜ ⎝0 0 ⎛ 0 ⎜0 ⎜ ⎜ Fπ2 = ⎜ 0 ⎜ ⎝2 0 ⎛ 0 ⎜0 ⎜ ⎜ Fπ3 = ⎜ 0 ⎜ ⎝0 2

0 0 3 0 0

0 0 0 2 0

2 0 0 0 0

0 0 3 0 0

0 9 0 0 0

2 0 0 0 0

0 0 3 0 0

0 9 0 0 0

2 0 0 0 0

⎞ 0 0⎟ ⎟ ⎟ 0⎟, ⎟ 0⎠ 7 ⎞ 0 0⎟ ⎟ ⎟ 0⎟, ⎟ 0⎠ 7 ⎞ 0 0⎟ ⎟ ⎟ 0⎟. ⎟ 2⎠ 0

π1 By Theorem 3.5 the matrix Fπ1 is not weakly 2-robust because 2 = c(Fπ1 ) < f32 and

H. Myšková, J. Plavka / Linear Algebra and its Applications 474 (2015) 243–259

Fπ1

1 2 3 4 1 2 3 4

⎛

0 ⎜2 ⎜ =⎜ ⎝0 0

0 0 3 0

0 0 0 2

257

⎞ 2 0⎟ ⎟ ⎟. 0⎠ 0

On the other side it is possible very easily to check that the matrices Fπ2 and Fπ3 are weakly 2-robust by Theorem 3.5 and hence the interval matrix A is possibly weakly 2-robust. 6. Universal weak λ-robustness of interval fuzzy matrices Let A be an interval matrix. By Theorem 4.1 we can suppose that A, A, CA have the forms



A11 A21 A 11 O , A= A= , (13) A 21 O A21 A22

1 2 ... k 1 2 ... k A 11 = A . (14) = CA , A11 = A 1 2 ... k 1 2 ... k Theorem 6.1. Let A be an interval matrix, λ = O and A, A have the form (13). Then A is universally weakly O-robust if and only if CA = A 11 contains no O columns and each off-diagonal element of A22 is equal to O. Proof. Suppose that CA = A 11 contains no O columns, each off-diagonal element of A22 is equal to O and A is not universally weakly O-robust, i.e., by Theorem 3.1 there is A ∈ A such that CA contains (say) jth column with O entries whereby j ∈ / {1, . . . , k} (hence it follows that ajj = O). Moreover each off-diagonal element of A22 of (13) is equal to O than we obtain that max aij = O. This is a contradiction with definition i∈N

of CA . To prove the converse implication suppose that each matrix A ∈ A is weakly O-robust, specially for A we get that CA = A 11 contains no O columns by Theorem 3.1. For the contrary suppose now that there are indices j, l such that ajl = O, j, l ∈ {k + 1, . . . , n} and j = l. Define an auxiliary matrix A˜ = (˜ auv ) as follows  a ˜uv =

auv , a uv ,

if (u, v) = (j, l) otherwise.

˜ and corresponding row of A˜ the submatrix After deleting jth column (O column of A) CA˜ contains lth column with entries equal to O which contradicts the weak O-robustness ˜ 2 of A. A square interval matrix A = (aij ) is called interval diagonal if all its diagonal entries are intervals [a ii , aii ] with a ii > O and off-diagonal entries are intervals [O, O]. An

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258

interval matrix obtained from an interval diagonal matrix by permuting the rows and/or columns is called an interval permutation matrix. Lemma 6.1. Let A be an interval matrix and λ > O. If A is universally weakly λ-robust then A is an interval permutation matrix. Proof. Suppose that A is universally weakly λ-robust and A is not interval permutation matrix. Then there exists a matrix A ∈ A which is not permutation matrix. This is a contradiction with Theorem 3.3. 2 Assume that A is an interval permutation matrix such that c(A) = a 1π(1) ⊗ · · · ⊗ a nπ(n) (= ap(A)), c(A) = a1π(1) ⊗ · · · ⊗ anπ(n) (= ap(A)) and A = (A 1 , . . . , A p ),

A = (A1 , . . . , Ap ).

Theorem 6.2. Let A be an interval matrix and λ > O. Then A is universally weakly λ-robust if and only if A is an interval permutation matrix such that matrices A, A are weakly λ-robust whereby A = (A 1 , . . . , A p ), A = (A1 , . . . , Ap ) and (∀A i ∈ B(s, s), 1 < s, 1 ≤ i ≤ p)[λ = c(A i ) = c(A) ⇒ A i = Ai ]. Proof. Suppose that A is an interval permutation matrix such that A, A are weakly λ-robust matrices (hence O < λ ≤ c(A)), A = (A 1 , . . . , A p ), A = (A1 , . . . , Ap ) and (∀A i ∈ B(s, s), 1 < s, 1 ≤ i ≤ p)[λ = c(A i ) = c(A) ⇒ A i = Ai ]. Further let A ∈ A be an arbitrary matrix. It is clear that A is a permutation matrix, A = (A1 , . . . , Ap ) and O < λ ≤ c(A) ≤ c(A). Then either λ < c(A) implies that A is weakly λ-robust or λ = c(A i ) = c(A) = c(A) together with the assumption A i = Ai = Ai implies that A weakly λ-robust too. To prove the converse implication suppose A is universally weakly λ-robust and then by Lemma 6.1 A is an interval permutation matrix. Moreover we can assume that each matrix A ∈ A has the form A = (A1 , . . . , Ap ). In particular for the matrices A and A we have that A = (A 1 , . . . , A p ) and A = (A1 , . . . , Ap ). Since matrices A, A are weakly λ-robust, by Theorem 3.5 we obtain (∀i ∈ {1, . . . , p})[λ < c(A i ) ∨ λ = c(A i ) = mA i ] ∧ [λ < c(Ai ) ∨ λ = c(Ai ) = mAi ]. Hence we get that O < λ ≤ c(A) ≤ c(A) ≤ c(A). For the contrary suppose now that there is an index i ∈ {1, . . . , p} and s > 1 such that A i ∈ B(s, s), λ = c(A) = c(A i ) and A i = Ai , i.e., there exist row index r and column index v of A i , Ai , r = v such that a rv < arv . Define the matrix A˜ = (˜ aut ) as follows  a ˜ut =

aut , a ut ,

if (u, t) = (r, v) otherwise.

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259

Thus λ = c(A) = c(A i ) = c(A˜i ) < aut ≤ mA˜i and by Theorem 3.5 it contradicts the ˜ 2 weak λ-robustness of A. In fact, Theorem 6.1 and Theorem 6.2 turn the problem of universal weak λ-robustness to the question whether the given interval matrix fulfills the necessary and sufficient conditions of the theorems. Now we show that this question can be answered by a simple O(n2 ) algorithm. It is based on the fact that we need O(n2 ) operations to find c(A) and to check the weak λ-robustness of A, A, O columns of CA , off-diagonal element of A22 and the condition (∀A i ∈ B(s, s), 1 < s, 1 ≤ i ≤ p)[λ = c(A i ) = c(A) ⇒ A i = Ai ]. Thus the complexity of checking universal weak λ-robustness of a given interval matrix is 6.O(n2 ) = O(n2 ). Acknowledgement The authors would like to thank the referee for his/her valuable remarks and suggestions that helped to increase the clarity of arguments in the paper. This work was supported by the Slovak Cultural and Education Agency No. 032TUKE-4/2013 and also by the Slovak Research and Development Agency under the contract No. APVV-0404-12. References [1] P. Butkovič, H. Schneider, S. Sergeev, Recognising weakly stable matrices, SIAM J. Control Optim. 50 (5) (2012) 3029–3051. [2] H.N. Gabow, R.E. Tarjan, Algorithms for two bottleneck optimization problems, J. Algorithms 9 (1988) 411–417. [3] M. Gavalec, K. Zimmermann, Classification of solutions to systems of two-sided equations with interval coefficients, Int. J. Pure Appl. Math. 45 (2008) 533–542. [4] V. Kreinovich, A. Lakeyev, J. Rohn, R. Kahl, Computational Complexity and Feasibility of Data Processing and Interval Computations, Kluwer Academic Publishers, Dordrecht, Boston, London, 1998. [5] V. Lacko, Š. Berežný, The color-balanced spanning tree problem, Kybernetika 41 (2005) 539–546. [6] M. Molnárová, J. Pribiš, Matrix period in max-algebra, Discrete Appl. Math. 103 (2000) 167–175. [7] M. Molnárová, H. Myšková, J. Plavka, The robustness of interval fuzzy matrices, Linear Algebra Appl. 438 (8) (2013) 3350–3364. [8] H. Myšková, Control solvability of interval systems of max-separable linear equations, Linear Algebra Appl. 416 (2006) 215–223. [9] J. Plavka, P. Szabó, On the λ-robustness of matrices over fuzzy algebra, Discrete Appl. Math. 159 (5) (2011) 381–388. [10] J. Plavka, On the O(n3 ) algorithm for checking the strong robustness of interval fuzzy matrices, Discrete Appl. Math. 160 (2012) 640–647. [11] J. Plavka, On the weak robustness of fuzzy matrices, Kybernetika 49 (1) (2013) 128–140. [12] E. Sanchez, Resolution of eigen fuzzy sets equations, Fuzzy Sets and Systems 1 (1978) 69–74. [13] E. Sanchez, Medical diagnosis and composite relations, in: M.M. Gupta, R.K. Ragade, R.R. Yager (Eds.), Advances in Fuzzy Set Theory and Applications, North-Holland, Amsterdam, New York, 1979, pp. 437–444. [14] Yi-Jia Tan, On the eigenproblem of matrices over distributive lattices, Linear Algebra Appl. 374 (2003) 96–106. [15] K. Zimmernann, Extremální algebra, Ekon. ústav ČSAV, Praha, 1976 (in Czech).