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C* Algebras, Volume 5: Selected Topics by Corneliu Constantinescu

By Corneliu Constantinescu

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Only if”: Let y be a solution. 25 w ≤ A∗ ⊗ u and so y = A ⊗ w ≤ A ⊗ A∗ ⊗ u = b. Since yk ∈ Z for k ∈ J we also have ˜ A ⊗ w = y ≤ b. 1 then ˆ l ≤ y = A ⊗ w ≤ A ⊗ A∗ ⊗ b˜ = x. 9 as above, hence xˆ is the greatest solution. 7)T and J = {1, 3} (l is not specified). 8, 4)T . 10 xˆ is the greatest solution to the BMISDI provided that l ≤ xˆ (otherwise there is no solution). 2 Max-algebra and Combinatorial Optimization There is a number of combinatorial and combinatorial optimization problems closely related to max-algebra.

3. 1. 4 enables us to compile the following algorithm. 5 BMISDI Input: B ∈ Rn×n , u, l ∈ Rn and J ⊆ N . 2) or an indication that no such vector exists. 1. 2. 3. 4. 5. A := (B), x := u xj := xj for j ∈ J z := A∗ ⊗ x, x := A ⊗ z If l x then stop (no solution) If l ≤ x and xj ∈ Z for j ∈ J then stop else go to 2. 6 [30] The algorithm BMISDI is correct and requires O(n3 + n2 L) operations of addition, maximum, minimum, comparison and integer part, where L= j ∈J uj − lj . 4. 26 and hence x = A ⊗ z ≤ u if it terminates at step 5.

The permanent plays a key role in a number of max-algebraic problems because of the absence of the determinant due to the lack of subtraction. It turns out that the structure of the set of optimal solutions is related to some max-algebraic properties, in particular to questions such as the regularity of matrices. 31 If ⎛ ⎞ 3 7 2 A = ⎝4 1 5⎠ 2 6 3 then maper(A) = 14, ap(A) = {(123), (1)(23), (12)(3)}. A very simple property, on which the Hungarian method is based, is that the set of optimal solutions to the assignment problem for A does not change by adding a constant to a row or column of A.

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