Permuted max-eigenvector problem is NP -complete P.Butkovi c - - PowerPoint PPT Presentation

Permuted max-eigenvector problem is NP-complete

P.Butkoviµ c University of Birmingham http://web.mat.bham.ac.uk/P.Butkovic/

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

De…nitions and basic properties

a b = max(a, b) a b = a + b a, b 2 R := R [ f∞g Properties (ε = ∞, a1 = a): a b = b a (a b) c = a (b c) a ε = a = ε a a b = b a (a b) c = a (b c) a ε = ε = ε a a 0 = a = 0 a a a1 = 0 = a1 a (a b) c = a c b c a b = a or b

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

De…nitions and basic properties

Extension to matrices and vectors: A B = (aij bij) A B =

• ∑

k aik bkj

• α A = (α aij)

diag(d1, ..., dn) = B B B B B B B @ d1 ... ε ... ε ... dn 1 C C C C C C C A I = diag(0, ..., 0)

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

De…nitions and basic properties

A B = B A (A B) C = A (B C) A ε = ε = ε A [not A B = B A] (A B) C = A (B C) A I = A = I A (A B) C = A C B C A (B C) = A B A C

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

A piece of magic ...

Invertibility of ! Idempotency of a a = a

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

A piece of magic ...

Invertibility of ! Idempotency of a a = a (a b)k = ak bk

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

A piece of magic ...

Invertibility of ! Idempotency of a a = a (a b)k = ak bk (A B)k 6= Ak Bk

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

A piece of magic ...

Invertibility of ! Idempotency of a a = a (a b)k = ak bk (A B)k 6= Ak Bk A B = ) A C B C for any compatible A, B, C

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Maximum cycle mean

The maximum cycle mean of A : λ(A) = max ai1i2 + ai2i3 + ... + aiki1 k ; i1, ..., ik 2 N

• P.Butkoviµ

c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Maximum cycle mean

The maximum cycle mean of A : λ(A) = max ai1i2 + ai2i3 + ... + aiki1 k ; i1, ..., ik 2 N

• A = (aij) 2 R

nn

! DA = (N, f(i, j); aij > ∞g, (aij)) ... associated digraph (N = f1, ..., ng)

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Maximum cycle mean

The maximum cycle mean of A : λ(A) = max ai1i2 + ai2i3 + ... + aiki1 k ; i1, ..., ik 2 N

• A = (aij) 2 R

nn

! DA = (N, f(i, j); aij > ∞g, (aij)) ... associated digraph (N = f1, ..., ng) A is irreducible i¤ DA strongly connected

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Maximum cycle mean

The maximum cycle mean of A : λ(A) = max ai1i2 + ai2i3 + ... + aiki1 k ; i1, ..., ik 2 N

• A = (aij) 2 R

nn

! DA = (N, f(i, j); aij > ∞g, (aij)) ... associated digraph (N = f1, ..., ng) A is irreducible i¤ DA strongly connected A is irreducible = ) λ(A) is the unique eigenvalue of A and all eigenvectors are …nite

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Transitive closures

For A = (aij) 2 R

nn :

Γ(A) = A A2 A3 ... ... metric matrix

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Transitive closures

For A = (aij) 2 R

nn :

Γ(A) = A A2 A3 ... ... metric matrix ∆(A) = I Γ(A) = I A A2 A3 ... ... Kleene star

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Transitive closures

For A = (aij) 2 R

nn :

Γ(A) = A A2 A3 ... ... metric matrix ∆(A) = I Γ(A) = I A A2 A3 ... ... Kleene star Ar ... greatest weights of paths of length r

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Transitive closures

For A = (aij) 2 R

nn :

Γ(A) = A A2 A3 ... ... metric matrix ∆(A) = I Γ(A) = I A A2 A3 ... ... Kleene star Ar ... greatest weights of paths of length r λ(A) 0 = ) Ar A A2 ... An for every r 1

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Transitive closures

For A = (aij) 2 R

nn :

Γ(A) = A A2 A3 ... ... metric matrix ∆(A) = I Γ(A) = I A A2 A3 ... ... Kleene star Ar ... greatest weights of paths of length r λ(A) 0 = ) Ar A A2 ... An for every r 1 λ(A) 0 = ) Γ(A) = A A2 ... An

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Transitive closures

For A = (aij) 2 R

nn :

Γ(A) = A A2 A3 ... ... metric matrix ∆(A) = I Γ(A) = I A A2 A3 ... ... Kleene star Ar ... greatest weights of paths of length r λ(A) 0 = ) Ar A A2 ... An for every r 1 λ(A) 0 = ) Γ(A) = A A2 ... An λ(A) 0 = ) ∆(A) = I A A2 ... An1

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Multi-machine interactive production process (MMIPP)

(R.A.Cuninghame-Green) Machines M1, ..., Mn work interactively and in stages

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Multi-machine interactive production process (MMIPP)

(R.A.Cuninghame-Green) Machines M1, ..., Mn work interactively and in stages xi(r) . . . starting time of the rth stage on machine Mi (i = 1, . . . , n; r = 0, 1, ...)

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Multi-machine interactive production process (MMIPP)

(R.A.Cuninghame-Green) Machines M1, ..., Mn work interactively and in stages xi(r) . . . starting time of the rth stage on machine Mi (i = 1, . . . , n; r = 0, 1, ...) aij . . . time Mj needs to prepare the component for Mi

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Multi-machine interactive production process (MMIPP)

(R.A.Cuninghame-Green) Machines M1, ..., Mn work interactively and in stages xi(r) . . . starting time of the rth stage on machine Mi (i = 1, . . . , n; r = 0, 1, ...) aij . . . time Mj needs to prepare the component for Mi xi(r + 1) = max(x1(r) + ai1, . . . , xn(r) + ain) (i = 1, . . . , n; r = 0, 1, ...)

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Multi-machine interactive production process (MMIPP)

(R.A.Cuninghame-Green) Machines M1, ..., Mn work interactively and in stages xi(r) . . . starting time of the rth stage on machine Mi (i = 1, . . . , n; r = 0, 1, ...) aij . . . time Mj needs to prepare the component for Mi xi(r + 1) = max(x1(r) + ai1, . . . , xn(r) + ain) (i = 1, . . . , n; r = 0, 1, ...) xi(r + 1) = ∑

k aik xk(r) (i = 1, . . . , n; r = 0, 1, ...)

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Multi-machine interactive production process (MMIPP)

(R.A.Cuninghame-Green) Machines M1, ..., Mn work interactively and in stages xi(r) . . . starting time of the rth stage on machine Mi (i = 1, . . . , n; r = 0, 1, ...) aij . . . time Mj needs to prepare the component for Mi xi(r + 1) = max(x1(r) + ai1, . . . , xn(r) + ain) (i = 1, . . . , n; r = 0, 1, ...) xi(r + 1) = ∑

k aik xk(r) (i = 1, . . . , n; r = 0, 1, ...)

x(r + 1) = A x(r) (r = 0, 1, . . .)

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Multi-machine interactive production process (MMIPP)

(R.A.Cuninghame-Green) Machines M1, ..., Mn work interactively and in stages xi(r) . . . starting time of the rth stage on machine Mi (i = 1, . . . , n; r = 0, 1, ...) aij . . . time Mj needs to prepare the component for Mi xi(r + 1) = max(x1(r) + ai1, . . . , xn(r) + ain) (i = 1, . . . , n; r = 0, 1, ...) xi(r + 1) = ∑

k aik xk(r) (i = 1, . . . , n; r = 0, 1, ...)

x(r + 1) = A x(r) (r = 0, 1, . . .) A : x(0) ! x(1) ! x(2) ! ...

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

MMIPP: Steady state

The system is in a steady state if it is moving forward in regular steps

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

MMIPP: Steady state

The system is in a steady state if it is moving forward in regular steps Equivalently, if there is a λ such that x(r + 1) = λ x(r)

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

MMIPP: Steady state

The system is in a steady state if it is moving forward in regular steps Equivalently, if there is a λ such that x(r + 1) = λ x(r) Since x(r + 1) = A x(r) (r = 0, 1, . . .)

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

MMIPP: Steady state

The system is in a steady state if it is moving forward in regular steps Equivalently, if there is a λ such that x(r + 1) = λ x(r) Since x(r + 1) = A x(r) (r = 0, 1, . . .) x (0) should satisfy A x = λ x

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

MMIPP: Steady state

The system is in a steady state if it is moving forward in regular steps Equivalently, if there is a λ such that x(r + 1) = λ x(r) Since x(r + 1) = A x(r) (r = 0, 1, . . .) x (0) should satisfy A x = λ x One-o¤ process (b is the vector of completion times): A x = b

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Two basic problems

Problem (LINEAR SYSTEM [LS]) Given A 2 R

mnand b 2 R m …nd all x 2 R n satisfying

A x = b

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Two basic problems

Problem (LINEAR SYSTEM [LS]) Given A 2 R

mnand b 2 R m …nd all x 2 R n satisfying

A x = b Problem (EIGENVECTOR [EV]) Given A 2 R

nn …nd all x 2 R n, x 6= (ε, ..., ε)T (eigenvectors)

such that A x = λ x for some λ 2 R (eigenvalue)

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Permuted matrices and vectors

For A = (aij) 2 R

nn

x = (x1, ..., xn)T 2 R

n

π, σ 2 Pn de…ne A(π, σ) =

• aπ(i),σ(j)
• x (π)

=

• xπ(1), ..., xπ(n)

T

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Permuted basic problems

Problem (PERMUTED LINEAR SYSTEM [PLS]) Given A 2 R

mnand b 2 R m, is there a π 2 Pm such that

A x = b (π) has a solution?

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Permuted basic problems

Problem (PERMUTED LINEAR SYSTEM [PLS]) Given A 2 R

mnand b 2 R m, is there a π 2 Pm such that

A x = b (π) has a solution? Problem (PERMUTED EIGENVECTOR [PEV]) Given A 2 R

nn and x 2 R n, x 6= (ε, ..., ε)T , is there a π 2 Pn

such that A x (π) = λ x (π) for some λ 2 R?

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Permuted basic problems - integer versions

Problem (INTEGER PERMUTED LINEAR SYSTEM [IPLS]) Given A 2 Zmn and b 2 Zm, is there a π 2 Pm such that A x = b (π) has a solution?

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Permuted basic problems - integer versions

Problem (INTEGER PERMUTED LINEAR SYSTEM [IPLS]) Given A 2 Zmn and b 2 Zm, is there a π 2 Pm such that A x = b (π) has a solution? Problem (INTEGER PERMUTED EIGENVECTOR [IPEV]) Given A 2 Znn and x 2 Zn, is there a π 2 Pn such that A x (π) = x (π)?

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Permuted basic problems - integer versions

Problem (INTEGER PERMUTED LINEAR SYSTEM [IPLS]) Given A 2 Zmn and b 2 Zm, is there a π 2 Pm such that A x = b (π) has a solution? Problem (INTEGER PERMUTED EIGENVECTOR [IPEV]) Given A 2 Znn and x 2 Zn, is there a π 2 Pn such that A x (π) = x (π)? Theorem Both IPEV and IPLS are NP-complete.

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

BANDWIDTH

Problem (BANDWIDTH) Given an undirected graph G = (N, E) and a positive integer K n, is there a π 2 Pn such that jπ(u) π(v)j K for all uv 2 E? Equivalently:

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

BANDWIDTH

Problem (BANDWIDTH) Given an undirected graph G = (N, E) and a positive integer K n, is there a π 2 Pn such that jπ(u) π(v)j K for all uv 2 E? Equivalently: Problem (BANDWIDTH - MATRIX VERSION) Given an n n symmetric 0 1 matrix M = (mij) with zero diagonal, and a positive integer K n, is there a π 2 Pn such that ji jj K whenever mπ(i),π(j) = 1?

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

BANDWIDTH

K

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Solvability of linear systems

Let A 2 Rmn and b 2 Rm, M = f1, ..., mg , N = f1, ..., ng A x = b

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Solvability of linear systems

Let A 2 Rmn and b 2 Rm, M = f1, ..., mg , N = f1, ..., ng A x = b S (A, b) = fx 2 Rn; A x = bg

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Solvability of linear systems

Let A 2 Rmn and b 2 Rm, M = f1, ..., mg , N = f1, ..., ng A x = b S (A, b) = fx 2 Rn; A x = bg xj = max

i

(aij bi), j 2 N

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Solvability of linear systems

Let A 2 Rmn and b 2 Rm, M = f1, ..., mg , N = f1, ..., ng A x = b S (A, b) = fx 2 Rn; A x = bg xj = max

i

(aij bi), j 2 N x = (x1, ..., xn)T

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Solvability of linear systems

Let A 2 Rmn and b 2 Rm, M = f1, ..., mg , N = f1, ..., ng A x = b S (A, b) = fx 2 Rn; A x = bg xj = max

i

(aij bi), j 2 N x = (x1, ..., xn)T Mj = fk 2 M; akj bk = max

i

(aij bi)g, j 2 N

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Solvability of max-linear systems

Theorem (R.A.Cuninghame-Green) Let A 2 Rmn, b 2 Rm and x 2 Rn. Then x 2 S (A, b) if and

• nly if

(a) x x and (b)

[

xj=x j

Mj = M Corollary (1) The following three statements are equivalent:

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Solvability of max-linear systems

Theorem (R.A.Cuninghame-Green) Let A 2 Rmn, b 2 Rm and x 2 Rn. Then x 2 S (A, b) if and

• nly if

(a) x x and (b)

[

xj=x j

Mj = M Corollary (1) The following three statements are equivalent:

(I) S (A, b) 6= ∅

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Solvability of max-linear systems

Theorem (R.A.Cuninghame-Green) Let A 2 Rmn, b 2 Rm and x 2 Rn. Then x 2 S (A, b) if and

• nly if

(a) x x and (b)

[

xj=x j

Mj = M Corollary (1) The following three statements are equivalent:

(I) S (A, b) 6= ∅ (II) x 2 S (A, b)

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Solvability of max-linear systems

Theorem (R.A.Cuninghame-Green) Let A 2 Rmn, b 2 Rm and x 2 Rn. Then x 2 S (A, b) if and

• nly if

(a) x x and (b)

[

xj=x j

Mj = M Corollary (1) The following three statements are equivalent:

(I) S (A, b) 6= ∅ (II) x 2 S (A, b) (III) S

j2N Mj = M

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Unique solution to a max-linear system

Corollary (2) S (A, b) = fxg if and only if

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Unique solution to a max-linear system

Corollary (2) S (A, b) = fxg if and only if

(a) S

j2N Mj = M and

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Unique solution to a max-linear system

Corollary (2) S (A, b) = fxg if and only if

(a) S

j2N Mj = M and

(b) S

j2N 0 Mj 6= M for any N0 N, N0 6= N.

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Unique solution to a max-linear system

Corollary (2) S (A, b) = fxg if and only if

(a) S

j2N Mj = M and

(b) S

j2N 0 Mj 6= M for any N0 N, N0 6= N.

Corollary (3) If m = n then S (A, b) = fxg if and only if there is a π 2 Pn such that Mπ(j) = fjg for all j 2 N. Equivalently ai,π(j) bi < aj,π(j) bj for all i, j 2 N, i 6= j.

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Strong regularity

A = (aij) 2 Rnn is strongly regular i¤ (9b 2 Rn) jS (A, b)j = 1

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Strong regularity

A = (aij) 2 Rnn is strongly regular i¤ (9b 2 Rn) jS (A, b)j = 1 Linear assignment problem for A : Find a π 2 Pn maximising w(A, π) = ∑

i2N

ai,π(i)

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Strong regularity

A = (aij) 2 Rnn is strongly regular i¤ (9b 2 Rn) jS (A, b)j = 1 Linear assignment problem for A : Find a π 2 Pn maximising w(A, π) = ∑

i2N

ai,π(i) ap(A) = fσ 2 Pn; w(A, σ) = max

π2Pn w(A, π)g

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Strong regularity

A = (aij) 2 Rnn is strongly regular i¤ (9b 2 Rn) jS (A, b)j = 1 Linear assignment problem for A : Find a π 2 Pn maximising w(A, π) = ∑

i2N

ai,π(i) ap(A) = fσ 2 Pn; w(A, σ) = max

π2Pn w(A, π)g

(PB + Hevery) A = (aij) 2 Rnn is strongly regular if and

• nly if jap(A)j = 1

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Strong regularity

A = (aij) 2 Rnn is strongly regular i¤ (9b 2 Rn) jS (A, b)j = 1 Linear assignment problem for A : Find a π 2 Pn maximising w(A, π) = ∑

i2N

ai,π(i) ap(A) = fσ 2 Pn; w(A, σ) = max

π2Pn w(A, π)g

(PB + Hevery) A = (aij) 2 Rnn is strongly regular if and

• nly if jap(A)j = 1

What are those b if A is strongly regular?

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Strong regularity

A = (aij) 2 Rnn is strongly regular i¤ (9b 2 Rn) jS (A, b)j = 1 Linear assignment problem for A : Find a π 2 Pn maximising w(A, π) = ∑

i2N

ai,π(i) ap(A) = fσ 2 Pn; w(A, σ) = max

π2Pn w(A, π)g

(PB + Hevery) A = (aij) 2 Rnn is strongly regular if and

• nly if jap(A)j = 1

What are those b if A is strongly regular? A 2 Rnn ! SA = fb 2 Rn; A x = b has a unique solutiong

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Strong regularity

SA = fb 2 Rn; A x = b has a unique solutiong

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Strong regularity

SA = fb 2 Rn; A x = b has a unique solutiong SA ... the simple image set (of the mapping x 7 ! A x)

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Strong regularity

SA = fb 2 Rn; A x = b has a unique solutiong SA ... the simple image set (of the mapping x 7 ! A x) A = (aij) 2 Rnn is normalised i¤

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Strong regularity

SA = fb 2 Rn; A x = b has a unique solutiong SA ... the simple image set (of the mapping x 7 ! A x) A = (aij) 2 Rnn is normalised i¤

λ(A) = 0 (A is de…nite) and

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Strong regularity

SA = fb 2 Rn; A x = b has a unique solutiong SA ... the simple image set (of the mapping x 7 ! A x) A = (aij) 2 Rnn is normalised i¤

λ(A) = 0 (A is de…nite) and aii = 0 for all i 2 N (A is increasing, A I)

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Strong regularity

SA = fb 2 Rn; A x = b has a unique solutiong SA ... the simple image set (of the mapping x 7 ! A x) A = (aij) 2 Rnn is normalised i¤

λ(A) = 0 (A is de…nite) and aii = 0 for all i 2 N (A is increasing, A I)

A normalised = ) ∆(A) = Γ(A) = A A2 ... An1 and I A A2 ..., hence ∆(A) = Γ(A) = An1 = An = An+1 = ...

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Strong regularity

SA = fb 2 Rn; A x = b has a unique solutiong SA ... the simple image set (of the mapping x 7 ! A x) A = (aij) 2 Rnn is normalised i¤

λ(A) = 0 (A is de…nite) and aii = 0 for all i 2 N (A is increasing, A I)

A normalised = ) ∆(A) = Γ(A) = A A2 ... An1 and I A A2 ..., hence ∆(A) = Γ(A) = An1 = An = An+1 = ... A normalised = ) Im (A)

• Im
• A2 Im
• A3 ...
• Im
• An1 = Im (An) = Im
• An+1 = ... = V (A)

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The simple image set

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The simple image set

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The simple image set

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The simple image set

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The simple image set

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The simple image set

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The simple image set

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The simple image set

Theorem If A 2 Rnn is normalised and strongly regular (that is SA 6= ∅) then V (A) = cl(SA) Corollary If A 2 Rnn is normalised and strongly regular then

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The simple image set

Theorem If A 2 Rnn is normalised and strongly regular (that is SA 6= ∅) then V (A) = cl(SA) Corollary If A 2 Rnn is normalised and strongly regular then

1

A b = b for every b 2 SA

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The simple image set

Theorem If A 2 Rnn is normalised and strongly regular (that is SA 6= ∅) then V (A) = cl(SA) Corollary If A 2 Rnn is normalised and strongly regular then

1

A b = b for every b 2 SA

2

For every b 2 V (A) there is a sequence fb(k)g∞

k=0 SA such that

b(k) ! b

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Normalised and strongly regular matrices

b(k) 2 SA means

• S
• A, b(k)
• = 1

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Normalised and strongly regular matrices

b(k) 2 SA means

• S
• A, b(k)
• = 1

If m = n : jS (A, b)j = 1 ( ) (9π 2 Pn) Mπ(j) = fjg for all j 2 N

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Normalised and strongly regular matrices

b(k) 2 SA means

• S
• A, b(k)
• = 1

If m = n : jS (A, b)j = 1 ( ) (9π 2 Pn) Mπ(j) = fjg for all j 2 N Equivalently ai,π(j) b(k)

i

< aj,π(j) b(k)

j

for all i, j 2 N, i 6= j.

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Normalised and strongly regular matrices

b(k) 2 SA means

• S
• A, b(k)
• = 1

If m = n : jS (A, b)j = 1 ( ) (9π 2 Pn) Mπ(j) = fjg for all j 2 N Equivalently ai,π(j) b(k)

i

< aj,π(j) b(k)

j

for all i, j 2 N, i 6= j. If A is normalised and strongly regular then π = id, hence aij b(k)

i

< b(k)

j

for every i, j 2 N, i 6= j and k = 0, 1, ....

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Normalised and strongly regular matrices

If A is normalised and strongly regular then aij b(k)

i

< b(k)

j

for every i, j 2 N, i 6= j and k = 0, 1, ....

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Normalised and strongly regular matrices

If A is normalised and strongly regular then aij b(k)

i

< b(k)

j

for every i, j 2 N, i 6= j and k = 0, 1, .... Let A = (aij) 2 Rnn be normalised, strongly regular and b 2 Rn. Then b 2 V (A) if and only if aij bi bj for every i, j 2 N

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

Normalised and strongly regular matrices

If A is normalised and strongly regular then aij b(k)

i

< b(k)

j

for every i, j 2 N, i 6= j and k = 0, 1, .... Let A = (aij) 2 Rnn be normalised, strongly regular and b 2 Rn. Then b 2 V (A) if and only if aij bi bj for every i, j 2 N Let A = (aij) 2 Znn be normalised, strongly regular, b 2 Zn and π 2 Pn. Then b(π) 2 V (A) if and only if aπ(i),π(j) bi bj for every i, j 2 N

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The NP-completeness result

Theorem IPEV is NP-complete for the class of normalised, strongly regular matrices. Proof M = (mij), 0 < K n ... an instance of BANDWIDTH.

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The NP-completeness result

Theorem IPEV is NP-complete for the class of normalised, strongly regular matrices. Proof M = (mij), 0 < K n ... an instance of BANDWIDTH. Let A = (aij) 2 Znn: aij = 8 < : K if mij = 1 n if mij = 0, i 6= j 0 if i = j

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The NP-completeness result

Theorem IPEV is NP-complete for the class of normalised, strongly regular matrices. Proof M = (mij), 0 < K n ... an instance of BANDWIDTH. Let A = (aij) 2 Znn: aij = 8 < : K if mij = 1 n if mij = 0, i 6= j 0 if i = j A is normalised, strongly regular

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The NP-completeness result

Set b = (1, ..., n)T

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The NP-completeness result

Set b = (1, ..., n)T The answer to IPEV for A and b is "yes" ( ) 9π 2 Pn: aπ(i),π(j) i j for all i, j 2 N

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The NP-completeness result

Set b = (1, ..., n)T The answer to IPEV for A and b is "yes" ( ) 9π 2 Pn: aπ(i),π(j) i j for all i, j 2 N ( ) K i j if mπ(i)π(j) = 1

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The NP-completeness result

Set b = (1, ..., n)T The answer to IPEV for A and b is "yes" ( ) 9π 2 Pn: aπ(i),π(j) i j for all i, j 2 N ( ) K i j if mπ(i)π(j) = 1 ( ) K ji jj if mπ(i)π(j) = 1

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

The NP-completeness result

Set b = (1, ..., n)T The answer to IPEV for A and b is "yes" ( ) 9π 2 Pn: aπ(i),π(j) i j for all i, j 2 N ( ) K i j if mπ(i)π(j) = 1 ( ) K ji jj if mπ(i)π(j) = 1 ( ) "Yes" to BANDWIDTH

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

More on regularity

v1, ..., vn 2 R

m are called

WLD i¤ for some k and αj 2 R vk = ∑

• j6=k αj vj

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

More on regularity

v1, ..., vn 2 R

m are called

WLD i¤ for some k and αj 2 R vk = ∑

• j6=k αj vj

Gondran-Minoux LD i¤ for some U, V f1, ..., ng , U \ V = ∅, U, V 6= ∅ and αj 2 R

∑

• j2U αj vj = ∑
• j2V αj vj

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

More on regularity

v1, ..., vn 2 R

m are called

WLD i¤ for some k and αj 2 R vk = ∑

• j6=k αj vj

Gondran-Minoux LD i¤ for some U, V f1, ..., ng , U \ V = ∅, U, V 6= ∅ and αj 2 R

∑

• j2U αj vj = ∑
• j2V αj vj

Strongly LD i¤

∑

• j=1,...,n vj xj = v

does not have a unique solution for any v 2 Rm

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

More on regularity

v1, ..., vn 2 R

m are called

WLD i¤ for some k and αj 2 R vk = ∑

• j6=k αj vj

Gondran-Minoux LD i¤ for some U, V f1, ..., ng , U \ V = ∅, U, V 6= ∅ and αj 2 R

∑

• j2U αj vj = ∑
• j2V αj vj

Strongly LD i¤

∑

• j=1,...,n vj xj = v

does not have a unique solution for any v 2 Rm SLI = ) GMLI = ) WLI

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

More on regularity

A = (aij) 2 Rnn, A = (A1, ..., An)

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

More on regularity

A = (aij) 2 Rnn, A = (A1, ..., An) A is called:

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

More on regularity

A = (aij) 2 Rnn, A = (A1, ..., An) A is called:

Weakly regular i¤ the following is not true for any k and αj 2 R : Ak = ∑

• j6=k αj Aj

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

More on regularity

A = (aij) 2 Rnn, A = (A1, ..., An) A is called:

Weakly regular i¤ the following is not true for any k and αj 2 R : Ak = ∑

• j6=k αj Aj

Gondran-Minoux regular i¤ the following is not true for any U, V f1, ..., ng , U \ V = ∅, U, V 6= ∅ and αj 2 R :

∑

• j2U αj Aj = ∑
• j2V αj Aj

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

More on regularity

A = (aij) 2 Rnn, A = (A1, ..., An) A is called:

Weakly regular i¤ the following is not true for any k and αj 2 R : Ak = ∑

• j6=k αj Aj

Gondran-Minoux regular i¤ the following is not true for any U, V f1, ..., ng , U \ V = ∅, U, V 6= ∅ and αj 2 R :

∑

• j2U αj Aj = ∑
• j2V αj Aj

Strongly regular i¤

∑

• j=1,...,n Aj xj = b

has a unique solution for some b 2 Rn

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

More on regularity

A = (aij) 2 Rnn, A = (A1, ..., An) A is called:

Weakly regular i¤ the following is not true for any k and αj 2 R : Ak = ∑

• j6=k αj Aj

Gondran-Minoux regular i¤ the following is not true for any U, V f1, ..., ng , U \ V = ∅, U, V 6= ∅ and αj 2 R :

∑

• j2U αj Aj = ∑
• j2V αj Aj

Strongly regular i¤

∑

• j=1,...,n Aj xj = b

has a unique solution for some b 2 Rn

Gondran-Minoux regularity = ) Weak regularity

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

More on regularity

Theorem (Gondran-Minoux) A = (aij) 2 Rnn is regular if and only if all permutations in ap(A) are of the same parity. Corollary: Strong regularity = ) Gondran-Minoux regularity

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

More on regularity

Theorem (Gondran-Minoux) A = (aij) 2 Rnn is regular if and only if all permutations in ap(A) are of the same parity. Corollary: Strong regularity = ) Gondran-Minoux regularity The problem "Given A, are all permutations in ap(A) of the same parity?" is equivalent to the Even Cycle Problem in digraphs

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

More on regularity

Theorem (Gondran-Minoux) A = (aij) 2 Rnn is regular if and only if all permutations in ap(A) are of the same parity. Corollary: Strong regularity = ) Gondran-Minoux regularity The problem "Given A, are all permutations in ap(A) of the same parity?" is equivalent to the Even Cycle Problem in digraphs SR |{z}

O(n3)

= ) GMR | {z }

Even Cycle

= ) WR |{z}

O(n3)

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

THANK YOU

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

WHAT IS MAX-ALGEBRA

Key players: The principal solution

B B B B @ 2 2 2 5 3 2 ε ε 3 3 3 2 1 4 ε 1 C C C C A

• @

x1 x2 x3 1 A = B B B B @ 3 2 1 5 1 C C C C A

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

WHAT IS MAX-ALGEBRA

Key players: The principal solution

B B B B @ 2 2 2 5 3 2 ε ε 3 3 3 2 1 4 ε 1 C C C C A

• @

x1 x2 x3 1 A = B B B B @ 3 2 1 5 1 C C C C A

• aij b1

i

= B B B B @ 5 1 1 3 1 ε ε 2 3 3 2 4 1 ε 1 C C C C A

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

WHAT IS MAX-ALGEBRA

Key players: The principal solution

B B B B @ 2 2 2 5 3 2 ε ε 3 3 3 2 1 4 ε 1 C C C C A

• @

x1 x2 x3 1 A = B B B B @ 3 2 1 5 1 C C C C A

• aij b1

i

= B B B B @ 5 1 1 3 1 ε ε 2 3 3 2 4 1 ε 1 C C C C A M1 = f2, 4g , M2 = f1, 2, 5g , M3 = f3, 4g

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

WHAT IS MAX-ALGEBRA

Key players: The principal solution

B B B B @ 2 2 2 5 3 2 ε ε 3 3 3 2 1 4 ε 1 C C C C A

• @

x1 x2 x3 1 A = B B B B @ 3 2 1 5 1 C C C C A

• aij b1

i

= B B B B @ 5 1 1 3 1 ε ε 2 3 3 2 4 1 ε 1 C C C C A M1 = f2, 4g , M2 = f1, 2, 5g , M3 = f3, 4g x = (3, 1, 2)T is a solution since S

j=1,2,3 Mj = M

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

WHAT IS MAX-ALGEBRA

Key players: The principal solution

B B B B @ 2 2 2 5 3 2 ε ε 3 3 3 2 1 4 ε 1 C C C C A

• @

x1 x2 x3 1 A = B B B B @ 3 2 1 5 1 C C C C A

• aij b1

i

= B B B B @ 5 1 1 3 1 ε ε 2 3 3 2 4 1 ε 1 C C C C A M1 = f2, 4g , M2 = f1, 2, 5g , M3 = f3, 4g x = (3, 1, 2)T is a solution since S

j=1,2,3 Mj = M

M2 [ M3 = M hence S(A, b) = n (x1, x2, x3)T 2 R

3; x1 3, x2 = 1, x3 = 2

• .

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

References

Historical remarks

R.A.Cuninghame-Green 1960 N.N.Vorobyov 1967, Litvinov, Maslov, Kolokoltsov, Sobolevski... M.Gondran, M.Minoux 1975 K.Zimmermann 1972 P.Butkoviµ c 1977 R.E.Burkard, U.Zimmermann 1981 H.Schneider G.Cohen, D.Dubois, J.-P.Quadrat, M.Viot 1983 K.Cechlárová, P.Szabó, J.Plávka 1985 G.-J.Olsder, C.Roos 1988, B.Heidergott 2000 R.D.Nussbaum 1991 M.Akian 1998, S.Gaubert 1992, C.Walsh 2001 R.B.Bapat, D.Stanford, P.van den Driessche 1993 M.Gavalec 1995 Tropical algebra from 1995: B.Sturmfels, M.Develin et al

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

F.L.Baccelli, G. Cohen, G.-J. Olsder and J.-P. Quadrat, Synchronization and Linearity, Chichester, New York: J.Wiley and Sons, 1992. R.E. Burkard and P. Butkovic: Finding all essential terms of a characteristic maxpolynomial, Discrete Applied Mathematics 130 (2003) 367-380. P.Butkoviµ c, Strong regularity of matrices – a survey of results, Discrete Appl. Math. 48, 45–68 (1994). P.Butkoviµ c, Regularity of matrices in min-algebra and its time-complexity, Discrete Appl. Math. 57, 121–132 (1995). P.Butkoviµ c, Simple image set of (max, +) linear mappings, Discrete Appl. Math. 105, 73–86 (2000). P.Butkovic, Max-algebra: the linear algebra of combinatorics? Lin.Alg. and Appl. 367 (2003) 313-335.

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

P.Butkovic, H.Schneider and S.Sergeev: Generators, Extremals and Bases of Max Cones, Linear Algebra and Its Applications 421 (2007) 394-406.

• P. Butkovic, S.Gaubert and R.A. Cuninghame-Green:

Reducible spectral theory with applications to the robustness

• f matrices in max-algebra, The University of Birmingham,

preprint 2007/16. G.Cohen, D.Dubois, J.-P.Quadrat, M.Viot, A Linear-System-Theoretic View of Discrete-Event Processes and Its Use for Performance Evaluation in Manufacturing, IEEE Transactions on Automatic Control, Vol. AC-30, No3, 1985. G.Cohen, S.Gaubert and J.-P.Quadrat, Duality and separation theorems in idempotent semimodules. Tenth Conference of the International Linear Algebra Society. Linear Algebra and Its Applications 379 (2004), 395–422. R.A.Cuninghame-Green, Minimax Algebra, Lecture Notes in Economics and Math. Systems 166, Berlin: Springer, 1979. R.A. Cuninghame-Green and P. Butkoviµ c, The equation Ax =

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)

R.A.Cuninghame-Green, Minimax algebra and applications, in Advances in Imaging and Electron Physics, Vol. 90, pp. 1–121, Academic Press, New York, 1995. S.Gaubert and R.Katz, The Minkowski Theorem for max-plus convex sets, Linear Algebra and Its Applications 421(2007)356-369. B.Heidergott, G.J.Olsder and J. van der Woude (2005), Max Plus at Work: Modeling and Analysis of Synchronized Systems, A Course on Max-Plus Algebra, PUP.

• G. J.Olsder and C. Roos, Cramér and Cayley-Hamilton in the

max algebra. Linear Algebra and Its Applications 101 (1988) 87–108. M.Plus, Linear systems in (max,+) algebra, in: Proceedings of 29th Conference on Decision and Control Honolulu, 1990.

• B. Sturmfels, F. Santos and M. Develin, On the tropical rank
• f a matrix, in Discrete and Computational Geometry, (eds.

J.E. Goodman, J. Pach and E. Welzl), Mathematical Sciences

P.Butkoviµ c University of Birmingham Permuted eigenvector (Manchester 20 May 2008)