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Math. Control Signals Systems (1993) 6: 1- 99 1993 Springer-Verlag Lon don Limited M a t h e m a t ic s o f C o n t r o lS i g n a ls a n d S y s t e m s

C h e c k i n g R o b u s t N o n s in g u l a r it y i s N P - H a r dS v a t o p l u k P o l j a k ~ - a n d J i f i R o h n t

Abstract W e consider the following problem: given k + 1 square m atrices withrational entries, Ao, A1, ..., Ak, decide ifA 0 + r x A 1 + . . . + r k A k is nonsingular fo rall possible choices of real numbers rx .. .. rk in the interval [0, 1]. We show thatthis question, which is closely related to the robust stability problem, is NP-hard.The p roo f relies on the new concept o f radius o f nonsinoulari ty of a square m atrixand on the relationship between computing th is radius and a graph-theoreticproblem.K e y w a r d s Robu stness, NP-co mplete problem s, Robust nonsingularity, Intervalmatrices.

1 I n t r o d u c t i o n

I t is n a t u r a l t o re q u i r e t h a t a c o n t r o l s y s t e m p e r f o r m s s a t is f a c to r i ly e v e n u n d e ru n k n o w n v a r i a t i o n s o f s y s t e m p a r a m e t e r s i n a s p ec i f ie d r a n g e , i.e ., t h a t i t i s r o b u s t .T h e m o s t i m p o r t a n t p e r f o r m a n c e is su e, n a m e l y r o b u s t s t a b i l i t y , h a s b e e n e x t e n si v e l ys t u d i e d r e c e n t ly ; w e r ef e r t o t h e s u r v e y p a p e r b y M a n s o u r [ M ] f o r a d e ta i l e d l is t o fr e f e r e n c e s .

I n t h is p a p e r w e a r e c o n c e r n e d w i th t h e p r o b l e m o f r o b u s t n o n s i n g u l a r i t y . T o b em o r e p r ec i se , fo r a n y t w o g i v e n n x n m a t r i c e s A a n d A , w i t h A n o n n e g a t i v e , w ei n t r o d u c e t h e r a d i u s o f n o n s i n g u l a r i t y d A , A ) a s th e m i n i m u m e > 0 f o r w h i c h t h e r ee x i st s a s i n g u l a r m a t r i x A ' s a t i s f y i n g A - - ~ A _< A ' < A + e A . T h e c o n c e p t o f t h er a d i u s o f n o n s i n g u l a r i t y i s s e e m i n g l y c l o s e ly re l a t e d t o D o y l e ' s " s t r u c t u r e d s i n g u l a rv a l u e " i n t r o d u c e d i n [ D 2 ] a s a t o o l fo r th e a n a l y s i s o f f e e d b a c k s y s t e m s w i t hs t r u c t u r e d u n c e r t a i n ti e s , b u t w e d o n o t p u r s u e t h i s c o n n e c t i o n i n t h is p a p e r . T h ec o n c e p t m a y a l s o p r o v e u s e f u l in t h e s e n s it i v i ty a n a l y s i s o f l i n e a r s y s t e m s [ - D 1 ].

W e n o w s u m m a r i z e t h e m a i n r e su l ts . T h e k e y r e s u l t ( T h e o r e m 2 .1 ) g i ve s a ne x p l i c i t f o r m u l a f o r d A , A ). I n o r d e r t o s h o w t h a t c o m p u t i n g d A , A ) is N P - h a r d ,w e c o n s i d e r t h e s p e c i a l c a s e o f A = J ( t h e m a t r i x w h o s e a l l e n t r i e s a r e o n e s ) a n d w es h o w i n T h e o r e m 2 .2 t h a t

1d A , J ) - r A _ l ) ,

* Da te received : July 3, 1990. Da te revised: M ay 7, 1992.~" Depa rtment of App lied Mathem atics, Faculty of Mathematics and Physics, Charles U niversity,M alostransk6 n/tm . 25, 118 00 Pra ha 1, Czecho slovakia.

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2 s . P o~ ak and J . Rohnw h e r e r (B) i s d e f i n e d b y

r (B ) : m a x { z t B y l z , y ~ { - 1, 1} }( z t d e n o t e s t h e t r a n s p o s e o f z ). S i n c e r i s a m a t r i x n o r m , w e f i rs t g iv e s o m e u p p e ra n d l o w e r b o u n d s o n it . T h e n , b y e s t a b l is h i n g a c o n n e c t i o n b e t w e e n r a n d t h em a x i m u m c u t i n a n a s s o c ia t e d g r a p h , w e s h o w t h a t c o m p u t i n g r (B) i s N P - h a r d f o rm a t r i c e s B w i t h r a t i o n a l e n t r ie s . A s a c o n s e q u e n c e o f t h e a b o v e r e s u l t w e s h o w t h a tt h e p r o b l e m o f t e s ti n g s i n g u l a r i t y o f i n t e r v a l m a t r i c e s i s N P - c o m p l e t e . W e r e c a l lt h a t a n i n t e r v a l m a t r i x A I = { A ' I A - A < A ' < A + A } i s c a l l e d s ingular i f i t c o n -t a i n s a s i n g u l a r m a t r i x ; i .e ., A I i s s i n g u l a r i f a n d o n l y i f d ( A , A) < 1 .

S o m e N o t a t i o nW e w o r k w i t h s q u a r e m a t r i c e s o f s iz e n n w i t h re a l e n tr ie s . W e d e n o t e b y Q = Q ,t h e n - d i m e n s i o n a l d i s c r e t e c u b e Q = { - 1 , 1 } , a n d b y e t h e v e c t o r e = ( 1, . . . , 1 ) .F o r e a c h y ~ Q , w e d e n o t e b y T t h e d i a g o n a l m a t r i x w i t h t h e v e c t o r y a s i ts d i a g o n a l(i.e ., (T r ) u = y i a n d (T r) ~i = 0 f o r i # j ). F o r a n a r b i t r a r y n x n m a t r i x A w e d e n o t e

po(A) = m a x { [ 2 l ] A x = 2 x f o r s o m e x # 0 , 2 r e a l} ,i.e ., a n a n a l o g u e o f t h e s p e c t r a l ra d i u s , w i t h m a x i m u m b e i n g t a k e n o n l y o v e r r e a le i g e n v a l u e s ; w e s e t g m ( ~ A= 0 i f n o r e a l e i g e n v a l u e e x i st s. W e u s e t h e f o l lo w i n g m a t r i xn o r m s : p ( A ) = x / p o ( A t A ) ( th e sp e c t r a l n o r m ) a n d s(A ) = ~.i , j lao[.

2 Radius of NonsingularityF o r a n n x n m a t r i x A a n d a n o n n e g a t i v e n x n m a t r i x A , w e d e fi n e t h e radius o fnons ingular i ty b y

d ( A , A ) = m i n { e > 0 1 A - e A < Z ' < A + e A f o r s o m e s i n g u l a r A ' } .O b v i o u s l y , d ( A , A ) = 0 i f a n d o n l y i f A i s s i n g u l a r . O n t h e o t h e r h a n d , i t c a ns o m e t i m e s b e in f i n it e . A s a n e x a m p l e , c o n s i d e r t h e m a t r i c e s

A = ( 0 1 ~ ) , A = ( ~ 0 0).H e r e e a c h A ' w i t h A - ~ A < A ' < A + e A sa t i s fi e s d e t A ' = - 1 , h e n c e d ( A , A ) i si n f i n i t e .

S i n c e t h e c a s e o f A s i n g u l a r i s t r iv i a l , w e a s s u m e A t o b e n o n s i n g u l a r i n w h a tf o ll o w s . I n t h is c a se , u si n g t h e n o t a t i o n i n t r o d u c e d i n th e p r e v i o u s s e c t i o n , w e d e r i v et h e f o l lo w i n g e x p l ic i t f o r m u l a f o r d ( A , A ) ( w e e m p l o y t h e c o n v e n t i o n ~ = 0 o):

Theorem 2 1 Let A be nons ingular and A > O. Then we haved ( A , A ) = m a x { p o ( A _ l T r A T z ) l y ' z ~ Q } 1 )

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Checking Robust Nonsingularity is NP-H ard 3P r o o f . F i r s t c o n s i d e r t h e c a s e o f d ( A , A) f i n it e . Fo r a g iv en e > 0 , t he ex i s t ence o fa s i n g u l a r m a t r i x A ' s a t is f y i n g A - e A < A ' < A + e A i s e q u i v a l e n t t o s i n g u l a r i t yo f th e i n t e r v a l m a t r i x [ A - c A, A + c A ], w h i c h , a c c o r d i n g t o a s s e r t i o n ( C 3 ) o fT h e o r e m 5 .1 i n [ R ] , i s t h e c a s e if a n d o n l y i fpo(A-1TrsATz ) > 1h o l d s f o r s o m e y , z ~ Q , i.e ., if a n d o n l y i f

s . m ax { po( A-1 TyA T z) ly, z ~ Q } >_ 1 .H e n c e t h e m i n i m u m v a l u e o f s i s g i v e n b y ( 1).

I f d (A , A ) = ~ , t h e n b y t h e s a m e r e su l t i n [ R ] w e h a v e e . p o ( A - 1 T r A T z ) < 1 fo re a c h y , z ~ Q a n d e a c h ~ > 0 , h e n c e po(A -1Tr A T~) = 0 fo r e ac h y , z ~ Q an d (1) ag a inh o l d s . 9

W e s h o w t h a t c o m p u t in g d ( A , A ) f o r a g i v e n i n s t a n c e A a n d A is N P - h a r d . F o rt h i s p u r p o s e , w e c o n s i d e r t h e s p e c i a l c a s e A = J - - e e t, a n d w e w r i t e d ( A ) i n s t e a d o fd ( A , J ). W e h a v e t h e f o l l o w i n g r e s u l t.T h e o r e m 2 . 2 . L et A be nonsingular . Th en

d ( A ) - r ( A _ l ) , ( 2 )w h e r e r ( A - 1 ) = m a x { z t A - l y [ z , y ~ Q } .P r o o f . F o r A = e e ' , w e h a v e A -1 Ty A T~ = A - l y z t f o r e a c h y , z ~ Q . I f 2 i s a n o n z e r or e a l e i g e n v a l u e o f A - l y z t , t h e n f r o m

A - l y z t x = 2 xw e h a v e ztx v~ O. P r e m u l t i p l y i n g t h is e q u a t i o n b y z t a n d d i v id i n g b y z t x g ives2 = z t A - l y . T h u s p o ( A - l y z t) = [ z t A - l y [ . T h e n T h e o r e m 2 . 1 g i v e s d ( A ) = 1 I t ( A - i ) ,w h e r e

r ( A - 1 ) = m a x { l z t A - l y l ] z , y e Q } = m a x { z t A - X y l z , y ~ Q } . 9T h e m a p p i n g

A ~---~ (A ) = m a x {z tA y[z , y ~ Q }i s o b v i o u s l y a m a t r i x n o r m (i.e., r (A ) > O, r (A) = 0 i f an d on ly i f A = 0 , r ( A + B ) c n - 1 /2 s ( A ) ,i=1w h e r e z is t h e s i g n v e c t o r o f A y .

T h e p r o o f o f t h e u p p e r b o u n d is t ri v ia l. 9

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Checking Robust Nonsingularity is NP Hard 5

L e t u s n o t e t h a t t h e o r i g i n a l p u r e l y p r o b a b i l i s t ic p r o o f o f t h e l o w e r b o u n d o f (3 )f r o m [ B S ] c a n b e m o d i f i e d t o a n a l g o r i t h m i c o n e . T h u s , f o r a g i v e n ___ 1 - m a t r i xA , w e c a n c o n s t r u c t ( in p o l y n o m i a l t im e ) a p a i r y , z ~ Q o f v e c to r s s u c h t h a tz t A y > c n 3 /2 w h e r e c is th e a b o v e c o n s t a n t .I n t h e r e s t o f t h is s e c t i o n w e s t u d y a r e l a t i o n b e t w e e n r ( A ) a n d t h e m a x - c u tp r o b l e m . A g a i n , s u c h a re l a t i o n is n o t q u i t e n e w s i n c e t h e m a x - c u t p r o b l e m h a sa l r e a d y b e e n u s e d f o r r e fo r m u l a t io n o f q u a d r a t i c o p t i m i z a t i o n p r o b l e m s o f t y p ex t A x -t- c t x ; see , e .g . , [B ] .

T h e M a x C u t P r o b l e m . L e t G = ( N , E ) b e a g r a p h a n d l e t c : E --* R b e a w e i g h tf u n c t i o n o n e d g e s . T h e m a x i m u m c u t m c ( G ) in t h e g r a p h G w i t h r e s p e c t t o c i s d e f in e das

m c ( G ) = m a x c ( 6 S ) ,S ~ N

w h e r e 6 S is th e s e t o f e d g e s w i t h o n e e n d v e r t e x in S a n d t h e o t h e r o n e i n N \ S , a n dc ( F ) = ~ f ~ r c ( f ) f o r a s ubs e t F c E .

I n o r d e r t o r e d u c e c o m p u t i n g r ( A ) t o t h e m a x - c u t p r o b l e m , w e d e f in e th e b i p a r t i t eg r a p h B a o f a m a t r i x A a s t h e w e i g h t e d b i p a r t i te g r a p h BA = ( Y t 3 Z ) w h e r e Y a n dZ a r e t w o c o p i e s o f { 1, . . . , n } a n d E = { i j [a o # 0 ) . T h e w e i g h t o f a n e d g e i j is aij .T h e o r e m 2.5. W e h a v e r ( A ) = 2 m c ( B a ) - e t A e .P r o o f . G i v e n y , z ~ Q , d e f i n e t h e s e t

S = { i ~ Y [ y i = 1 } t 3 { j ~ Z l z j = - l } .W e h a v e

Y tA z = .~ . a iJyizJ = E ai j - Y ' . a i~ = 2 E ai~ - .~ . ao = 2c( 6S ) - e t A e,t 3 Yi =Zj Yi 5~ Zj yi =z j l J

a n d t a k i n g t h e m a x i m u m o n b o t h s id e s g iv e s t h e re s u lt .T h e m a x - c u t p r o b l e m is a k n o w n N P - h a r d p r o b l e m ( se e I-G ] ) . S i n ce i t is d i ff ic u lt

t o f in d a n e x a c t s o l u t io n , w e m a y u s e s o m e h e u r is t ic s . W e n e x t s u r v e y s o m e o f th e m .L o w e r B o u n d s o n M a x C u t .

(i) [ P T ] I f G = ( N , E ) is a w e i g h t e d c o n n e c t e d g r a p h , t h e nm c ( G ) > 8 9+ t h e m i n i m u m w e i g h t o f a s p a n n i n g t r ee .

A c u t 6 S s a t i s fy i n g t h e a b o v e i n e q u a l i t y c a n b e f o u n d i n O ( n 3 ) t im e .(ii) L i e b e r h e r r a n d S p e c k e r h a v e i m p l i c it l y s h o w n i n [ S] t h e b o u n d

nm c ( G ) > c ( E ) 2 n - - 1

I t is e a s y to e s t a b l is h t h e a b o v e b o u n d b y a p r o b a b i l i s ti c m e t h o d . T h e m e r i to f [ L S ] is a p o l y n o m i a l - t i m e a l g o r i t h m a c h i e v i n g i t.

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6 S, Poljak and J. Ro hnU p p e r o u n d s o n M a x - C u t . A n u p p e r b o u n d o n m a x - cu t w a s g iv en b y M o h a r a n dP o l j a k [ - M P ] :

n

m c ( G) < ~ sw h e r e ~ 'm ax s t h e m a x i m u m e i g e n v a l u e o f t h e Lap lac ian ma tr i x L G = ( li j) g i v e n b y

c i j i f i j ~ E , i # j ,l o = 0 i f i j ~ E , i # j ,

2 k C i k i f i = j .F u t h e r i m p r o v e m e n t o f t h e e ig e n v a lu e b o u n d o n t h e m a x - c u t p r o b l e m is g i ve n

b y D e l o r m e a n d P o l j a k [ D P ] . A n o t h e r w a y to o b t a i n so m e b o u n d s o n t h e m a x - c u tis v ia a n a s s o c i a t e d s y s t e m o f l i n e a r i n e q u a l it i e s; s e e [ D L ] f o r a s u r v e y .

W e h a v e s h o w n t h a t c o m p u t i n g r ( A ) c a n b e r e d u c e d to t h e m a x - c u t. N o w w ep r e s e n t a n o p p o s i t e r e d u c t i o n , i n o r d e r t o e s ta b l is h t h a t c o m p u t i n g r ( A ) is N P - h a r d .W e r e c a ll t h a t t h e cardinaI i ty vers ion o f th e m a x - c u t , i .e ., w h e n a l l t h e w e i g h t s c o a r e0 o r 1, is a l r e a d y N P - h a r d ( se e p r o b l e m G T 2 5 , p . 1 96 , o f [ G J ] ) . T h e c a r d i n a l i t yv e r s i o n i s s o m e t i m e s c a l l e d t h e maximum bipar t i te subgraph problem.T h e o r e m 2.6. Com put ing r (A) i s NP -ha rd fo r a ma tr i x A w i th ra tiona l en t ri e s.

P r o o ~ L e t G = ( N , E ) b e a g r a p h . D e f i n e a m a t r i x A b yif i j a E , i ri f i j 6 E , i e j ,i f i = j ,

w her e M i s a s u f f i c i en t l y l a r ge i n t e ge r ( M > 21El is su f f ic i en t) . L e t r (A ) = z t A y f o rs o m e z , y ~ Q . I t is e a s y t o s e e t h a t z = y b e c a u s e o f t h e c h o i c e o f M . F o r e a c h y ~ Q ,wi th S -= {i[y~ = 1}, we ha ve

y t A y = ~ . a o y i y = ~ ( _ 8 9 i _ 35)2 _ 2)t J LJ= --89 .~. ai~(Yl -- 35)2 + .~. ai = M n + 413S1 - 21E l

l~J t J

h e n c e r ( A ) = M n + 4 m c ( G ) - 2 1E l. T h u s , a n e x i s t e n c e o f a p o l y n o m i a l - t i m e a lg o -r i th m t o c o m p u t e r ( A ) w o u l d y i e ld a p o l y n o m i a l - t im e a l g o r i t h m t o c o m p u t e m c (G ) .S i nc e th e l a tt e r is a n N P - h a r d p r o b l e m , c o m p u t i n g r (A) i s N P - h a r d a s w e ll . 9

A n i m m e d i a t e c o r o l l a r y is t h e s t a t e m e n t f o r m u l a t e d i n t h e a b s t r a c t.Cor o l l a r y 2 . 7 . The fo l low ing prob lem i s NP -hard .I n s t anc e : k + 1 square matr ices hav ing rat ional entr ies , Ao, A1, . . . A k .Q u e s t i o n : Is the m atr ix A o + r l A1 + "" + rkAk nons ingular for a l l poss ib le choicesof real numbers r l , . . . , rk in the in terval I-0, 1]?

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Checking Robust Nonsingularity s NP-H ard 7P r o o f . W e s h o w t h a t th e p r o b l e m o f c o m p u t i n g t h e r a d iu s o f n o n s i n g u l a r it y c a nb e r e d u c e d t o i t. A s s u m e t h a t w e w a n t t o d e c i d e w h e t h e r t h e r a d iu s o f n o n s i n g u l a r i t yo f a g i v e n ( r a t i o n a l ) m a t r i x A i s a t l e a s t a g i v e n e > 0 . T h e n c o n s i d e r A o = A ,k = 2 n 2 , a n d d e f i n e t h e m a t r i x A i j w h o s e / j t h e n t r y is e a n d a l l o t h e r e n t r i e s 0 ,a n d A }j = - A i j f o r i , j = 1 . . . , n . C l e a r l y , t h e r a d i u s o f n o n s i n g u l a r i t y o f A i sa t l e a s t e i f a n d o n l y i f t h e p r o b l e m f o r m u l a t e d i n t h e c o r o l l a r y h a s a p o s i ti v ea n s w e r . 9

L e t u s r e m a r k t h a t t h e p r o b l e m f o r m u l a t e d i n C o r o l l a r y 2 . 7 i s a l g o r i t h m i c a l l yd e c i d a b le b y th e w o r k o f T a r s k i I T ] . H o w e v e r , w e d o n o t k n o w w h e t h e r o r n o t t h ep r o b l e m b e l o n g s t o th e c la s s N P ( t h o u g h w e c o n j e c t u r e t h a t t h e a n s w e r i s y e s ) . T h ed i f f ic u l t y a r i se s f r o m t h e f a c t t h a t A o + r l A 1 + + r k A k m a y b e s i n g u l a r o n l y f o r( r 1 . . . , r k) i r r a ti o n a l , a s s h o w n i n t h e f o l lo w i n g e x a m p l e . L e t k = 1, a n d

T h e n d e t~ A o + t A ~ ) = t 2 + 3 t + 1 , an d A o + t A 1 i s s i n g u l a r i f a n d o n l y i f t =( - 3 + , , /5 ) /2 .

T h u s , t h e s i n g u l a r i t y o f A o + f l a x + . . + r k A k c a n n o t b e c e r ti fi e d b y a d i r e c tc h e c k f o r c o n c r e t e v a l u e s o f ( r 1 . . . . r k) . T h e m e t h o d u s e d b y T a r s k i is i n d ir e c t, b a s e do n t h e S t u r m t h e o r e m a n d i t s g e n e r a l i z a ti o n s f o r m o r e v a r i a b le s . H o w e v e r , h isc e r t if i c a ti o n r e q u i re s c r e a t i n g a h u g e f a m i l y o f a u x i l i a ry p o l y n o m i a l s , a n d h e n c e i tis n o t p o l y n o m i a l - t i m e b o u n d e d i n t h e s iz e o f t h e i n p u t d a t a .F i n a l ly , w e s h o w t h a t a r e l a t e d p r o b l e m o f si n g u l a r it y o f i n t e r v a l m a t r i c e s isN P - c o m p l e t e . I n c o n t r a s t t o C o r o l l a r y 2 .7 , w e a r e a b le t o e s ta b l is h t h e m e m b e r s h i pi n t h e cl as s N P . A s q u a r e i n t e r v a l m a t r i x A = {A'I_A < A' < . , t } i s ca l l ed s i n g u l a ri f i t c o n t a i n s a s i n g u l a r m a t r ix . C o n s i d e r t h e d e c i s i o n p r o b l e m :I n s t a n c e : S q u a r e i n t e r v a l m a t r i x A I, w h e r e b o t h A a n d . 4 a r e r a t i o n a l m a t r ic e s .Q u e s t i o n : I s A I s i ng u l a r ?T h e o r e m 2 8 Th e recogn i t ion prob lem of sin gu lar i t y of in terva l mat r i ces is N Pcomplete.P r o o f . I t is e a s y t o s ee t h a t c o m p u t i n g r ( A ) c a n b e r e d u c e d t o t h e p r o b l e m w h e t h e ra n i n t e r v a l m a t r i x i s s i n g u la r , a n d h e n c e t h e p r o b l e m i s N P - h a r d . I t r e m a i n s t o s h o wt h a t i t b e l o n g s t o t h e c la s s N P , t h e c la s s o f n o n d e t e r m i n i s t i c - p o l y n o m i a l - t i m ep r o b l e m s . W e c l a i m t h a t i f a n i n t e r v a l m a t r i x A x = { A'I_ A < A ' < .4 } is s i n g u l a r ,t h e n t h e r e e x i s ts a s i n g u l a r m a t r i x A ' in t h e i n t e r v a l s u c h t h a t a l l t h e e n t r i e s o f A 'a r e r a t i o n a l n u m b e r s w h o s e s iz es a re b o u n d e d b y a p o l y n o m i a l in t h e s iz e s o f t h ee n t r i e s o f _A a n d .4 . S u c h a m a t r i x A ' c a n b e " g u e s s e d " (i.e ., g e n e r a t e d b y a n o n -d e t e r m i n i s t ic a l g o r it h m ) , a n d t h e n i t c a n b e c h e c k e d d e t e r m i n i s ti c a l ly in p o l y n o m i a lt i m e t h a t A ' is si n g u la r , s in c e G a u s s i a n e l i m i n a t i o n is k n o w n t o b e p o l y n o m i a l t i m en o t o n l y in t h e n u m b e r o f a r i t h m e t i c o p e r a t i o n s , b u t a l s o t h a t t h e s iz es o f t h en u m b e r s t h a t o c c u r d u r i n g t h e e li m i n a t io n r e m a i n p o l y n o m i a l l y b o u n d e d . ( A d e -t a i l e d a n a l y s i s c a n b e f o u n d i n I S ] . ) T h i s g i v e s t h e r e q u i r e d n o n d e t e r m i n i s t i c -p o l y n o m i a l - ti m e a l g o ri th m .

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8 s. PoUak and J. RohnT h e v a l i d i t y o f t h e c l a i m f o l lo w s f r o m a r e s u l t o f t h e s e c o n d a u t h o r ( se e p a r t C 7

o f T h e o r e m 5 .1 o f I- R ]) , w h o p r o v e d t h a t i f A I is s i n g u l a r, t h e n t h e r e e x i s t s a s i n g u l a rm a t r i x A = (A ~ ) ~ A t w i t h t h e f o l l o w i n g p r o p e r t y : t h e r e i s a p a i r ( k , m ) o f s u b s c r i p t ss u c h t h a t , f o r e v e r y i , j ) v~ k , m) , en tr y A~j sat is f ies ei th er A~i = _A~j o r Aij = -4i~.H o w e v e r , s i n c e A is s in g u l a r , t h e e x c e p t i o n a l e n t r y A k , c a n b e e x p r e s s e d a s a li n e a rc o m b i n a t i o n o f s u b d e t e r m i n a n t s o f A u s i n g th e L a p l a c e e x p a n s i o n . H e n c e t h e si zeo f a l l e n t r i e s o f A is b o u n d e d b y a p o l y n o m i a l i n t h e s iz e s o f e n t r ie s o f _A a n d A . T h i sp r o v e s t h e c l a im . 9

3 . C o n c l u d in g C o m m e n t sW e c o n c l u d e b y m e n t i o n i n g t w o p o s s i b l e a p p li c a t io n s o f t h e r a d i u s o f n o n -s i n g u l a r i t y .D a t a u n c e r t a i n t y . A s s u m e t h a t w e h a v e o b t a i n e d e n t r ie s o f a m a t r i x A = (a~ j) a s ar e s ul t o f a n e x p e r i m e n t w h e r e t h e d a t a w e r e m e a s u r e d b y a d e v i ce e n s u ri n g s o m e( u n i f o rm ) p r e c i s i o n 6. T h i s m e a n s t h a t it i s g u a r a n t e e d t h a t t h e ( u n k n o w n ) a c t u a lv a l u e i s i n t h e i n t e r v a l [ a i i - 6 , a 0 + 6 ] . N o w , w e h a v e t o d e c i d e w h e t h e r A i ss u i t a b l e f o r f u r t h e r n u m e r i c a l p r o c e s s i n g , o r w h e t h e r t h e e x p e r i m e n t s h o u l d b er e p e a t e d w i t h b e t t e r p r e c is i o n, w h i c h m a y b e m o r e c o st ly . O u r d e c is i on w i ll d e p e n do n w h e t h e r 6 > d A ) ( a n e w e x p e r i m e n t i s n e c e s s a r y ) o r 6 < d A ) ( t h e d a t a a r esuf f ic ien t ly prec ise) .R o u n d i n g in f i x e d - p o i n t a r it h m e ti c . A s s u m e t h a t a m a t r i x A w i t h p o s s i b ly i rr a t io -n a l e n t ri e s is g i v en . S u c h a s i t u a t i o n m a y o c c u r w h e n t h e d a t a a r e d e r i v e d f o r m a l l y ,e .g ., v / 2 m a y a r i s e a s a d i s ta n c e . I f w e i n t e n d t o a p p l y a n u m e r i c a l a l g o r i t h m , w eh a v e t o r o u n d o f f e a c h e n t r y t o s o m e n u m b e r p o f d e c i m a l d i gi ts . L e t . 4 d e n o t e t h em a t r i x o f r o u n d e d e n t r ie s , c a l le d a r e p r e s e n t a t i o n m a t r i x . I f I[A - -411 > d(.~), th isi n d i c a t e s a p o t e n t i a l l y d a n g e r o u s s i t u a ti o n , s i n c e t h e p r e s e n c e o f a s i n g u l a r m a t r i xw i t h in t h e p r e c i s io n o f A m a y m e a n t h a t A d o e s n o t r ef le c t w e ll t h e p r o p e r t i e s o f A .Acknowledgment T h e a u t h o r s t h a n k t w o a n o n y m o u s r ef er ee s f o r v a l u a b le c o m -m e n t s a n d h e lp f u l su g g e s t io n s t o i m p r o v e t h e t e x t o f t h e p a p e r .

References[B] F. Barahona, A solvable case of quadratic 0-1 programming, Discrete Ap pl. Ma th. , 13 (1986),24-36.[BS] T. A. B rown an d J. Sp encer, M inimization of _+ 1-matrices un de r line shifts, Colloq. Math.Poland), 23 (1971), 165 -171.[D1] A. Deif, Sensit ivi ty Analysis in L inear Systems, Springer-V erlag, Berlin, 1986.[DP ] Ch. Delorme and S. Poljak, Laplacian Eigenvalues and the M aximum Cut Problem, Technical

Re port 599, L.R.I., Un iversit6 Paris-S ud, 1990.[DL ] M. Deza and M . Laurent, Facets for the cut cone, I, M ath. Programming,56 (to appear).[-D2] J .C . Doy le, Analysis of feedback systemswith structured uncertainties, Proc. IEE E, 129 (1982),242-250.

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C he c k i ng R obus t Nons i ngu l a r i t y i s NP -H a rd 9[ S][ G J ][ L S ][ M ]

[ M P ][ P R S ]

[ P T ][ R ]IS ][ T ]

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