2 MARTIN GAVALE, GÜNTER ROTE R Ψ 2 3 I 4 6 R Ψ 5 Figu I ff R Ψ lcm(3; 4; 6) = 12. Th componnts a non-compaabl, in th sns of th o

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1 REAHABILITY OF FUZZY MATRIX PERIOD MARTIN GAVALE, GÜNTER ROTE Abstact. Th computational complxity of th matix piod achability (MPR) poblm in a fuzzy algba B is studid. Givn an n n matix A with lmnts in B, th poblm is to dcid whth th is an n-vcto x such that th squnc of matix pows A; A 2 ;A 3 ;::: has th sam piod lngth as th squnc Ax; A 2 x; A 3 x;::: of itats of x. In gnal, th MPR poblm is NP-complt. Two conditions a dscibd, which both togth imply that MPR can b solvd in O(n 2 ) tim. If only on of th conditions is satisfid, th poblm mains NP-complt. 1. Intoduction Pow squncs of matics in fuzzy algba w studid by R. A. uningham- Gn [2]. Th convgnc and piodicity of spcial classs of matics w studid by M. G. Thomason [9], and subsquntly by many oth authos. Li Jian-Xin [7, 8] considd th piodicity of fuzzy matics in th gnal cas and gav an upp stimat fo th piod of a matix. Th convgnc of th pow squnc of a squa matix in fuzzy algba was studid using digaphs by K. chláová [1], and a ncssay and sufficint condition was givn fo a matix A to b stationay. Th computational complxity of sval poblms connctd with finding th xact valu of th matix and obit piod in a fuzzy algba B was considd in [3]. It was shown that th piod of a matix A is th last common multipl of th piods of at most n non-tivial stongly connctd componnts in thshold digaphs G(A; h) fo som thshold lvls h and an algoithm was suggstd which nabls to comput th matix piod in O(n 3 ) tim. On th oth hand, th matix piod p(a) is th last common multipl of th obit piods p(a; x) fo all vctos x 2B(n). In [4], th qustion of aching th matix piod by som obit piod is discussd. Th aching matix piod (MPR) poblm is shown to b NP-complt in [5]. Th aim of this pap is to discuss som conditions und which th MPR poblm can b solvd in polynomial tim. To illustat th situations that can occu in this poblm, w stat with th xampls. W f th ad to Sction 2 fo th pcis dfinition of all tms. Exampl 1. Lt B = f0; 1g and lt A 2 B(n; n) b th adjacncy matix of th digaph G in Figu 1. Th digaph consists of n = 13 vtics 0; 1;:::;12 in 3 disjoint cycls, 0 ; 1 ; 2, of lngths 3, 4 and 6. Th digaph G is th thshold digaph of matix A fo thshold lvl h = 1, and th cycls 0 ; 1 ; 2 a its only (non-tivial) stongly connctd componnts. Th componnts piods a 3; 4; 6. Thfo, th matix piod p(a) is qual to Dat: Apil 4, Ky wods and phass. piod of a matix, obit of a vcto, minimax algba, fuzzy algba, NP-compltnss. 1

2 2 MARTIN GAVALE, GÜNTER ROTE R Ψ 2 3 I 4 6 R Ψ 5 Figu I ff R Ψ lcm(3; 4; 6) = 12. Th componnts a non-compaabl, in th sns of th oding of componnts of th digaph G, i.., thy a not connctd by any walk in G. As a consqunc of th non-compaability, th vcto x 2 B(n) dfind on th vtx st by putting x(i) = 1 fo i = 0; 3 and x(i) = 0 othwis, has its obit piod p(a; x) = lcm(3; 4) = 12. That mans that this instanc of th MPR poblm has th positiv answ. Lt us mak that th sam sult w can obtain by putting,.g., x 0 (i) = 1 fo i = 4; 7 and x 0 (i) = 0 othwis. In this cas, th obit piod is p(a; x 0 ) = lcm(4; 6) = 12. In gnal, w can choos two vtics, i (o i ) and i and put x(i) =1foi=i 0 ;i 1 and x(i) = 0 othwis R Ψ I 6 R Ψ 5 Figu 2 Exampl 2. A is th adjacncy matix of th digaph G on Figu 2. Th digaph consists of n = 7 vtics 0; 1;:::;6 in 2 disjoint cycls 0 ; 1, of lngths 3 and 4. Th cycls a connctd with an ac lading fom th vtx 2 to 3, i.., 0 < 1, in th oding of componnts. Th cycls 0 ; 1 a th only non-tivial stongly connctd componnts. Th componnt piods a 3 and 4 and th matix piod is p(a) = lcm(3; 4) = 12. Howv, in this cas th is no vcto x 2 B(n) with th popty p(a; x) = p(a). Th ason is that th piods of both cycls hav th gatst common diviso qual to 1, which implis that vy lag nough intg can b xpssd as a lina combination with positiv cofficints, of componnt piods 3; 4. As a consqunc, fo any vtx i 2 0 and j 2 1, th a walks bginning in i and nding in j, of any givn lngth gat than som valu. If x(j) = 1 fo som j, thn th i-th coodinat in th obit of vcto x has vntually constant valu 1, s Lmma 2.1 blow. That mans that th numb 3 will not occu as a coodinat-obit

3 REAHABILITY OF MATRIX PERIOD 3 piod of x. Thus, th obit piod p(a; x) will not b a multipl of 3. On th oth hand, if x(j) = 0 fo vy vtx j 2 1, thn 4 cannot occu as th coodinat piod and th obit piod p(a; x) will not b amultipl of 4. Exampl 3. In this xampl th digaph G on Figu 3 consists of n =10vtics 0; 1;:::;9 in 2 disjoint cycls 0 ; 1, of lngths 4 and 6. Th cycls a connctd with an ac lading fom th vtx 2 to 4, i.., 0 < 1. Th situation diffs fom th pvious xampl by th fact that th lngths of th cycls hav a non-tivial common diviso R 7 I R - Ψ ff I Ψ Figu 3 Again, th cycls 0 ; 1 a th only non-tivial stongly connctd componnts. Th componnt piods a 4 and 6 and th matix piod is p(a) = lcm(4; 6) = 12. In this cas, w can find a vcto x 2 B(n) with th popty p(a; x) = p(a). W choos two vtics, i 2 0 and j 2 1 and put x(i) = x(j) = 1 and x(k) = 0 othwis. If th is a walk of odd lngth fom i to j, thn all walks fom i to j hav odd lngths and, by Lmma 2.1(ii), th i-th coodinat in th obit squnc of x on odd positions vntually has constant valu 1, whil on vn positions it oscillats btwn valus 0 and 1. Thus, th i-th coodinat-obit piod is p(a; x; i) = 4. On th oth hand, p(a; x; j) = 6 and, by Thom 2.2(ii), th obit piod is p(a; x) = 12. W may notic that, if w choos th vtics i 2 0 and j 2 1 in such a way that th walks fom i to j hav vn lngths, thn by a simila agumnt as abov, th i-th coodinat in th obit of vcto x vntually has constant valu 1 on vn positions and 0 on odd positions, thfo p(a; x; i) = 2. Thus, th numb 4 dos not occu as a coodinat-obit piod of x and th obit piod is p(a; x) =6. Ou th xampls indicat that in saching fo a vcto solution of th quation p(a; x) = p(a), w hav to find a systm of stongly connctd componnts with suitabl piods and to choos a suitabl systm of vtics, on in ach of ths componnts. Th xact fomulation of this ida can b found in Thom 3.5. It can b asily sn that if th componnts can b found in suchaway that thy a paiwis non-compaabl, thn any choic of vtics (on vtx in ach componnt) inducs a solution of th MPR poblm. If th xist compaabl componnts in th thshold gaph G(A; h), thn th solvability of th poblm dpnds on th gatst common divisos of th piods fo compaabl pais of componnts. In th pap, two conditions a dscibd which togth imply that MPR poblm is polynomially solvabl (Thom 4.1). W show that if only on of th conditions is fulfilld, thn MPR is NP-complt (Thoms 4.2 and 4.3).

4 4 MARTIN GAVALE, GÜNTER ROTE 2. Matix and obit piods In this sction w dfin th notions mntiond infomally in th intoduction. Fo simpl notation of indx sts w shall us th convntion by which any natual numb n is considd as th st of all small natual numbs, i.., n = f0; 1;:::;n1g. By N w dnot th st of all non-ngativ intgs. Th gatst common diviso and th last common multipl of a st S N a dnotd by th abbviations gcd S and lcm S, spctivly. If G = (V; E) is a digaph (dictd gaph), thn by a stongly connctd componnt of G w man a subdigaph K = (K; E K 2 ) gnatd by a non-mpty subst K V such that vy vtx x 2 K is achabl fom vy oth vtx y 2 K, and K is a maximal subst with this popty. Th vtx sts K of th stongly connctd componnts fom a patition of V. Fo i 2 V,wdnot by K[i] th uniqu stongly connctd componnt to which i blongs. A stongly connctd componnt K is calld non-tivial if th is a cycl of positiv lngth in K; othwis, is is calld tivial. By S Λ (G) w dnot th st of all non-tivial stongly connctd componnts of G. Fo any K2S Λ (G), th piod p(k) is dfind as gcd of th lngths of all cycls in K. If K is tivial, thn p(k) =0. Dfinition 2.1. Th fuzzy algba B is a tipl (B; Φ; Ω), wh B is a linaly odd st and Φ; Ω a th binay opations of maximum and minimum, spctivly, onb. Fo any natual n>0, B(n) dnots th st of all n-dimnsional column vctos ov B, and B(n; n) dnots th st of all squa matics of od n ov B. Fo x 2B(n);A=(a ij ) 2B(n; n), w dnot by μx and A μ th input sts f xi ; i 2 n g and f a ij ; i; j 2 n g, spctivly. Th matix opations ov B a dfind fomally in th sam mann (with spct to Φ; Ω) as th matix opations ov a ing. Fo any A 2 B(n; n);x 2 B(n), th obit of x gnatd by A is th squnc x (0) ;x (1) ;x (2) ;:::, wh x () := A x. Dfinition 2.2. Lt A 2B(n; n);h 2B. Thn th thshold digaph G(A; h) is th digaph G = (n; E), with th vtx st n = f0; 1;:::;n1g and with th ac st E = f (i; j) :i; j 2 n; a ij h g. Fo any natual and fo any two vtics i; j of a gaph G, w wit i! j if th is a walk of lngth in G, bginning in vtx i G and nding in j. If th gaph G is undstood fom th contxt, w will simply wit i! j. Th following lmma, which is du to chláová [1], givs a connction btwn th xistnc of walks in thshold gaphs and valus of matix pows A and obit vctos x (). Lmma 2.1. Lt A 2B(n; n); x 2B(n);h2B;2N;1;i; j 2 n. Thn (i) (A ) ij h, i! j (ii) (A x) i h, (9j 2 n) h x j h ^ i Poof. By induction on. i! j Dfinition 2.3. Lt A 2 B(n; n); x 2 B(n). Th piod of an infinit squnc a 1 ;a 2 ;::: is th smallst positiv numb p fo which th is an R such that fo all > R w hav a = a +p, if such a numb xists. Th matix piod, th obit piod and th i-th coodinat-obit piod of x with spct to A, in notation: p(a),

5 REAHABILITY OF MATRIX PERIOD 5 p(a; x) and p(a; x; i), a dfind as th piods of th squncs A, x (),ox () i (=1;2;:::), spctivly. Th piod of a st R N is th piod of its chaactistic squnc 8 < 1 if 2 R a := : 0 othwis. Rmak 2.1. By linaity of B, any lmnt of any pow of th matix A is qual to som lmnt of A. Thfo, th squnc of pows of A contains only finitly many diffnt matics. As a consqunc, th piods p(a); p(a; x); p(a; x; i) a always wll-dfind. Th connction btwn matix piods, obit piods and coodinat-obit piods is dscibd by th following thom. Thom 2.2. Lt A 2B(n; n); x 2B(n). Thn (i) p(a) = lcm p(a; x) x2b(n) (ii) p(a; x) = lcm p(a; x; i) i2n Poof. It is asy to s that p(a) is a multipl of p(a; x) fo any x 2 B(n) and thfo, p(a) isamultipl of lcm p(a; x). x2b(n) Fo th convs lation, lt us assum that p is a common multipl of p(a; x) fo all x 2B(n), i.., fo any x 2B(n)th is a numb R x such that (8 >R x )x () = x (+p) :In gnal, th a infinitly many vctos x 2B(n), but w can stict ou considation to only finitly many ofr x, bcaus th obit piod of a givn vcto x 2B(n) dpnds only on th way in which th lmnts of μx a compaabl with all lmnts of A μ and th a only finitly many possibilitis fo placing th lmnts of μx btwn th lmnts of A. μ Thfo, th a only finitly many quivalnc classs in B(n) with spct to th compaability with lmnts of A μ (in th sns that x; x 0 2 B(n) a quivalnt if and only if (8i; j; k 2 n)[ a ij» x k, a ij» x 0 k ] ). Thus, w may conclud that (9R 2 N)(8 >R)(8x 2B(n)) x () = x (+p) : By th simpl fact that A = B, (8x 2B(n)) Ax = Bx, w gt (9R 2 N)(8 2 R) A () = A (+p) i.., p is a multipl of p(a). Th asstion (ii) is povd analogously. Dfinition 2.4. Lt A 2 B(n; n); h 2 B, lt G(A; h) b a thshold digaph of A. Rcall that S Λ G(A; h) dnots th st of all non-tivial stongly connctd componnts of G(A; h). (i) S Λ (A) := S fs Λ G(A; h) :h2 μ Ag (ii) S min (A) :=fk2s Λ (A) :Kis minimal in (S Λ (A); ) g Th piod of a matix in max-min (fuzzy) algba is chaactizd by th piods of th non-tivial stongly connctd componnts in th thshold gaphs of th matix: Thom 2.3. [3] Lt A 2B(n; n). Thn (i) p(a) = lcm f p(k); K2S Λ (A) g (ii) p(a) = lcm f p(k); K2S min (A) g

6 6 MARTIN GAVALE, GÜNTER ROTE 3. Rachability of th matix piod In viw of Thom 2.2(i), a natual qustion aiss: und which conditions is th valu of th matix piod achivd by som obit piod [5]? Dfinition 3.1. Matix Piod Rachability Poblm (MPR) Givn a matix A 2B(n; n), is th x 2B(n)such that p(a) =p(a; x)? In this sction w show that if an instanc of MPR poblm has a solution, thn th solution can b psntd in a standad fom, which will b dscibd in Thom 3.5. This sult is in accodanc with ou obsvations mad by th xampls dscibd in th intoduction. Th following dfinition and lmma a cucial fo invstigating th piodicity of th obit squnc ( x () ; = 1; 2;:::), sinc thy stablish a connction with th piods of paths and componnts in a digaph. Dfinition 3.2. Fo vtics i and j in a digaph G, w dnot by W (i; j) th st of all lmntay paths fom i to j (i.., walks with distinct vtics) in a digaph. Fo any walk w w dfin and w dnot p(w) = gcdf p(k) :K2S Λ (G) ^K w6=;g R(i; j; w) :=f2n : th isawalk w 0 fom i to j with acs and w w 0 g: Rmak 3.1. If th walk w mts no non-tivial componnt K 2 S Λ (G), thn by Dfinition 3.2, p(w) = 0 holds tu and R(i; j; w) contains th only lmnt jwj: Th notation of p(w) as th piod of th walk w is justifid by pat (ii) of th following lmma. Lmma 3.1. Fo any digaph G, th a numbs R 1 and R 2 fo which th following holds. (i) If a vtx i is containd in a componnt K, thn (8 >R 1 ) h i 0 mod p(k), i! i (3.1) (ii) Fo any two vtics i and j and fo any w 2 W (i; j), w hav (8 >R 2 ) h jwjmod p(w), 2 R(i; j; w) i (iii) Fo any two vtics i and j and fo any walk w fom i to j, w hav (8 >R 2 ) h i jwjmod p(w) ) i! j Poof. In pat (i), th xistnc of R 1 fo any spcific vtx i follows fom th dfinition of p(k). As th numbofvtics is finit, w can tak R 1 lag nough such that (3.1) holds fo all vtics i in G. Fo pat (ii), w fist not that if w dos not mt any K 2 S Λ (G), thn, by mak 3.1, p(w) = 0 and th statmnt of th lmma is tivially fulfilld. Now, w show that whnv K2S Λ (G) is a non-tivial componnt with K w6=;,thn 2 R(i; j; w) ^ c 2 N ^ c p(k) R 1 ) + c p(k) 2 R(i; j; w): If k 2K wfo som K2S Λ (G), thn, by pat (i) of th lmma, th is a cycl though k with c p(k) acs whnv c p(k) R 1. This cycl may b addd to w to obtain a walk of lngth + c p(k) fom i to j. Sinc p(w) is th gatst

7 REAHABILITY OF MATRIX PERIOD 7 common diviso of p(k) fo all non-tivial componnts K 2 S Λ (G) which mt w, thn, by a wll-known thom of numb thoy, any big nough multipl of p(w) can b xpssd as a lina combination, with positiv cofficints, of th cosponding compomnt piods p(k). Thfo, if th st R(i; j; w) contains an lmnt in som congunc class modulo p(w), thn all lag lmnts in that congunc class a also containd in R(i; j; w). Thus, R(i; j; w) contains ith all lmnts of any giv congunc class xcpt a finit numb, o non at all. Sinc th a only finitly many congunc classs, finitly many vtics i and j, and finitly many lmntay paths w, R 2 can b slctd lag nough so that th statmnt of th lmma is fulfilld. Fo pat (iii), not fist that w may assum w.l.o.g. that w is an lmntay path. If w contains a vtx k twic, th cycl that w foms btwn th fist and last visit to k has a lngth which is a multipl of p(k[k]). Thus, if w clip out of w, th lngth mod p(w) mains unchangd. Futhmo, dos not touch any additional componnts, by th dfinition of stongly connctd componnts. Thus, th st of componnts K in th dfinition of p(w) is also unchangd. By patdly dlting cycls out of w, w vntually aiv at an lmntay path with lss that n acs that has th sam piod p(w) and th sam lngth modulo p(w) as th oiginal walk. Th statmnt of th lmma is now a simpl consqunc of pat (ii). Th following squnc of lmmas will allow us to convt a givn vcto x into anoth vcto that has th sam obit piod p(a; x) and satisfis additional containts. Fo a givn vcto x 2B(n)and fo givn h 2B,wdnot S(x; h) :=fi2n:x(i)hg: Fo i 2 n, w dnot by K[i; h] th uniquly dtmind stongly connctd componnt in G(A; h) containing th vtx i. Lmma 3.2. Lt A 2 B(n; n); x 2 B(n). Thn th is vcto x 0 2 B(n) with p(a; x 0 ) = p(a; x), such that any vtx i 2 S(x 0 ;h) is containd in a non-tivial componnt of th thshold digaph G(A; h), fo any h 2B. Poof. Lt H = f h k ; k 2 s g = A μ [ μx b th union of th input sts of A and x, in dscnding od. W shall pocd by cusion though H. W tak h = h k 2 H and w assum that fo any h l > h, th asstion of lmma holds tu. Thn w modify x to x 0 by th following two uls (all walks mntiond in th poof a in digaph G(A; h)). 1. Fo any j 2 S(x; h) with K[j; h] tivial, w dfin x 0 (j) =h k+1 (i.., j is lft out of S(x; h), but it mains in S(x; h l ) fo any h l 2 H; h l <h). s 2. Fo any j 2 S(x; h) with K[j; h] tivial and fo any w : i! j such that i is th only vtx in w with non-tivial componnt K[i; h], w choos a vtx s j w 2K=K[i; h] such that i! j w. As K is non-tivial, such a vtx j w can always b found in K. Thn w st x 0 (j w ) = max(h; x(j w )), (i.., j w is addd to S(x; h) but not to S(x; h l ) fo h l 2 H; h l >h), if it has not alady bn th). All maining valus x(j) a unchangd at this stag of cusion. W may notic that th uls 1, 2 apply only to vtics j with x(j) =h. Namly, ifx(j)<h,thn j 62 S(x; h) and if x(j) >h,thn, by cusion assumption, K[j; h] is non-tivial. In viw of Lmma 2.1(ii) and Thom 2.2, it is sufficint to vify that fo any i 2 n and fo any big nough, th quality (A x) i =(A x 0 ) i holds tu, i.., th following

8 8 MARTIN GAVALE, GÜNTER ROTE two fomulas a quivalnt: (9j 2 n) h x(j) h ^ i! j i ; (9j 0 2 n) h x 0 (j 0 ) h ^ i! j i: 0 Lt th fist fomula b fulfilld, in oth wods, lt th b j 2 n such that x(j) h and w : i! j. If is big nough, thn th walk w cannot b an lmntay path, i.., w must mt som non-tivial componnt in G(A; h). If th componnt K[j; h] is non-tivial, thn w hav x 0 (j) = x(j) and th scond fomula holds tu. If th componnt K[j; h] is tivial, thn w dnot by i 1 th last vtx in w with non-tivial K[i 1 ;h] = K and w dnot w 1 := w(i 1 ;j) (th subwalk of w fom i 1 to j). By th ul 2 of th dfinition of x 0, th is a vtx j 0 = j w1 2 K and a walk w 0 s 1 : i 1! j 0 such that s = jw 1 j and x 0 (j 0 ) h. If w dnot w 0 := w(i; i 1 ), thn th walks w = w 0 w 1 and w 0 = w 0 w 0 1 a of th sam lngth and, thfo, th scond fomula holds tu. onvsly, lt th scond fomula b fulfilld, i.. lt j 0 2 n such that x 0 (j 0 ) h and w : i! j 0. By th dfinition of x 0, th componnt K := K[j 0 ;h] is non-tivial. If x(j 0 ) h, thn th fist fomula holds tu. If x(j 0 ) < h, thn th a vtics s i 1 2K;j62 K and walks w 1 : i 1! j; w 0 1 :! s j 0 such that x(j) h and i 1 is th only vtx in w 1 with non-tivial componnt K[i; h] =K. Wchoos a walk u : j 0 t! i 1, u K. Th concatnation c = uw 0 1 is a cycl in K and, thfo, th lngth s + t of c is a multipl of p(k) and s + t mod p(k). As is big nough, w can assum that s>r 2,and as p(k) is a multipl of p(w) =p(wu), w hav s + t mod p(wu). By Lmma 3.1(iii) applid to th concatnation wu of s lngth + t w gt a walk w 0 : i! i 1. Thn w 0 w 1 : i! j and th fist fomula holds tu. Th ky notions in Thom 3.5 a th notion of dominanc of vtics in a digaph G and th notion of dciding componnts in S Λ (G). Dfinition 3.3. Lt i; j 2 n; h 2 B. notation: i μ h j), if (9R 2 N)(8 >R)(8k 2 n) W say that j dominats i at lvl h (in " k #! i ) k! j : Rmak 3.2. Fo fixd h, th lation μ h is flxiv and tansitiv (i.., μ h is a quasi-od on n). Not that th thshold R := R ijh in Dfinition 3.3 may dpnd on i; j and h. By taking th maximum R of th finitly many numbs R ij, (fo i; j 2 n; h 2 A)whav μ a global constant R with th popty: " # i μ h j ) (8 >R)(8k 2 n) k! i ) k! j : (3.2) Rmak 3.3. If th componnt K = K[i; h] is non-tivial, thn, in viw of Lmma 3.1, i μ h j holds tu if and only if " # (8 >R) i! i ) i! j which can b quivalntly xpssd as " (8 >R) 0 mod p(k) ) i #! j o,

9 (9 >R) REAHABILITY OF MATRIX PERIOD 9 " 0 mod p(k) ^ i! j # (3.3) Lmma 3.3. Lt A 2 B(n; n); x 2 B(n). Thn th is vcto x 0 2 B(n) such that p(a; x 0 ) = p(a; x) and any vtx i 2 n with x 0 (i) =h>min( μ A) is not dominatd at lvl h by any vtx j 6= i with x 0 (j) x 0 (i). Poof. If th a vtics i 6= j such that min( A) μ < h = x(i)» x(j) and i μ h j, thn w dnot by h 0 th pcdsso of h in A μ [ μx and dfin 8 < h 0 if k = i x 0 (k) := : x(k) othwis. In viw of Lmma 2.1(ii), th vctos x; x 0 hav th sam obits and, thfo, p(a; x) = p(a; x 0 ). Th pocdu is patd finitly many tims, until x 0 has th dsid popty. Rmak 3.4. As K[i; h] K[i; h 0 ] holds tu fo h h 0, th pocdu in th abov poof psvs th popty of vcto x 0, fomulatd in Lmma 3.2. Th following lmma povids a low bound fo th piod of a squnc, if th piod of som subsqunc is known. Lmma 3.4. If a squnc a 1 ;a 2 ;::: has piod p, thn th piod of th subsqunc a d ;a 2d ;a 3d ;::: fomd by taking vy d-th lmnt divids p= gcd(p; d). Poof. Sinc p divids lcm(p; d) =pd= gcd(p; d), w hav a id = a id+pd= gcd(p;d) = a (i+p= gcd(p;d))d ; fo lag nough i, and hnc p= gcd(p; d) is a multipl of th piod of th subsqunc. Dfinition 3.4. Lt p(a) = p ff 0 0 p ff 1 1 :::p ff k1 k1 b a dcomposition of th matix piod p(a) into pows of distinct pims. Thn a subst D S Λ (A) which contains a componnt K with p fft t j p(k), fo all t =0;::: ;k1, is calld a dciding st of componnts. Thom 3.5. Fo any A 2B(n; n), th following statmnts a quivalnt. (i) Th is x 2B(n)such that p(a; x) = p(a): (ii) Th isadciding st D of paiwis disjoint componnts at lvls H = f h K ; K2 D^K2S Λ (G(A; h K )) g and a st I = f i K 2 K : K 2 D g of vtics, with on vtx chosn fom ach componnt in D, such that, if h K» h L ; K 6= L, thn th vtx i L dos not dominat i K at lvl h K. Poof. (ii) ) (i): W dfin avcto x 2B(n)in th following way: w put 8 < h K if i = i K fo som K2D x(i) := : min( A) μ othwis. By Thom 2.2(ii), it is sufficint to pov that (8t 2 k)(9i 2 n) p fft t j p(a; x; i): Lt t 2 k b fixd and lt K 2 D b a maximal componnt with th popty j p(k) (maximal in th sns of th oding inducd by G(A; h K )). W dnot p fft t

10 10 MARTIN GAVALE, GÜNTER ROTE by i := i K th vtx chosn by th systm I. All th walks in this pat of th poof a undstood in th thshold digaph G := G(A; h K ). W will fist show that (9R 2 N)(8 >R) h ( 0 mod p(k)=p t ) ) ( x () (i) =h K, 0 mod p(k)) i : By Lmma 2.1 and by Lmma 3.1(i) w can conclud that p(k) j implis x () (i) h K, fo big nough. By th non-dominanc assupmtion and by (3.3) w gt x () (i) = h K. Now, if w had x () (i) = h K fo som with (p(k)=p t ) j and p(k) =j, this can only happn if i! j fo som j 2 I which was slctd fom anoth componnt K 0. By th maximality of K w know that p fft t =j p(k 0 ), and hnc, applying Lmma 3.1(iii), w conclud that i! 0 j fo all lag nough 0 with 0 mod p(w). Sinc p fft t =j p(w), w hav gcd(p fft t ; p(w)) j p fft1 t j (p(k)=p t ), and th two quations 0 mod p(w), 0 0 mod p(k) hav infinitly many solutions. So w hav i! 0 j fo som 0 0 mod p(k). This mans that i μ hk j, contadicting th assumption of non-dominanc of vtics in I. W hav sn that th subsqunc (x () (i)) of all lmnts with 0mod p(k)=p t has piod p t. Applying Lmma 3.4 with incmnt d = p(k)=p t, w obtain that p t gcd(p(k)=p t ; p(a; x; i)) = gcd(p(k);p t p(a; x; i)) divids p(a; x; i), and hnc gcd(p fft t ;p t p(a; x; i)) j p(a; x; i), fom which it follows that p fft t j p(a; x; i). (i) ) (ii): Lt x 2 B(n); p(a) = p(a; x): By Lmma 3.3, w may assum that x fulfills th non-dominanc quimnt: any vtx i 2 n with x(i) = h > min( A) μ is not dominatd at lvl h by any vtx j 6= i with x(j) x(i). By Lmma 3.2 and Rmak 3.4, w may assum that any vtx i 2 S(x; h) is containd in a non-tivial componnt of G(A; h). W only hav to nsu that, fo vy t 2 k, th is a lvl h 2 A μ [ μx such that S(x; h) contains a vtx j in a componnt K 2 S Λ (G(A; h)) with p fft t j p(k). W pov this by contadiction. Suppos that th is a t such that, fo all h 2 A μ [ μx and fo all j 2 S(x; h), w hav p fft t =j p K[j; h]. By Lmma 2.1, fo any h 2 A μ [ μx and i 2 n w hav (th notation W (i; j; h); R(i; j; w; h) p(w; h) is usd fo objcts W (i; j); R(i; j; w) p(w) dfind with spct to th thshold digaph G = G(A; h)): x () (i) h, (9j 2 S(x; h)) i! j, (9j 2 S(x; h))(9w 2 W (i; j; h)) 2 R(i; j; w; h), 2 R(i; h) := [ [ R(i; j; w; h) j2s(x;h) w2w (i;j;h) Und th abov assumption, w havp fft t =jp(w; h), fo all possibl paths w in this statmnt, sinc p(w; h) is dfind as th gcd of th piods of ctain componnts which includ th componnt K[j; h], whos piod is not a multipl of p fft t. By Lmma 3.1(ii), th st R(i; j; w; h) is piodic with piod p(w; h). It follows that th st R(i; h), bing a finit union of sts R(i; j; w; h), is also piodic, and its piod divids lcmf p(w; h) :j2s(x; h);w2w(i; j; h) g; which is not a multipl of p fft t. It is asy to s that p(a; x; i) divids th lcm of th piods of sts R(i; h) fo h 2 A μ [ μx and, thfo, it is not a multipl of p fft t, fo abitay i. Thn, by Thom 2.2(ii), th obit piod p(a; x) = lcm p(a; x; i) i2n is not a multipl of p fft t. Hnc, p(a; x) is diffnt fom p(a), which contadicts to ou assumption (i).

11 REAHABILITY OF MATRIX PERIOD 11 Now, in viw of th facts that two componnts K; L2S Λ (A) a ith disjoint o compaabl, and if K L,thn p(l) j p(k), w can asily choos a dciding systm D of paiwis disjoint componnts and th sts H, I satisfying th statmnt (ii). 4. Th computational complxity of th MPR poblm It was shown in [5] that, in gnal, th MPR poblm is NP-complt. W dscib h two conditions und which th poblm is polynomially solvabl. By Thom 3.5, th solving of th poblm MPR is quivalnt to pfoming two choics: w hav to choos a suitabl dciding st of componnts in such awaythat it is possibl to choos a st of paiwis non-dominating vtics, by on fom ach componnt. In spit of th fact that th poblm of pfoming ths two choics is NP-complt, w may hop that und som stictiv conditions it can b solvd in polynomial tim. Simila situation occus with th classical satisfiability poblm (SAT) fo disjunctiv boolan fomulas. If th numb of vaiabls in ach disjunctiv claus is stictd to 2, th poblm 2-SAT is polynomially solvabl. Howv, it is a wllknown fact, that th stiction to 3-disjunctiv fomulas is not sufficint: th poblm 3-SAT mains NP-complt. Thom 4.1. MPR poblm with two stictiv conditions (i) th matix A has a uniqu minimal dciding st of componnts, (ii) th isadciding st D of componnts such that fo any componnts K; L2D at lvls h» h 0 with K < L in G(A; h), gcd(p(k); p(l))» 2 holds tu, is solvabl in tim O(n 2 ) fo a givn matix A 2B(n; n). Poof. It is asy to show using Thom 2.3, that th is always at last on minimal dciding st D of componnts, which is ncssaily disjoint. Thus, condition (i) ducs th poblm to th choic of paiwis non-dominating vtics in D. ondition (ii) implis that, fo any compaabl componnts K; L 2 D, w hav gcd(p(k); p(l)) = 1, o gcd(p(k); p(l)) = 2. If th fist cas occus fo som h» h 0 and fo K; L 2 D; K 2 S Λ (G(A; h)); L2S Λ (G(A; h 0 )); with K < L in G(A; h), thn fo any i 2K;j 2L,th vtx i is dominatd by j at lvl h and th instanc of MPR has no solution. If th scond cas taks plac fo all compaabl pais K; L 2 D, thn th dominancy i μ h j fo i 2 K 2 S Λ (G(A; h)); j 2 L 2 S Λ (G(A; h 0 )); h» h 0 dpnds only on th paity of th positions of th vtics i; j and on th xistnc o nonxistnc of a walk w in digaph G(A; h), of vn lngth and conncting points of th sam paity in componnts K; L, o of odd lngth and conncting points of diffnt paitis in K and L. Thfo, th choic of paiwis non-dominating vtics in D is quivalnt to th choic of paiwis compatibl psntativs fom a systm of n two-lmnt sts. It is shown in [6] that such a choic can b pfomd in tim O(n 2 ). Th conditions (i), (ii) in Thom 4.1 may sm ath stictiv. Howv, th nxt two thoms show that ach of th conditions alon is too wak to mak th MPR poblm polynomially solvabl. Thom 4.2. Th MPR poblm with th stictiv condition (i) fom Thom 4.1 is NP-complt.

12 12 MARTIN GAVALE, GÜNTER ROTE Poof. As a spcial cas of MPR, th poblm blongs to NP. W show that th wll-known NP-complt poblm of 3-colouing fo gaphs (3-OL) polynomially tansfoms to MPR(i) poblm (i.., MPR with th stictiv condition (i)). W dscib a polynomial algoithm which assigns, to any instanc of 3-OL, such an instanc of MPR(i) that both instancs a quivalnt, i.., thy hav th sam answ ys o no. Lt G 0 =(V 0 ;E 0 ) b an instanc of 3-OL, lt us dnot m 0 = jv 0 j; n 0 = je 0 j. W choos m 0 + n 0 distinct pims (p v ; v 2 V 0 ); (q ; 2 E 0 ) and w dfin a digaph G =(V; E) consisting of m 0 + n 0 disjoint ointd cycls ( v ; v 2 V 0 ); (D ; 2 E 0 ) of lngths j v j =3p v ;jd j=3q fo any v 2 V 0 ; 2 E 0 : Th vtics in ach cycl a numbd bginning with 0, in th sns of th ointation. Th vtics will b fd to by th notation v (i), D (j). Bsids th acs containd in th cycls, th a additional acs btwn cycls in G, dfind as follows: w fix a lina oding» of vtics in V 0 and fo any dg =(u; v) 2 E 0, u» v w add th acs (D (0); u (0)), (D (0); u (2)), (D (0); v (1)) to E AK O A u A A A ff 0 I Λ ΛΛ 0 A AA D ff AAU AK Λ Λν A v A A Λ ΛΛ A ff 0 Λ ΛΛ Figu 4 Th additional acs a schmatically dawn on Figu 4. Fo th sak of simplicity, th cycls u ; v ; D, a dpictd with 3 vtics only. This simplification is basd on th fact that th gatst common diviso of th piods of th cycls is qual to 3. Th matix A is dfind as th adjacncy matix of th digaph G. Dnoting B = f0; 1g and n = 3(m 0 + n 0 ), w hav A 2 B(n; n). laly, A has xactly on dciding st of componnts, namly th st of all cycls f v : v 2 V 0 g and f D : 2 E 0 g. Thfo, A is an instanc of MPR(i). By th wll-known poptis of pim numbs, th constuction of A is polynomial with spct to th siz of G 0. In th following w show that th instancs A and G 0 a quivalnt.

13 REAHABILITY OF MATRIX PERIOD 13 Lt A b a ys instanc of MPR(i). By Thom 3.5, th is a st I = f i u 2 u : u 2 V 0 g[fj 2D :2E 0 g of paiwis non-dominating vtics in G (at lvl 1). W shall show that th gaph G 0 is a ys-instanc of 3-OL, i.., th is a colouing F : V 0! f0; 1; 2g with F (u) 6= F (v) fo vy = (u; v) 2 E 0. Th colouing F is dfind fo any u 2 V 0, k 2 N by th fomula 8 >< 0 if i u = u (3k) F (u) := 1 if i u = u (3k +1) >: 2 if i u = u (3k +2) Lt us suppos that F (u) =F(v) fo som =(u; v) 2 E 0, u» v. If F (u) =F(v)= 0, thn th vtx i u = u (3k) dominats all th vtics of th fom D (3l + 1), D (3l +2) and th vtx i v = v (3k) dominats all th vtics D (3l). As a consqunc, th vtx j is dominatd ith by i u o by i v. Th assumption F (u) = F (v) = 1, o F (u) = F (v) = 2 lads to contadiction in a simila way. Thfo, F (u) 6= F (v) holds tu fo any adjacnt vtics of th gaph G 0. onvsly, lt G 0 b a ys-instanc of 3-OL with th colouing F : V 0!f0;1;2g. Thn w dfin i u = u (F (u)) fo any u 2 V 0 and j = D (F (u)) fo any =(u; v) 2 E 0, u» v. By th abov asoning, th vtx i u dos not dominat j. Th adjacncy popty of F implis that th vtx i v dos not dominat j,aswll. Thfo by Thom 3.5, th matix A is a ys-instanc of MPR(i). Rmak 4.1. To undlin th analogy with 2-SAT and 3-SAT, w may notic that Thom 4.2 holds tu vn if th modifid condition (ii), with 3 instad of 2, is addd. Thom 4.3. Th MPR poblm with th stictiv condition (ii) fom Thom 4.1 is NP-complt. Poof. It is shown in [5] that th NP-complt poblm of satisfiability fo 3-disjunctiv boolan fomulas (3-SAT) polynomially tansfoms to MPR poblm. It is asy to vify that th constuction dscibd in [5] satisfis th condition (ii) of Thom 4.1. Rfncs [1] K. chláová, On th pows of matics in bottlnck/fuzzy algba, Ppint 21/93, Univsity of Bimingham. [2] R. A. uningham-gn, Minimax algba, Lctu Nots in Econom. and Math. Systms 166, Sping-Vlag, Blin, [3] M. Gavalc, omputing matix piod in max-min algba, Disc. Appl. Math. (to appa). [4] M. Gavalc, Piodicity of matics and obits in fuzzy algba, Tata Mountains Math. Publ. 6 (1995), [5] M. Gavalc, Raching matix piod is NP-complt, Tata Mountains Math. Publ. (to appa). [6] D. E. Knuth, A. Raghunathan, Th Poblm of ompatibl Rpsntativs, SIAM J. Disc. Math. 5 (1992), [7] Li Jian-Xin, Piodicity of pows of fuzzy matics (finit fuzzy lations), Fuzzy Sts and Systms 48 (1992), [8] Li Jian-Xin, An upp bound on indics of finit fuzzy lations, Fuzzy Sts and Systms 49 (1992), [9] M. G. Thomason, onvgnc of pows of a fuzzy matix, J. Math. Anal. Appl. 57 (1977),

14 14 MARTIN GAVALE, GÜNTER ROTE Dpatmnt of Mathmatics, Faculty of Elctical Engining and Infomatics, Tchnical Univsity, Hlavná 6, Ko»sic, Slovakia addss: gavalcccsun.tuk.sk Institut fü Mathmatik, Tchnisch Univsität Gaz, Stygass 30, Gaz, Austia addss: otopt.math.tu-gaz.ac.at