Appendix Generalized Pseudo-Boolean Programming* By I VO ROSENBERG 1. Let us put, for any integer k > 1, (1) B,,={O,I, ... ,k-l} and let (2) be the Cartesian product of the sets B"J (j = 1, ... , n). In the sequel we shall be concerned only with functions which map the set K into the field R of reals. For any integer k> 1 and any i E B", we define, as in GR. C. MorSIL [21, the so-called "Lagrangean functions" /_ x(x - 1) •.. (x - i + 1) (x - i - 1) ..• (x - k + 1) (3) x"= i(i-l) ... l.(-l) ... (i--k+l) mapping B" into B 2 • Obviously, i'''=I, x"'=O for xEB", x =\= i. For instance, for k = 3 we have (4) x30=~x2-h+l, X31 =-x2 +2x, X32=~X2_h. Any function f (Xl' . _ ., xn) has a Lagrangean development (5) f(Xl,' .. ,xn)= ~ f((1I, ... ,(1n)x~,u, ... x~na". (u". ,an) E f{ We see, from (3), that each x~JaJ is a polynomial with real coefficients of degree kr - 1. Therefore, formula (5) shows that each function f: K -+ R is a polynomial with real coefficients, having as degree of the j-th variable at most kJ - 1 (j = 1, ... , n). A point ~ = (~l' ... , ~,.) E K will be called a minimizing point of the function f, if for each ((11, .. , (1n) E K, relation (6) holds. * See 1. ROSENBERG [1]. HammerjRudeallu, Boolean Methods 20 302 Appendix In the sequel we shall be concerned with the determination of one (all) minimizing point(s) of a function I: K -+ R. When kl = ... = k n = 2, then the problem becomes one of pseudo-Boolean programming. 2. For any i = 1, ... , n, we put (7) We also put /1=/' Let us assume that for i(l~i:::;:n), we have already determined a function IJ (xJ' XJ +1, ..• , Xn), mapping KJ into R. We define an auxiliary function <PJ (XJ +1, ... , Xn) mapping K J +1 into BkJ , as follows: To each (aJ +1, ... , an) E K J +1 we associate an tX E BkJ , defined by the relation (8. i) In other terms, tX is one of the values in BkJ for which the function IJ (x, aJ + 1, ... , an), considered as a function of the single variable X, reaches its minimum (as x belongs to the finite set Bkj , it is obvious that one or several such tX do exist). This tX will be chosen as a value of <PJ (aJ +1, ... , (Tn). In 3 we shall deal with the practical determination of the function <PJ . Using <PJ' we define the function IJ +1 (XJ +1, ... , X n ), mapping KJ +1 into R, as follows: (9. j) 1;+dxJ+l, ... , Xn) = IJ (<pJ (XJ +1, ... , Xn), XJ +1, ... , X,,). It follows from (8. j) that (10. j) Relation (10. j) shows that the value IJ +l(a;t1, . .. , an) does not depend on the choice of tX, in case that the function IJ (x, aJ +1, ..• , an) reaches its minimum for several values x E Bkj • We obtain thus a sequence 11, I~, ... , j n' According to the definitions, <pn is a constant for which In (xn) reaches its minimum. The repeater1 application of (10. j) for i = n, n - 1, ... , 1, shows that (11) /n+l=/n(<Pn)= min ( min ( ... ( min I({JI, ... ,{In-l,{Jn)) ... )). {3nEBkn {3n-,EB'''_l {3,EBkl Therefore, (12) I n+l = ({3" m,~) E J(I ({Jl,' ., (In), i.e., In+l is the sought minimum of the function /. Now, it remains to determine Generalized Pseudo-Boolean Programming 303 Obviously, if (X~, ... , x~) is a minimizing point of the function I, then In+l = fn(qJn) = fn(x~) = fn-l(qJn-dx~), x~) = fn-dx~-I' x~) = ... = 11(x~"", X;,-b x~) = I(x~, ... , x~). In certain cases it is necessary to determine the set M I of all the minimizing points. We can proceed as follows: First we determine (14. n) then (14. n - 1) M n- I = {(Xn-I, xn) E K n- I I Xn E Mn, fn-dxn-I, xn) = fn+l}, finally (14.1) Ml = {(XI," 0 ,xn)EKI = K I (X2," 0 ,xn)EM2, fdxI,' 0 0 ,xn) = In+l}' 3. Let us now examine the problem of the determination of the functions qJj (1 :S::: i;:; n). For v E Bkj we define the sets (15) P j = {(Xj+I," 0, xn) E Kj+l I fj(v, Xj+l, 0 0 0, xn):S::: :S::: Ij (Xj, Xj+l, . 0 0, xn) for each Xj E Bkj}. Thus P~ is the set of those (aj +1, . 0 0, an) E Kj +1 for which ex = v satisfies (8. j). Obviously, (16) We choose a partition {B 1 , ••• , Bt} of Kj + 1 , so that for each r = 1, 2, . 0 ., t, there exists an mr E BkJ so that Br ~ Pfr". Further, we define the characteristic functions Pr(Xj+l,' .. , xn) of the sets Br (r = 1, .. 0' t), that is the functions which take on Br the value 1, and on K j +l - Br the value O. It is easy to see, that we can choose as functions qJj (Xj +1, .. 0, xn), the functions t (17) 1: mr Pr (Xj +1, .. 0, X,,). r-I The sets P~ may be determined as follows. We express the function Ij (Xj' Xj +1, . 0 0, Xn) as a polynomial; namely, we put u (18) fj (Xj , Xj+l,' 0., xn) = 1: xJ hp, p~o where xJ are powers of Xj; ho, hi, . 0 0, hu are polynomials in the vari- ables Xj +l' • 0 ., X n. It is easy to see that pj is the set of all (Xj +l' • 0 ., xn) E Kj +1 for which the system of kj - 1 inequalities u (19) 1: (f-lP - vP) hp (Xj +1, ... , X,,) > 0 p~1 (f-l = 0, 1, . 0 ., v-I, v + 1, . 0 ., kj - 1) has a solution. 20* 304 Appendix 4. In order to illustrate the above described methods, let us consider the following simple _example. Let n = 3, kl = k2 = ka = 3, (20) f(XI, X2, x3) = 2x~ X2 Xa - 3xI x~ x; + 7x~ x~ x; - 3x~ + The values of this function are shown in Fig. 1. aJ.J 9 13 17 5/ 30/ 80/ -3L 100/ 389/ 3 7 11 1 OL 10V 28 / -9 / ,",1/ 103 / -3 1 5 -5L z/ -zl/ ([;2 -15 V -11 / -7 / ([;, Fig. 1. We put II = I = xi(2xix3 + 7x~x~ - 3) + xd-3x~xi) + 4X2 + + 6X3 - 3. P~ is the set of all the solutions (X2' X3) of the inequalities (21) (02 - 12) (2X2 X3 + 7x~ x~ - 3) + (0 - 1) (-3x; x~) > 0, (22) (22 -12) (2X2X3 + 7x;xi - 3) + (2 -1) (-3x~x;) > O. Relation (21) may also be written in the form - 2x~ xi - 4X2 X3 + 3 ~ O. Hence X2 X3 = 0; but for X2 X3 = 0, the inequality (22) has no solutions. Therefore pi = 0. P~ is the set of all the solutions (X2' X3) of the inequalities (23) (02 - 22) (2X2 X3 + 7x~ x; - 3) + (0 - 2) (-3x; x;) 2: 0, (24) (l2 - 22) (2X2 X3 + txi xi - 3) + (1 - 2) (-3x~ xi) 2: o. It follows from (23) that X~ X3 = O. If X2 X3 = 0, then (24) is fulfilled too. Hence P~ is the set of all (X2 x3) E B~ for which X2 X3 = O. If we consider the partition {L~ - PL P~}, then 'P2 = (X2 X3)30 is obviously the characteristic function of the set P~. According to (4), we have 'P2 = t(x;xi - 3X2X3 + 2) and by (17), CPI = 2'P2. Since 'P2 takes on only the values 0 and 1, we have 'P~ = 'P2 and therefore cpi = 4'P2· Generalized Pseudo·Boolean Programming 305 We replace Xl by f(!l in 11 and obtain 12 = PI [4 (2X2 Xa + 7x~ xi - 3) - 3x: xi] + 4X2 + 6xa - 3 = ! (x: xi - 3x2 Xa + 2) [25x: xi + 8x2 Xa - 12] + 4X2 + 6xa - 3 = ! (25x~ x: - 67 x~ x~ + 14x: x: + 52x2 Xa - 24) + 4X2 + 6xa - 3 P~ is the set of all Xa for which (25) (04 - 14) 225 x: + (oa - 13) (- 6; X~) + (02 - 12) 7x: + + (0- 1) (26xa + 4) > 0, (26) (24 - 14) 2: X~ + (2a _ 13) (- 6; xi) + (22 - 12) 7x: + + (2 - 1) (26xa + 4) :::; O. These inequalities may be written in the simpler forms 25 4 67 s . 2 (27) - 2 X3 + 2 Xs - 7 Xs - 26xa - 4 :::; 0, 375. 4 469 s 2 (28) -2- Xs - 2 xs +21xa +26xa +4:::;0. It is easy to check that (27) has no solutions for Xa E Ba. Therefore, P~ = 0. P: is the set of all Xa for which (29) -200x~ + 268x~ - 28xi - 52xa - 8:::; 0, 375 4 469 2 2 (30) --2-xs +~xa - 21xa - 26xa - 4:::; O. It is easy to verify here too, that (29) has no solutions for Xa E Ba. We can therefore choose f(!2 (xa) identically equal to 0; fa = 6xa - 15. The minimum of la is reached for Xs = O. Therefore f(!a = 0 and 14 = fa (f(!a) = fa (0) = -15. The mmmuzmg point has x~ = f(!a = 0, x~ = f(!2 (x~) = 0, Xl, = f(!l ('X2' Xs') = X212 Xs12 - 3"x2 xa + 2 =.2 Thus the mllllmlzmg.... pomt. 306 Appendix is (2, 0, 0). According to the concluding remark in 3, this is the unique minimizing point. 5. Our methods are also valid for polynomials of higher degrees.