Polynomial unconstrained binary optimisation Part 1

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1 232 Int. J. Metaheuristis, Vl. 1, N. 3, 2011 Plynmial unnstrained binary timisatin Part 1 Fred Glver* OtTek Systems, In., th St., Bulder, CO 80302, USA glver@ttek.m Jin-Ka Ha Labratire d Etude et de Reherhe en Infrmatique (LERIA), Université d Angers, 2 Bulevard Lavisier, Angers Cedex 01, Frane jin-ka.ha@univ-angers.fr Gary Khenberger Shl f Business Administratin, University f Clrad at Denver, Denver, CO 80217, USA gary.khenberger@udenver.edu Abstrat: The lass f rblems knwn as quadrati zer-ne (binary) unnstrained timisatin has rvided aess t a vast array f mbinatrial timisatin rblems, allwing them t be exressed within the setting f a single unifying mdel. A ga exists, hwever, in addressing lynmial rblems f degree greater than 2. T bridge this ga, we rvide methds fr effiiently exeuting re searh resses fr timisatin rblems in the general lynmial unnstrained binary (PUB) dmain. A variety f searh algrithms fr quadrati timisatin an take advantage f ur methds t be transfrmed diretly int algrithms fr rblems where the bjetive funtins invlve arbitrary lynmials. In this Part 1 aer, we give fundamental results fr arrying ut the transfrmatins. We als desribe ding and deding redures that are relevant fr effiiently handling sarse rblems, where many effiients are 0, as tyially arise in ratial aliatins. In a sequel t this aer, Part 2, we rvide seial algrithms and data strutures fr taking advantage f the basi results f Part 1. We als dislse hw ur designs an be used t enhane existing quadrati timisatin algrithms. Keywrds: zer-ne timisatin; unnstrained lynmial timisatin; metaheuristis; mutatinal effiieny. Referene t this aer shuld be made as fllws: Glver, F., Ha, J-K. and Khenberger, G. (2011) Plynmial unnstrained binary timisatin Part 1, Int. J. Metaheuristis, Vl. 1, N. 3, Cyright 2011 Indersiene Enterrises Ltd.

2 Plynmial unnstrained binary timisatin Part Bigrahial ntes: Fred Glver is a Distinguished Prfessr at the University f Clrad and is the Chief Tehnlgy Offier fr OtTek Systems, In. He has authred r -authred mre than 400 ublished artiles and eight bks in the fields f mathematial timisatin, muter siene and artifiial intelligene. He is the reiient f the Distinguished vn Neumann Thery Prize. He is an eleted member f the Natinal Aademy f Engineering and has reeived hnrary awards and fellwshis frm the Amerian Assiatin fr the Advanement f Siene (AAAS), the NATO Divisin f Sientifi Affairs, the Miller Institute f Basi Researh in Siene and numerus ther rganisatins. Jin-Ka Ha is a Full Prfessr in the Cmuter Siene Deartment f the University f Angers (Frane) and is urrently the Diretr f the LERIA Labratry. His researh lies in the design f effetive heuristi and metaheuristi algrithms fr slving large-sale mbinatrial searh rblems. He is interested in varius aliatin areas inluding biinfrmatis, telemmuniatin netwrks and transrtatin. He has -authred mre than 100 eer-reviewed ubliatins in internatinal jurnals, bk haters and nferene reedings. Gary Khenberger is a Full Prfessr f Deisin Siene at the University f Clrad at Denver and is a C-diretr f the Deisin Siene rgramme. His researh fuses n designing and testing metaheuristis methds fr large sale timisatin rblems. He has -authred mre than 70 refereed aers and three bks. 1 Intrdutin 1.1 Prblem reresentatin We frmulate the lynmial unnstrained binary (PUB) timisatin rblem as PUB : Minimise x = + F : P xbinary The vetr x = (x 1,x 2,,x n ) nsists f binary variables, x i {0,1} fr i N = {1,,n}, and the effiients fr P = {1,, } are nn-zer salars. F is a rdut f mnents f the x vetr given by F =Π x : i N where N N. i Eah variable x i in the rdut defining F aears nly ne, nting that xi = xi fr x i binary, whih renders wers h f x i ther than h = 1 irrelevant. A seemingly mre general frmulatin arises by relaing the sets N, P by vetrs N, P and writing F = x i1 x i2 x ih fr ( i1, i2,, ih) = N, whih allws different effiients fr different ermutatins f the same set f indexes. Hwever, as nted in the fllwing bservatin, suh a frmulatin in terms f ermutatins an be readily nverted int a frmulatin in terms f sets, with the advantage f reduing the number f nn-zer effiients (and thereby inreasing the sarsity f the rblem reresentatin). h

3 234 F. Glver et al. Remark 1: In a lynmial reresentatin based n ermutatins, where tw N i, i,, i N = j, j,, j, are ver the same set f ermutatins = ( ) and 1 2 h q indexes, and the assiated sts and q are bth nn-zer, an equivalent rblem results by redefining : = + and = 0, thus eliminating the term fr the vetr N q. q q The validity f the remark fllws simly frm the fat that the rdut x i1 x i2 x ih has the same value as the rdut x j1 x j2 x jh. By summing the sts fr different ermutatins as indiated, the remark gives a basis fr a reressing ste enabling any ermutatin-based frmulatin f PUB t be nverted int the frm shwn. Suh a reressing ste may als be viewed as equivalent t rduing a restrited ermutatin frmulatin where eah ermutatin N = ( i1, i2,, ih) has the asending index rerty i 1 < < i h. T see hw the PUB frmulatin relates t mre lassial frmulatins that embdy the asending index rerty, nsider the ubi lynmial given by ( qx i i : i N) + ( qij xx i j : i< j N) + ( qijk xx i j xk : i< j< k N). Let N(1) = {i N: q i 0}, N(2) = {(i,j): i < j N: q ij 0}, N(3) = {(i,j,k): i < j < k N: q ijk 0}. Then the reresentatin is mleted by assigning the index the nseutive integer values = 1 t = N(1) + N(2) + N(3), and letting eah nseutive take the assiated value q i r q ij r q ijk, tgether with N = {i} r {i,j} r {i,j,k}, fr i N(1), (i,j) N(2), (i,j,k) N(3). The next remark is useful t failitate ertain eratins f ur methd. Remark 2: We assume P j = {j} fr j N withut regard t the value f j, thus rviding an exetin t the rule f nly inluding terms with nn-zer effiients in the PUB reresentatin Illustratin f the PUB reresentatin Cnsider the PUB rblem whse bjetive funtin is given by x = 7+ 5x + 2 x 3 x 4x x + 5 x x 2x x + 3 x x x. First, sine x = x2, x1 = x1 and x1x2 = x2x1, we an re-write x in asending index ntatin, inluding nly the nn-zer effiients, as x = 7+ 5 x x 4xx + 3xx + 3 xx x By Remark 2, we inlude the x 3 term in the reresentatin, even thugh it has a 0 effiient, t give x = 7+ 5 x x + 0 x 4x x + 3x x + 3 x x x Assigning indexes = 1 t 6 t the terms in sequene, we identify the indexes f the variables in these terms by N = {1}, N = {2}, N = {3}, N = {1,3}, N = {1,2}, N = {1,2,3} h

4 Plynmial unnstrained binary timisatin Part The st effiients assiated with these terms, inluding the nstant term, are given by = 7, = 5, = 1, = 0, = 4, = 3, = (Nte that while we assume the indexes f any set N are in asending rder, we d nt assume the sets themselves are rganised in a lexigrahi asending rder. Hene, in the examle it is aetable t give a smaller index t the set {1,3} than t the set {1,2}.) 1.2 Aliatins and mtivatin The seial ase where the lynmial bjetive f PUB is a quadrati funtin (hene the lynmial has a degree f 2) has been widely studied. In this ase, PUB already reresents a brad range f imrtant rblems, inluding thse frm sial syhlgy (Harary, 1953), finanial analysis (Laughunn, 1970; MBride and Yrmak, 1980), muter aided design (Kraru and Pruzan, 1978), traffi management (Gall et al., 1980; Witsgall, 1975), mahine sheduling (Alidaee et al., 1994), ellular radi hannel allatin (Chardaire and Sutter, 1994) and mleular nfrmatin (Phillis and Rsen, 1994). Mrever, many mbinatrial timisatin rblems ertaining t grahs suh as determining maximum liques, maximum uts, maximum vertex aking, minimum verings, maximum indeendent sets, and maximum indeendent weighted sets are knwn t be aable f being frmulated by PUB in the quadrati ase as dumented in aers f Pardals and Rdgers (1990), and Pardals and Xue (1994). A review f additinal aliatins and frmulatins an be fund in Khenberger et al. (2004). The mre general PUB frmulatin given here is f interest fr its ability t enmass a signifiantly exanded range f rblems. The ubi ase, fr examle, ermits PUB t reresent the imrtant lass f satisfiability rblems knwn as 3-SAT, and ffers the advantage f rviding a reresentatin whse size des nt deend n the number f lgial lauses, whih stands in ntrast t the ase fr ustmary binary integer rgramming frmulatins f 3-SAT (see Khenberger, 2010). The availability f redures fr effiiently handling and udating mve evaluatins fr instanes f PUB invlving lynmials f degree greater than 2, as identified in the fllwing setins, rvides an imetus t unver additinal rblems that an be usefully given binary lynmial frmulatins. Our redures, whih aly t mves that fli (mlement) the values f ne r mre variables x j in rgressing frm ne slutin t anther, nstitute a generalisatin f redures fr generating 1-fli mves desribed in Glver et al. (1998) and extended t 2-fli and multi-fli mves Glver and Ha (2010a, 2010b). Imrtant reent ntributins f a similar nature fr the quadrati rblem are rvided in Hanafi et al. (2010). A rinile utme f ur develment is that a variety f searh methds whih urrently inrrate redures t evaluate fli mves fr the quadrati rblem an relae these redures by the methds desribed here, thereby rduing methds aable f slving general PUB rblems withut any ther hanges in their struture.

5 236 F. Glver et al. 2 Preliminary relatinshis We start by making sme basi bservatins that diretly generalise bservatins given in Glver and Ha (2010a, 2010b). These bservatins rvide a basis fr a mre enmassing framewrk given in the next setin whih is artiularly useful fr exliting sarsity. Let x and x " reresent tw binary slutins and define ( : ), where ( : ) = =Π k x F P F x k N " " " " = =Π k x F : P, where F x : k N. " Δ x = x x The bjetive funtin hange Δx thus dislses whether the transitin (mve) frm x t x" will ause x t imrve r deterirate (resetively, derease r inrease) relative t the minimisatin bjetive. We identify the variables x i that are mlemented in ging frm the slutin x t " the slutin x ", and the subsets fr whih x = 1 and 0 by defining { : " i 1 i} N = i N x = x " { } N (1) = i N : x i = 1 " { } N (0) = i N : x i = 0 = { } " " Observatin 1: If xx i i 0( xi 0 r xi 0) PM P: N M, where Mis an arbitrary subset f N. = = = fr eah i N, then i Δ x = ( : P( N (1))) ( : P( N (0))) " Prf: The identity Δ x = ( F : P) ( F : P) may be rewritten as " " ( ( F F ) : P ( N )) + ( ( F F ) : P P ( N )). If P P(N ), " k xk then N must ntain at least ne k suh that x = = 0, and hene " " F = F = 0. Cnsequently, Δ x = ( ( F F) : P( N )). We rewrite the latter as " " ( ( F F) : P( N (1))) ( ( F F) : P( N (0))) " ( ( F F) : P( N ) P( N (1)) P( N (0))). + +

6 Plynmial unnstrained binary timisatin Part Fr eah in the range f this latter summatin, N is nt a subset f N (1) and als nt a subset f N " (0), whih imlies N must ntain a air {j,k} suh that x = 1, x = 0 and x " k xk = 0, = 1. " Thus, F = F = 0 fr eah f the terms f this latter summatin, and we are left with " " ( ( ): ( (1))) ( ( ) : ( (0))). But Δ x = F F P N + F F P N F = 0 fr P(N " (1)) while F = 0 and 1 j j " F = 1 and F = fr P(N (0)). This mletes the rf. In eah f the fllwing tw rllaries, we make the mre restritive assumtin that " x = 0. Let N" = { i N : xi = 1} ( hene P( N" ) = { P: N N" }) t identify thse " sets N suh that x i = 1 fr all i N. Thus, if N" = { i1,, i h }, then PN ( ") is the index set fr all thse sets N msed f ne r mre f the elements i 1,,i h. Crllary 1.1: If 0, ( ) x = then Δ x = : P( N" ) Prf: The result fllws diretly frm Observatin 1, by nting that fr x = 0 we " have N" N (1) N (1) = i N : x i = 1. At the same time, the definitin = and { } " = { = } whih imlies N { i N x i } N (0) i N : x i 0, emty. (0) = : = 1, shwing that N (0) is T simlify the statement f the next result, it is useful t islate the effiients f the rdut terms F that refer nly t a single variable x j. As reviusly nted, we suse N = {} fr N, althugh = 0 is ssible fr sme f these indexes. Hene, the rdut term F fr N is the single variable term F = x. If we dente indexes N instead by j N t nfrm t the ratie f referring t variables x j fr j N, ur indexing nventin identifies the x j mnent f the bjetive funtin x t be just j x j. Crllary 1.2: If x = 0, and x " results frm Δx = j. x by fliing the single variable x j, then Prf: This result is an immediate nsequene f Crllary 1.1 where N" = { j}. " Crllary 2 an als be established diretly by nting that the slutin x" = x + xjej " " yields F = 0 fr all P exet fr = j, sine eah term F ther than fr = j " " ntains a variable x k suh that xk = xk = 0. Hene, Δx redues t x j j x j j, whih is just j, as stiulated. In the ntext f Crllaries 1.1 and 1.2, j gives the hange in x that results frm fliing a single variable x j, while ( : P ( N ")) is the hange that results frm fliing all variables x i fr i N". Hene, Crllary 1.2 gives an evaluatin f a 1-fli mve and Crllary 1.1 gives an evaluatin f a q-fli mve by letting the set N " reresent the indexes fr a seleted set f q variables that are flied frm 0 t 1.

7 238 F. Glver et al. 3 Exliting and udating rblem transfrmatins The reeding rllaries have an imrtant imliatin: rvided we start frm a slutin x = 0, the amunt f effrt t evaluate a mve invlving any number f flis, frm 1 t q, is the same fr any lynmial f degree d fr d q (the degree d may be defined by d = max( N : P). Thus, by this stiulatin, the wrk t evaluate a 1-fli is the same fr all lynmials, the wrk t evaluate a 2-fli is the same fr all lynmials f degree 2 r greater, and s n. It als fllws that the wrk t evaluate a q-fli fr a lynmial f degree d q nly requires a single additin eratin beynd the wrk t evaluate a q-fli fr a lynmial f degree d = q 1. Cnsequently, a 3-fli in a lynmial f degree 3 r larger an be evaluated by using nly ne additin eratin beynd that required t evaluate a 3-fli in a lynmial f degree 2. Crllaries 1.1 and 1.2 therefre rvide a natural mtivatin t maniulate the frmulatin f PUB s that a urrent slutin x may always be treated as if it were the 0 slutin. T d this we use the mmn devie f transfrming the x vetr int anther binary vetr y by mlementing seleted mnents f x; in this ase, seifially mlementing thse mnents f x suh that x j = 1, thus ausing the assignment x = x t yield a rresnding assignment y = y fr whih y = 0. Fr greater reisin, we refer t the frmulatin PUB as PUB(x), and nsider the alternative equivalent frmulatin PUB(y) based n the transfrmed vetr y. T exress PUB(y), let { i } { i } N(1) = i N : x = 1 and N(0) = i N : x = 0 and define the relatinshi between y and x by y = 1 x, i N(1) i i y = x, i N(0) j j This transfrmatin f variables auses the bjetive fr PUB(y) t inlude different terms than the bjetive f PUB(x). We demnstrate hw this urs by demsing the transitin frm the frmulatin PUB(x) t the frmulatin PUB(y) int a series f 1-fli stes, eah nsisting f imlementing the mlementatin eratin fr a single variable, i.e., fr a single index i N(1), and then reeating until the eratin has been mleted fr all i N(1). The amended terms F fr PUB(y) require altering the identity f the set P, in rder that it may ntinue t refer stritly t sets N suh that 0 with the exetin that we inlude the sets N j = {j} fr j N even if j = 0 as nted earlier. In ther wrds, the methds we desribe will result in hanging P by identifying ertain terms F fr whih the effiient fr > n will hange frm = 0 t 0, and ertain ther terms fr whih the effiient fr > n will hange frm 0 t = 0. Thus, in the frmer ase will nt belng t P, and will be added t P (by setting P := P {}) while in the latter ase will be dred frm P (by setting P := P {}). The maintenane f P in this manner, s that it always refers t terms having nn-zer effiients fr > n, is imrtant fr exliting sarsity, whih is a

8 Plynmial unnstrained binary timisatin Part harateristi feature f many PUB rblems, artiularly thse f mderate t large size. T treat sarsity effetively we imliitly refer t an additinal set f indexes, dented by P α, whih inludes relevant indexes fr whih = 0, i.e., seifying that the lletin {N : P α } nsists f all subsets N f N ntaining frm 1 t d indexes f N. Later, we rvide a mat ding mehanism that makes it ssible t identify elements f P α that are relevant fr algrithmi udates withut needing t rely n searhes t arry ut this identifiatin. T braden the generality f ur results, we intrdue a seial set N and a rresnding rdut term F assiated with the bjetive funtin variable x, where we stiulate that N =. By the standard nventin that the rdut f variables ver the emty set equals 1, we have F = 1 (alying the definitin F = Π(x k : k N ) t the ase where N = N ). This yields F =, and hene F is just the nstant term assiated with the bjetive funtin x. These nventins allw us t exress hanges in x using the same ntatin emlyed t exress hanges in general terms f the frm F. Cnsequently, we understand that P α inludes the index = 0, in rder t inlude referene t x and. The relevane f these stiulatins will beme lear in an illustratin subsequently rvided. T address the 1-fli ase we dente the variable that is flied by x j, hene yielding y j = 1 x j. Then we define the fllwing fr eah j N: { } P( j) = P: j N α [ j] = the unique index in P suh that N = N { j} [ j] ( hene, F[ j] =Π( xk : k N { j} ), and [ j] = 0 if [ j] P). F j = yjf[ j] ( F j F yj xj) [ ] hene, [ ] is the same as exet that relaes. We bserve that P(j), whih identifies the index set fr all sets N that ntain j, is effetively a seial ase f P(M) = { P: N M}, by taking M = {j}. 3.1 Illustratin f the set P(j) The set P(j) lays a ivtal rle in several arts f ur develment, and hene we illustrate its msitin by referene t the examle lynmial used earlier t illustrate the ntatinal nventins underlying the PUB reresentatin. After alying these nventins the lynmial tk the frm x = 7+ 5 x x + 0 x 4x x + 3x x + 3 x x x. whih gave rise t the sets N = {1}, N = {2}, N = {3}, N = {1,3}, N = {1,2}, N = {1,2,3} Then the sets P(j) fr j N = {1,2,3} are given by P(1) = {1, 4,5,6}, P(2) = {2,5,6}, P(3) = {3, 4,6}. (The terms [j] and F [j] are readily understd by referene t the msitin f P(j).)

9 240 F. Glver et al. Observatin 2: Fliing x j t relae x j by y j = 1 x j rdues the fllwing hanges fr eah index P(j): ( ) : = + hanging F t beme + F. [ j] [ j] [ j] [ j] [ j] [ j] : = F : = F [ j] hanging F t beme F [ j] fr the new value. Mrever, these hanges are indeendent, s that the hange fr ne index P(j) des nt affet the hange fr anther P(j). Prf: The term F fr P(j) an be written as [ ] F = x Π x : k N { j} = x F. j k j j Fliing x j, whih rresnds t substituting 1 y j fr x j thus transfrms F int (1 y j ) F [j] = F [j] y j F [j] = F [j] F [j]. Hene, the term ( [j] + )F [j] relaes the term [j] F [j], defined ver the index set N [j] = N {j}, and after setting := the term F [j] relaes the revius term F, defined ver the same index set N. The indeendene f these hanges fllws frm the fat that eah set N fr P(j) is unique and hene eah set N [j] = N {j} is als unique. It is imrtant t bserve that sme r all f the indexes [j] may nt belng t P (as urs when [j] = 0). If [j] P, then exet fr the seial ase where N ntains the index f a single variable, P imlies 0, and hene the new effiient [j] := [j] + must be nn-zer. This mels P t be enlarged fr the reresentatin PUB(y) by setting P := P {[j]}. On the ther hand, if [j] P, then it is ssible that the new value [j] + f [j] may beme 0, and in this ase P must be redued by setting P := P {[j]}. Again, we later give resses fr handling suh eratins effiiently. The udate rdued by Observatin 2 is mleted by redefining x t be y (hene, redefining x j := 1 x j ) s that we may treat the urrent slutin as x = 0, and aly Observatin 2 again t reeat the ress. A rerd f the true slutin x is ket thrughut these eratins, but the algrithm des nt have t knw this slutin, and by means f the urrently udated frmulatin nly sees a urrent lletin f terms F and their assiated sets N fr P (referring t the urrent P). We larify these mments by means f the fllwing examle. We ntinue t emly the nventin N = {} fr N, inluding the ase where may nt belng t P (i.e., when = 0). 3.2 Illustratin f Observatin 2 Let j = 3 dente the index f the variable x j that is flied, and let { } P( j) = P(3) = 3,30, 40,50,60

10 where 1 Plynmial unnstrained binary timisatin Part N = {3}, N = {3,7}, N = {1,2,3}, N = {2,3,5}, N = {1,2,3,5,7} Frm the definitin N [j] = N {j}, we have N = { }, N = {7}, N = {1, 2}, N = {2,5}, N = {1,2,5,7}. 3[3] 30[3] 40[3] 50[3] 60[3] In ardane with nventins mentined earlier, we refer t the seial ase N [], here N 3[3], by N = { }, taking the index 3[3] t be 0. T assign seifi indexes t the remaining indexes [j] = [3], suse 30[3] = 7, 40[3] = 15, 50[3] = 25 and 60[3] = 55. Nte that 30[3] = 7 satisfies the nventin N 7 = {7}, i.e., N = {} fr n, given that N 30[3] = {7}. The remaining indexes = 15, 25 and 55 are hsen simly fr urses f illustratin. Ging thrugh the sets N and N [3] in the sequene given by P(3) = {3,30,40,50,60} generates the fllwing hanges. The symbl is used t dente bemes, and the variables are listed in asending index rder. The first ase belw, fr N 3 = {3}, utilises the nventin that F = 1. The stiulatins fr this ase may be understd mre learly by maring them t the stiulatins fr the send ase where N 30 = {3,7}. fr N 3 = {3} and N 3[3] = N = { } ( : ) F = + = F = x y y : = fr N 30 = {3,7} and N 30[3] = N7 = {7} ( : ) F = x + x x = F = x x y x y x : = fr N 40 = {1,2,3} and N 40 [3] = N 15 = {1,2} ( : ) F = x x + x x x x = F = x x x x x y x x y : = fr N 50 = {2,3,5} and N 50[3] = N 25 = {2,5} ( : ) F = x x + x x x x = F = x x x x y x x y x : = fr N 60 = {1,2,3,5,7} and N 60[3] = N 55 = {1,2,5,7} ( : ) F = xxxx + xxxx xxxx = F = xxxxx xxyxx xxyxx : =

11 242 F. Glver et al. One all hanges indiated in the reeding illustratin are imlemented, y 3 is redefined t be x 3, allwing the ress t be alied anew t fli additinal variables. The illustratin demnstrates hw the nventins N = { } and F = 1 ermit Observatin 2 t identify the udate t the bjetive funtin nstant. By this devie, Observatin 2 subsumes Crllary 1.2. Hwever, the simler statement f Crllary 1.2 is nvenient fr evaluating 1-flis. 3.3 Udate t handle multile flis simultaneusly We nw nsider the generalisatin f Observatin 2 that gives the frm f the udate fr erfrming an arbitrary lletin f flis. Let M be the index set fr the variables flied; i.e., M = {j N: y j = 1 x j }. We will generate a lletin f terms frm M and thse sets N having a nn-emty intersetin with M, i.e., fr whih the set M, defined by M = N M, is nn-emty. Eah term rdued frm airing M with a set N derives frm exanding the urrent term F as a result f mlementing the elements f M (whih is the reasn fr stiulating M ) and has the frm γ F [y], where γ = r as subsequently exlained, and F [y] is the rdut f a y mnent Π(y k : k N[y]) and an x mnent Π(x k : k R ), where R is defined t be the residual set given by R = N M. T identify these udates fr all relevant indexes, we define P (M) = { P: M } (hene, P(M) P (M) by the definitin P(M) = { P: N M}). The sets N[y] vary arding t rules that deend n the artiular term F exanded, generating a term γ F [y] as N[y] ranges ver the subsets f M. We inlude referene t the emty subset, whih yields Π(y k : k N[y]) = Π(y k : k ) = 1 (again by the nventin that a rdut ver the emty set equals 1). The set R an als be emty, whih urs in the ase where N = M, giving R = N N = (whih arises beause N itself is a subset f N ). Then we have Π(x k : k R ) = 1 fr this ase as well. The stiulatin f whether γ = r is determined by the number f elements in N[y], s that γ = if N[y] is even and γ = if N[y] is dd. We identify the different msitins f N[y] and the assiated effiients γ by letting S h (M ) dente the lletin f all h-element subsets f M, fr h = 0,1,, M. Then the utme f fliing the variables x j fr j M is identified by seifying the frm f F[y] as N[y] ranges ver the subsets ntained in S h (M ) fr eah h = 0 t M. The ase fr h = 0 is the ne that gives rise t a set N[y] =, sine there are n 0-element subsets f the set M. Thus, fr this ase we have S (M ) = { } (the set whse nly element is the emty set), and we nte that this identity is indeendent f the set M. Observatin 3: Fliing x j fr eah j M rdues the fllwing hanges fr eah set N suh that P (M), and given, fr eah q = 0 t M (where M = N M and P (M) = { P: M }). Let γ q = if q is even and γ q = if q is dd and let δ q = (γ q F q (K): K S q (M )) where F q (K) = Π(x k : k R ) Π(y k : k K)). Then the hange fr set N is given by θ ( δ : q 0 t M ) = = q and the ttal hange fr the entire lynmial is given by ( θ : P( M) )

12 Plynmial unnstrained binary timisatin Part Prf: The rf f Observatin 3 results by iteratively alying the analysis given in the rf f Observatin 2. The derivatin is straightfrward but rather umbersme and s we mit the details. The essential elements f Observatin 3 may be demnstrated by means f the fllwing examle. 3.4 Illustratin f Observatin 3 Let M = {1,2,3}, and fr simliity suse the sets N that have nn-emty intersetins with M are indexed s that P (M) = {1,2,3,4}, where the sets N 1,,N 4 (and their assiated sets M and R, = 1,2,3,4) are given by: N 1 = {1,2,3,4,5} M 1 = {1,2,3} R 1 = {4,5} N 2 = {1,2,4} M 2 = {1,2} R 2 = {4} N 3 = {2,3} M 3 = {2,3} R 3 = (Π x k : k R 3 ) = 1) N 4 = {1} M 4 = {1} R 4 = (Π x k : k R4) = 1) Index sets Assiated terms γ q F q (K): K S q (M ) Fr = 1 S (M 1 ) = 1 x 4 x 5 (γ q = ) S 1 (M 1 ) = {{1},{2},{3}} 1 y 1 x 4 x 5 1 y 2 x 4 x 5 1 y 1 x 4 x 5 (γ q = ) S 2 (M 1 ) = {{1,2},{2,3},{1,3}} 1 y 1 y 2 x 4 x y 2 y 3 x 4 x y 1 y 3 x 4 x 5 (γ q = ) S 3 (M 1 ) = {{1,2,3}} 1 y 1 y 2 y 3 x 4 x 5 (γ q = ) Fr = 2 S (M 2 ) = 2 x 4 (γ q = ) S 1 (M 2 ) = {{1},{2}} 2 y 1 x 4 2 y 2 x 4 (γ q = ) S 2 (M 2 ) = {{1,2}} 2 y 1 y 2 x 4 (γ q = ) Fr = 3 S (M 3 ) = 3 (γ q = ) S 1 (M 3 ) = {{2},{3}} 3 y 2 3 y 3 (γ q = ) S 2 (M 3 ) = {{2,3}} 3 y 2 y 3 (γ q = ) Fr = 4 S (M 4 ) = 4 (γ q = ) S 1 (M 4 ) = {{1}} 4 y 1 (γ q = ) As stiulated in Observatin 3, these quantities are aumulated and then added t the urrent effiients f the rresnding rdut terms. Observatin 3 allws the udate f a multi-fli mve t be arried ut with greater effiieny than the udate f a single fli mve, beause it avids udating sets P(i) that may lse r gain elements as a result f effiients that may hange frm nn-zer t zer r vie versa, when this may ur mre than ne fr the same index. Hwever, beause f the need fr exanded data strutures t kee trak f aumulated terms in suh a multi-fli udate, we antiiate that it is referable in mst irumstanes t

13 244 F. Glver et al. demse suh an udate int a series f mnent udates erfrmed by referene t Observatin 2. In ur fllwing develment, therefre, we fus n an imlementatin that is rganised seifially t exlit suh mnent udates. We nw turn t the questin f hw t erfrm these resses effiiently, giving artiular emhasis t the situatin in whih the lynmial is sarse; i.e., where P reresents nly a small subset f all ssible index sets (equivalently, when many terms F have assiated sts = 0 and hene d nt exliitly aear in the lynmial). 4 Overall struture f the algrithm We draw n Observatin 2 as the re bservatin fr evaluating and udating mves that transfrm ne slutin t anther. Cnsequently, we treat multi-fli mves as msed f a sequene f 1-fli mves rather than seeking t gain effiienies using Observatin 3. Fllwing suh a design, a generi methd fr the PUB rblem may be summarised as fllws. 4.1 Generi PUB methd 0 Create the initial PUB data strutures and selet a starting slutin. Transfrm the rblem reresentatin s that this slutin bemes the urrent 0 slutin. While a hsen terminatin nditin is nt satisfied 1 Selet ne r mre variables x j t be flied, by emlying the j values t identify Δx fr 1-fli mves as in Crllary 1.2, r the values t identify Δx fr multi-fli mves as in Crllary Aly Observatin 2 t udate the urrent rblem reresentatin by treating the seleted mves as a series f 1-flis. Maintain the arriate msitin f P, and the sets N and P(i), fr eah 1-fli relaing x j by 1 x j, exeuted as fllws fr * = [j] where * > n: If * = 0 then Create the set N * and add * t the set P(i) fr eah i N *. ElseIf * + = 0 ( * 0) then Remve referene t the set N * and dr * frm P by remving * frm the set P(i) fr eah i N *. Endif EndWhile The use f the j and values as riteria fr seleting a mve in Ste 1 abve may f urse be amanied by additinal riteria, suh as emlying tabu restritins and asiratin inentives derived frm reeny and frequeny memry in a tabu searh methd (see, e.g., Glver and Laguna, 1997). An alternative fundatin fr desribing and justifying the reeding algrithm is given in Aendix 1, whih des nt rely n emlying transfrmatins f variables. We

14 Plynmial unnstrained binary timisatin Part additinally nte that the literature rvides a different tye f transfrmatin mehanism with the gal f reduing a ubi r higher degree lynmial t a quadrati, and therefre enabling the rblem in rinile t be slved by a quadrati algrithm. We int ut mliatins that arise by using this quadrati redutin arah by marisn with using ur urrent arah in Aendix 2, ausing the quadrati redutin t require multi-fli evaluatins t ahieve the effet f 1-fli and 2-fli evaluatins when using ur urrent arah. A key hallenge fr imlementing the Generi PUB Algrithm is t seify the manner f struturing the sets P(i) = { P: i N ) fr i N. As intimated by the algrithm s desritin, the eratins f maintaining the entire rblem reresentatin redue t just the eratins f udating the sets P(i), tgether with maintaining a reresentatin f the sets N themselves. In artiular, exliit knwledge f the set P is unneessary, sine all relevant knwledge abut P is ntained in the sets P(i). In the next setin, we shw a way t identify the elements f the sets N withut exliitly listing them, while maintaining a list f values that is dramatially smaller than wuld be reated by allating a multidimensinal matrix t this task (by reating a st matrix C(i 1,i 2,,i d )). T amlish this requires a means t de vetrs f the frm (i 1,,i h ) as single numbers v, and t dede suh numbers v bak int the vetrs (i 1,,i h ) frm whih they were derived. 5 Cding and deding index vetrs fr rdut terms We reiterate the nventin that the indexes f rdut terms are written in the frm a vetr f indexes (i 1,i 2,,i h ) where i 1 < i 2 < i h (hene rresnding t the rdut term Π(x k : k = i r : r = 1,,h) where h takes a value between 1 and the degree d f the lynmial). Nte that we use the symbl i in this reresentatin beause eah element i r identifies an index rather than a variable suh as x k. We first disuss the redure fr ding eah suh vetr f indexes as a single value V. This rresnds t generating the set f indexes belnging t the set P α as disussed in Setin Cding redure Organisatin Partitin the terms int grus, G(1) t G(d), where eah gru is a lletin f vetrs (i 1,,i h ) fr h = 1,,d, identifying the indexes f all ssible terms ntaining h elements (the elements f these vetrs nstitute the rdered frm f the sets N that may tentially be reated, in a ase where the rblem is fully dense): G(1) : ( i1) : i1 = 1,, n G(2) : ( i1, i2) : i1 = 1,, n 1; i2 = i1+ 1,, n G(3) : ( i1, i2, i3) : i1 = 1,, n 2; i2 = i1+ 1,, n 1; i3 = i2 + 1,, n Gd :( i1,, id ): i1 = 1,, n ( d 1 ); i2 = i1+ 1,, n d; ; i = i + 1,, n 1; i = i 1 + 1,, n d 1 d 2 d d

15 246 F. Glver et al. Ste 1 Create a base ardinality Δ(h) and a umulative ardinality n(h) fr eah gru G(h), h = 1,,d (the umulative ardinality is the number f vetrs in gru G h added t the number f vetrs in all reeding grus). G(1) : Δ (1) = n; n(1) =Δ(1) G(2) : Δ (2) = n( n 1) / 2; n(2) = n(1) +Δ(2) G(3) : Δ (3) = n( n 1)( n 2) / 6; n(3) = n(2) +Δ(3) G(4) : Δ (4) = n( n 1)( n 2)( n 3) / 24; n(4) = n(3) +Δ(4) Gd : Δ ( d) = nn ( 1) ( n ( d 1))/ d! nd = nd ( 1) +Δ( d) Ste 2 Create the base ding v(h) fr an arbitrary vetr (i 1,,i h ), fr eah gru G(h): G(1) : v(1) = i 1 ( 2 )( 2 ) ( 3 )( 3 )( 3 ) G(2) : v(2) = i 1 i 2 / 2 + v(1) G(3) : v(3) = i 1 i 2 i 3 / 6 + v(2) G(4) : v(4) = i 1 i 2 i 3 i 4 / 24 + v(3) G( d): v( d) =Π i q : q = 1,, d)/ d! + v( d 1) d Ste 3 Create the full ding V(h), by adding the umulative ardinality n(h 1) t v(h) fr an arbitrary vetr (i 1,,i h ), fr eah gru G(h): ( 1) = ( = 1) ( 1 2) ( 1 2 3) = + G(1) : Fr i : V(1) v(1) i G(2) : Fr i, i : V(2) n(1) v(2) G(3) : Fr i, i, i : V(3) = n(2) + v(3) G(4) : Fr i, i, i, i : V(4) = n(3) + v(4) ( ) Gd : Fr i,, i : Vd = nd ( 1) + vd 1 d The umulative ardinality n(d) is the maximum value f P fr a lynmial f degree d, hene, the maximum number f nn-zer effiients when the lynmial is reresented in asending index rder. The ding eratin assigns a unique index = V[M] t eah set M = {i 1,,i h } fr h = 1,,d, where i 1 < < i h. We use the ntatin V[M] t distinguish the value rdued by ding M frm the value V(h) that reresents the ding value reviusly defined. Thus, in artiular, V[M] = V(h) fr the value V(h) alulated by referene t M = {i 1,,i h } in Ste 3 abve (beause f the asending index rdering i 1 < < i h, M may stritly seaking be nsidered a vetr, thugh we ntinue t refer t it as a set fr nveniene). The indexes = V[M] are exatly thse frm the set {1,2,,n(d)}.

16 Plynmial unnstrained binary timisatin Part Using the ding fr inutting initial rblem data The ding redure f Setin 5.1 is imlemented immediately un inutting the initial rblem data, thereby rviding a mat reresentatin t take advantage f situatins where the data is sarse. T handle this as art f the data inut redure is quite simle: Inut redure 1 Initialise = 0 fr = 1 t n(d) 2 Read rblem data and simultaneusly generate the ding: let M dente the urrent set f indexes inut frm the rblem data t beme a set N, and let dente the st assiated with M that is t beme the value attahed t the rdut term Π(x k : k N ) fr N = M. 3 Prdue the index = V[M] by alying the full ding rule t the elements f M, and let =. We next identify the eratin that is the inverse f the ding eratin, and whih is a bit mre subtle. 5.3 Deding a ded value = V t btain the index set M suh that V[M] = V Let V dente the (full) ded value f an index set M = {i 1,,i h }, where V is the index f the unknwn set N = M. We seek t identify eah mnent i 1,,i h f M (hene f N ) by erfrming arriate eratins n the value V. Fr an arbitrary real number z, let [z] dente the greatest integer z, and let <z> dente the least integer z (hene, when z is a sitive nn-integer value, then [z] runds z dwn and <z> runds z u). Then fr eah value r frm 2 t the degree d f the lynmial, we make referene t a nstant β(r) determined by the fllwing frmula β () r = r+ 1 < (!)1/ r r >. The runding u eratr < > is essential t the rretness f this frmula. The values β(r), fr r = 2,,d, may be muted in advane and stred, thus aviding the need t re-mute these values multile times when alying the deding algrithm. Fr examle, if d = 10, the relevant β(r) values btained by the reeding frmula are as fllws: β(2) = 1, β(3) = 2, β(4) = 2, β(5) = 3, β(6) = 4, β(7) = 4, β(8) = 5, β(9) = 5, β(10) = 6 (these values dislse that when r 5, β(r) an be muted frm the simler frmula (r) = [(r + 1)/2]). We nw state the full deding algrithm Deding algrithm Ste 1 (Identify the gru h assiated with V.) The gru is G(1) if V n(1); G(2) if n(1) < V n(2); and in general is G(h) if n(h 1) < V n(h).

17 248 F. Glver et al. Ste 2 (Cnvert V t a base de value v) v = V n( h 1) Ste 3 (Determine the vetr (i 1,i 2,,i h ) frm the base de value by generating its mnents i r in reverse rder, fr r = h, h 1,,1.) r while r > 1 = h If v = 1 Then ir = r Else w = (v 1)r! u = [w 1/r ] i* = u + β(r) (i r = i* r i* + 1, deending n whih f these gives the largest value f i r satisfying Π w fr Π = (i r 1)(i r 2) (i r r)) Π = (i* 1)(i* 2) (i* r + 1) Π 1 = (i* r) Π (the value f Π if i r = i*) Π 2 = i*π (the value f Π if i r = i* + 1) If Π 2 w then Π = Π 2 i r = i* + 1 Else Π = Π 1 i r = i* Endif v := v Π/r! (giving the residual v t determine the next mnent i r fr r := r 1.) Endif r := r 1 EndWhile i 1 = v We sketh the justifiatin f the deding methd as fllws: the validity f Stes 1 and 2 is aarent. The ratinale underlying Ste 3 derives frm the requirement that i r be the largest integer satisfying Π w fr Π= i 1 i 2 i r r r r This value must learly be unique and hene, if suessfully identified as valid, will uniquely determine eah mnent i r f the deded vetr. The fat that the iteratin ver r = h, h 1,,1 atually generates the rer values fr Π derives frm first bserving that setting i r = r gives i r the rret value fr v = 1, and then additinally bserving that i r is arriately determined fr v = 2 as a result f the definitin f β(r)

18 Plynmial unnstrained binary timisatin Part (whih alies in artiular fr v = 2 in assiatin with setting i r = i* + 1). The validity f the determinatin f i r fr larger values f v an then be established by indutin. The fllwing examle illustrates hw the ding and deding resses are imlemented Illustratin f the ding and deding resses Assume d = 4 and n = Cding We first de the set M = {i 1,i 2,i 3 } = {4,6,11}, in rder t btain the index = V[M] (hene giving the index f the set N = M). Alying Ste 1, t determine the n(h) values gives: G1: n(1) = n = 50 G2 : n(2) = n(1) + n( n 1) / 2 = ,225 = 1,275 G3 : n(3) = n(2) + n( n 1)( n 2) / 6 = 1, ,600 = 296,500 G4 : n(4) = n(3) + n( n 1)( n 2)( n 3) / 24 = 296, ,300 = 526,800 Alying Ste 2 t give the base ding vetr v(3) fr {4,6,11} we btain: v(1) = i1 = 4 v(2) = v(1) + ( i2 1)( i2 2 )/ 2 = / 2 = 14 v(3) = v(2) + i 1 i 2 i 3 / 6 = / 6 = Sine {i 1,i 2,i 3 } is in the gru G(3), by Ste 3 we btain the full ding V(3) f {4,6,11} frm V(3) = n(2) + v(3) = 1, = 1,409. The value V(3) = 1,409 is therefre the value = V[M] that is sught Deding Fr this art f the illustratin we dede the full ding value V = 1298 t find the set M suh that V[M] = V. Ste 1 f the deding methd heks fr membershi in ne f the grus. Sine 1,298 is greater than n(2) = 1,275 and nt mre than n(3) = 296,500, we nlude h = 3, and hene V reresents a vetr in the gru G(3). Ste 2 generates the basi sre v = V n(2) = 1,298 1,275 = 23. We demse Ste 3 int its suessive iteratins. The first iteratin f the While l starts with v = 23 and r = h = 3. w= ( v 1) r! = (23 1)3! = 22 6= 132 1/ r 1/3 u = w = [1, 32 ] = 5 i* = u+ β( r) = 5 + β(3) = 5+ 2= 7 Π = ( i* 1)( i* 2) ( i* r+ 1) = ( i* 1)( i* 2) = 6 5 = 30 Π 1 = ( i* r) Π = 4 30 = 120 Π = i * Π = 7 30= 210 2

19 250 F. Glver et al. Sine Π 2 > w (210 > 132), we have Π = Π 1 = 120 and i 3 = i* = 7. Finally v := v Π/r! gives v = /6 = = 3. The send iteratin, fr r = 2, gives w= ( v 1) r! = (3 1)2! = 2 2 = 4 1/ r 1/2 u = w 4 = = 2 i* = u+ β( r) = 2 + β(2) = 2+ 1= 3 Π = ( i* 1)( i* 2) ( i* r+ 1) = ( i* 1) = 2 Π 1 = (* ) i r Π = 1 2= 2 Π = i * Π = 3 2= 6 2 Sine Π 2 > w (6 > 4), we have Π = Π 1 = 2 and i 2 = i* = 3. Then v := v Π/r! gives v = 3 2/2 = 2. Hene, n the third iteratin, fr r = 1, we have i 1 = v = 2 and the methd terminates. In summary, the index set generated is given by M = {i 1,i 2,i 3 } = {2,3,7} (hene, frm the riginal V = 1,298, we have N 1,298 = {2,3,7}). 5.4 Illustratin f memry required fr the st values and the assiated sets N The amunt f memry required t stre the sts and the assiated sets N using the freging resses is equal t the value n(d) fr a lynmial f degree d when using an asending index rder reresentatin. Fr urses f illustratin, we alulate these values fr the ases d = 1 t 5, assuming n = 100 in eah ase. Fr marisn, we shw the amunt f memry required by using a st matrix f the frm C(i 1,,i d ), whih entails a strage sae f n d. Values f d Cst matrix memry n d ,000 1,000, ,000,000 10,000,000,000 Cded memry n(d) 100 5, ,750 4,087,957 79,375,495 The raid grwth in the size f memry deited by this illustratin suggests that a lynmial f degree 4 r 5 may be the largest that is ratial t wrk with. Hwever, we nte that this memry des nt take aunt f the fat that the rblem will generally be sarse, s that nly a few erent f the ttal number f ssible rdut terms may atually exist at any given time in the rblem frmulatin. We address the hallenges f taking advantage f ur results within the ntext f exliting sarsity, whih is a hallmark f real wrld rblems, in Part 2 f this aer. 6 Cnlusins We have rvided basi relatinshis fr develing effiient algrithms t slve PUB rblems, and have identified ding and deding resses that give the raw materials

20 Plynmial unnstrained binary timisatin Part fr reating seial algrithms t exlit ratial rblems that ntain sarse matries. The sequel t this wrk in Part 2 will give seial tyes f memry strutures and assiated udating algrithms t take advantage f the fundatins laid in Part 1, t yield signifiant imrvements bth in memry nsumed and in the seed f exeuting an algrithm fr the PUB rblem. Referenes Alidaee, B., Khenberger, G. and Ahmadian, A. (1994) 0-1 quadrati rgramming arah fr the timal slutin f tw sheduling rblems, Internatinal Jurnal f Systems Siene, Vl. 25, Brs, E. and Hammer, P.L. (2002) Pseud-Blean timizatin, Disrete Alied Mathematis, Vl. 123, Ns. 1 3, Chardaire, P. and Sutter, A. (1994) A demsitin methd fr quadrati zer-ne rgramming, Management Siene, Vl. 41, N. 4, Gall, G., Hammer, P. and Simene, B. (1980) Quadrati knasak rblems, Mathematial Prgramming, Vl. 12, Glver, F. and Ha, J.K. (2010a) Effiient evaluatins fr slving large 0-1 unnstrained quadrati timizatin rblems, Internatinal Jurnal f Metaheuristis, Vl. 1, N. 1, Glver, F. and Ha, J.K. (2010b) Fast 2-fli mve evaluatins fr binary unnstrained quadrati timisatin rblems, Internatinal Jurnal f Metaheuristis, Vl. 1, N. 2, Glver, F. and Laguna, M. (1997) Tabu Searh, Kluwer Aademi Publishers. Glver, F., Khenberger, G., Alidaee, B. and Amini, M. (1998) Tabu searh with ritial event memry: an enhaned aliatin fr binary quadrati rgrams, in Vss, S., Martell, S., Osman, I.H. and Ruairl, C. (Eds.): Meta-Heuristis Advanes and Trends in Lal Searh Paradigms fr Otimizatin, , Kluwer Aademi Publishers. Hanafi, S., Rebai, A-R. and Vasquez, M. (2010) Several versins f the devur digest tidy-u heuristi fr unnstrained binary quadrati rblems, Wrking aer, LAMIH, Université de Valeniennes. Harary, F. (1953) On the ntin f balaned f a signed grah, Mihigan Mathematial Jurnal, Vl. 2, Khenberger, G. (2010) Ntes n 3-SAT and max 3-SAT, Wrking aer, University f Clrad, Denver. Khenberger, G., Glver, F., Alidaee, B. and Reg, C. (2004) A unified mdeling and slutin framewrk fr mbinatrial timizatin rblems, OR Setrum, Vl. 26, Kraru, J. and Pruzan, A. (1978) Cmuter aided layut design, Mathematial Prgramming Study, Vl. 9, Laughunn, D.J. (1970) Quadrati binary rgramming, Oeratins Researh, Vl. 14, MBride, R.D. and Yrmak, J.S. (1980) An imliit enumeratin algrithm fr quadrati integer rgramming, Management Siene, Vl. 26, Pardals, F. and Xue, J. (1994) The maximum lique rblem, The Jurnal f Glbal Otimizatin, Vl. 4, Pardals, P. and Rdgers, G.P. (1990) Cmutatinal asets f a branh and bund algrithm fr quadrati zer-ne rgramming, Cmuting, Vl. 45, Phillis, A.T. and Rsen, J.B. (1994) A quadrati assignment frmulatin f the mleular nfrmatin rblem, Jurnal f Glbal Otimizatin, Vl. 4, Witsgall, C. (1975) Mathematial methds f site seletin fr eletrni system (EMS), NBS Internal Rert.

21 252 F. Glver et al. Ntes 1 The indexing seleted fr this illustratin is based n susing that n < 30, t surt the nventin that N = {} fr n (as illustrated by N 3 = {3}). We have taken the liberty f hsing the indexes > n fr the reeding sets N by making them smewhat smaller than they wuld likely be under rdinary irumstanes. Aendix 1 Alternative fundatin fr the generi PUB methd The main results that surt the generi PUB methd an be based n a different fundatin, whih des nt rely n reating a rblem transfrmatin t reast eah iteratin as starting frm the slutin x = 0. T exress this, we intrdue rss rdut terms F (i) and values v(i) assiated with the sets P(i) = { P: i N } fr PUB as fllws: F () i =Π x : k N {} i k ( ) vi () = F (): i Pi () As befre, we begin ur analysis frm the 1-fli ersetive, and let x and x " reresent tw binary slutins where x " is btained frm x by fliing the value f a single variable x i frm 0 t 1 r frm 1 t 0. Similarly, we define x = ( F P) " where F =Π( x : i N) and x = ( F : P), where F =Π( xi i N) " :, " " :, and define Δ x = x x. Let F () i and v( i ) and be the instanes f F (i) and v(i) that result fr x = x and let F " () i and v"( i ) be the rresnding instanes that result fr x = x", hene, F() i =Π( xk : k N {}, i ) v( i) = ( F( i) : P( i) ), et. Prsitin 1: ( i) Prf: Write x in the frm Δ x = 1 2 x v( i), i M. ( : ) ( : ) x = F P i + F P P i Given that the sets N fr P(i) are reisely thse that ntain the index i, the freging may be re-written as ( (): ()) ( : ()) x = x F i P i + F P P i i " The value Δ x = x x therefre an be written " " " i i Δ x = x x = x F ( i): P( i) x F ( i): P( i)

22 Plynmial unnstrained binary timisatin Part " sine ( F : P P ( i )) = ( F : P P ( i )) as a nsequene f the fat that x i is nt a art f any F fr P P(i). Mrever, sine x i is nt reresented in either " F () i r F () i fr P(i), and x i is the nly variable that hanges its value, we " have F () i = F () i fr P(i). Hene, " Δ x = ( xi xi) ( F(): i P() i ) Finally frm v( i) = ( F( i) : P( i) ) we btain ( i) Δ x = 1 2 x v( i), i M, as stiulated. Prsitin 1, whih effetively nstitutes an alternative frmulatin f Crllary 2 in Setin 2, diretly generalises the rresnding result f Glver and Ha (2010a) fr quadrati rblems, and likewise shws that the amunt f effrt t mute Δ x is the same fr general lynmial bjetive funtins as fr quadrati bjetive funtins. Thus, in artiular, by this reresentatin, when the rblem data is first set u, and we have a starting slutin x, we mute and save the values v( i) = ( F( i) : P( i) ) fr eah i M. The ritial element fr exliting this rsitin in the mst effetive manner is t identify a way t udate the v( i ) values effiiently frm ne iteratin t the next, as x is udated t beme the slutin reviusly dented by x ". Next define { } { } =Π( ) ( ) Pi (: j) = Pi (): j N = P:, i j N = Pi () P() j F ( i: j) x : k N { i, j} fr P( i: j) k vi ( : j) = F ( i: j): Pi ( : j) We are interested in identifying the new value v"( i ) that relaes v( i ) when x " results " frm x by x = 1 x. j Prsitin 2: = + ( j ) j v"( i) v( i) 1 2 x v( i: j) () (): (), we may break the summatin int tw mnents t give Prf: Frm the definitin vi = ( F i Pi) ( ) ( ) vi () = F (): i Pi ( : j) + F (): i Pi () Pi ( : j) Sine the sets N fr P(i: j) are reisely thse fr P(i) that inlude the index j, the summatin n the left may be rewritten as x j ( F( i: j) : P( i: j) ). By the same tken, nne f the mnents f the summatin n the right will hange when relaing x by x ", hene the value Δ v() i = v"() i v() i will be given by

23 254 F. Glver et al. " " j j Δ v( i) = x F ( i: j): P( i: j) x F ( i: j): P( i: j). Hwever, by definitin, F (i: j) exludes referene t the index j, and hene " F (: i j) = F (: i j). Cnsequently, we btain: " ( j j) ( ) Δ v() i = x x F (: i j): P(: i j). and finally by the substitutin 1 x j fr " j x and v( i: j ) fr ( F (: i j ): P (: i j )) gives Δ = ( j ) whih yields the value v"( i) v( i) ( 1 2 xj ) v( i: j) v v( i) 1 2 x v( i: j) = + fr "( i ) stated in the rsitin. Prsitin 2 likewise generalises the rresnding result f Glver and Ha (2010a) fr quadrati rblems. By this Prsitin 2, we identify the value v( i: j ) and then btain the udated v"( i ) value that bemes v( i) n the next iteratin. If v( i: j ) itself has been maintained in udated frm, this likewise is a O(1) eratin. In general, nsider a sequene f terms P(i:j:k: ), F (i,j,k: ) and v(i:j:k: ), whih g as far as neessary t arry ut the udates needed at eah level. The struture fr ding this is as fllws. Let I be a subset f N that des nt ntain index j and let J = I {j}. Then define { } ( ) PJ = P: J N = PI P( j) F ( J) =Π x : k N J fr P( J) k vj = F( J): PJ (Nte P( ) = P, F ( ) = F and v( ) = v) If J ntains a single element, J = {j}, then I =, and we fr simliity we dente P(J) by P(j), hene, yielding P(j) = { P: j N }, F (j) = Π(x k : k N {j}), v j = ( F j P j ) :, rresnding t ur earlier definitin. We want t btain v"( i ) fr all i N after fliing x j. Cnsider first the sets J that are the maximal sets N ntaining j. Let P max (j) = { P: j N and N is maximal (n set N q stritly ntains N fr q P)}. Lemma: If J is a maximal set N ntaining j, i.e., * P max (j), and J = N *, then v(j) = * fr all x (hene, in artiular, v( J) = *). Prf: J = N * where * P max (j) imlies P(J) = {*}, sine N * is the unique set ntaining J. In turn, the definitin F (J) = Π(x k : k N J) imlies F (J) = 1, sine J is maximal and hene N J =. Finally, by the definitin vj = ( F ( J): PJ ) we have v(j) = * F * (J), whih gives v(j) = *. Nw we btain the fllwing.

24 Plynmial unnstrained binary timisatin Part General rsitin: v"( I) = v( I) + ( 1 2 xj ) v( J) Prf: Frm the definitin vi = ( F I PI) :, we may break the summatin int tw mnents t give ( ) ( ) vi = F( I): PJ + F( I): PI PJ Sine the sets N fr P(J) are reisely thse fr P(I) that inlude the index j, the summatin n the left may be rewritten as x j ( F ( J ): P ( J ) ). By the same tken, nne f the mnents f the summatin n the right will hange when relaing x by x ", hene the value Δ v( I) = v"( I) v( I) will be given by " " j j Δ v( I) = x F ( J): P( J) x F ( J): P( J). Hwever, by definitin, F (J) exludes referene t the index j, and hene " F ( J) = F ( J). Cnsequently, the exressin fr Δ v( I) redues t " ( j j) ( ) Δ v( I) = x x F ( J) : P( J). and finally by the substitutin gives and hene ( j ) Δ v( I) = 1 2 x v( J) ( j ) v"( I) = v( I) x v( J) fr v"( I) 1 x j fr x " j and v J fr ( F ( J ): P ( J ) ) as stated in the rsitin. The methds f this aer and f the Part 2 sequel an likewise use the reeding results as a starting int. Aendix 2 Transfrmatin fr reduing a higher degree lynmial t a quadrati Brs and Hammer (2002) rvide a enalty transfrmatin arah that an be alied iteratively t redue a higher degree lynmial t a quadrati. The mehanism an be deited by nsidering the ase where we seek t relae a ubi frmulatin by a quadrati frmulatin. The rules f the arah fr this ase are as fllws. a identify a ubi term x i x j x k fr redutin b hse a tw variable rdut term within the ubi term, say x i x j, whih will be relaed by a new binary variable x ij (Every ubi term x i x j x h that ntains the rdut x i x j will beme relaed by the assiated term x ij x k. Of urse, the