OLIGONUCLEOTIDE microarrays are widely used

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1 Evolution Strategy with Greedy Probe Seletion Heuriti for the Non-Unique Oligonuleotide Probe Seletion Problem Lili Wang, Alioune Ngom, Robin Gra and Lui Rueda Abtrat In order to aurately meaure the gene expreion level in miroarray experiment, it i ruial to deign unique, highly peifi and highly enitive oligonuleotide probe for the identifiation of biologial agent uh a gene in a ample. Unique probe are diffiult to obtain for loely related gene uh a the known train of HIV gene. The non-unique probe eletion problem i to find a mallet probe et that i able to uniquely identify target in a biologial ample. Thi i an NP-hard problem. We preent two approahe for finding nearminimal non-unique probe et. Eah approah ombine of a determiniti greedy probe eletion heuriti that elet good probe, with an evolution trategy that optimize the eleted probe et. The heuriti, guided by eletion funtion defined over a probe et, deide at eah moment whih probe are the bet to be inluded in, or exluded from, a andidate olution. Our method produe reult that are very loe to, and in many ae better than, thoe of the urrent tate-of-the-art approahe for the non-unique probe eletion problem, namely integer linear programming, optimal utting-plane and geneti algorithm approahe. I. INTRODUCTION OLIGONUCLEOTIDE miroarray are widely ued tool, in moleular biology, providing a fat and oteffetive method for monitoring the expreion of thouand of gene imultaneouly []. In order to meaure the expreion level of a peifi gene in a ample, one mut deign a miroarray ontaining hort trand of known DNA equene of 8 to bp, alled oligonuleotide probe, that are omplementary to the gene egment, alled target. Thee target, if preent in the ample, hould bind to their omplementary probe by mean of hybridization. Typially, the total length of a probe ued to hybridize a gene i only a mall fration of the length of the gene []. The ue of a miroarray experiment depend on how well eah probe hybridize to it target. Expreion level an only be aurately meaured if eah probe hybridize to it target only, given the target i preent in the biologial ample at any onentration. However, hooing good probe i a diffiult tak ine different equene have different hybridization harateriti. A probe i unique, if it i deigned to hybridize to a ingle target. However, due to hybridization error, there i no guarantee that unique probe will hybridize to their Lili Wang, Alioune Ngom, Robin Gra and Lui Rueda with the Shool of Computer Siene, Univerity of Windor, Windor, N9B P4, Ontario, Canada ( {wang111v, angom, rgra}@uwindor.a). Lui Rueda i with the Department of Computer Siene, Univeridad de Conepión, Edmundo Larena 1, Conepión, VIII Region, Chile ( lrueda@ude.l). Thi work wa upported by the Natural Siene and Engineering Reearh Counil of Canada under Reearh Grant RG-PIN 8117 intended target only. Many parameter uh a eondary truture, alt onentration, GC ontent, free energy and melting temperature alo affet the hybridization quality of probe [], and their value mut be arefully determined to deign high quality probe. It i partiularly diffiult to deign unique probe for loely related gene that are to be identified. Too many target will be imilar and hene hybridization error inreae ubtantially. An alternative approah i to devie a method that an make ue of nonunique probe, i.e. probe that are deigned to hybridize to at leat one target []. Alo, a maller probe et an be ued with non-unique probe than an be with unique probe. Minimizing the number of probe in a miroarray experiment i alo a reaonable objetive, ine it i proportional to the ot of the experiment. The non-unique probe eletion problem i to determine a mallet et of probe able to identify all target preent in a biologial ample. Thi i an NP-hard problem [1], and everal approahe to it olution have been propoed reently [1] [] [] [4] [] [6] [7] [8]. II. NON-UNIQUE PROBE SELECTION PROBLEM Given a target et, T = {t 1,..., t m }, and probe et, P = {p 1,..., p n }, an m n target-probe inidene matrix H = [h ij ] i uh that h ij = 1, if probe p j hybridize to target t i, and h ij = otherwie. Table I how an example of a matrix with m = 4 target and n = 6 probe. A probe p j eparate two target, t i and t k, if it i a ubtring of either t i or t k, that i, if h ij h kj = 1. For example, if t i = AGGCAATT and t k = CCATATTGG, then probe p j = GCAA eparate t i and t k, ine it i a ubtring of t i only, wherea probe p l = ATT doe not eparate t i and t k, ine it i a ubtring of both target []. Two target, t i and t k, are -eparated, 1, if there exit at leat probe uh that eah eparate t i and t k ; in other word, the Hamming ditane between row i and k in H i at leat. For example, in Table I target t and t 4 are 4-eparated. A target t i -overed, 1, if there exit at leat probe uh that eah hybridize to t. In Table I, target t i -overed. Due to hybridization error in miroarray experiment, it i required that any two target be min -eparated and any target be min -overed; uually, we have min and min. Thee two requirement are alled eparation ontraint and overage ontraint. Given a matrix H, the objetive of the non-unique probe eletion problem i to find a minimal ubet of probe that determine the preene or abene of peified target, and uh that all ontraint are atified. In Table 1, if min = min = 1 and auming that exatly one of t 1,..., t 4 i in the ample, then the goal i to elet a minimal et of probe that /8/$. 8 IEEE 4

2 TABLE I A 4 6 TARGET-PROBE INCIDENCE MATRIX. p 1 p p p 4 p p 6 t t t t allow u to infer the preene or abene of a ingle target. In thi ae, a minimal olution i {p 1, p, p } ine for target t 1, probe p 1 and p hybridize while p doe not; for target t, probe p 1 and p hybridize while p doe not; for target t, probe p and p hybridize while p 1 doe not; and finally for target t 4, only probe p hybridize. Thu, eah ingle target will be identified by the et {p 1, p, p }, if it i the only target preent in the ample; moreover, all ontraint are atified. For min = min =, a minimal olution that atifie all ontraint i {p, p, p, p 6 }. Of oure, the et {p 1,..., p 6 } i a olution but it i not minimal, and therefore i not oteffetive. Our prior aumption above, that only a ingle target i preent in the ample, reate ome diffiultie when in reality multiple target are in the ample. Let min = min = 1, then eleting {p 1, p, p } a before, reult in the hybridization of all probe when both target t 1 and t are in the ample. Hene, we annot ditinguih between the ae where pair (t 1, t ) i in the ample and where t i alo in the ample. Uing {p, p, p }, however, intead of {p 1,..., p 6 } will reolve thi experiment. Let min = min =, then any of the pair (t 1, t ), (t 1, t 4 ), (t, t ), and (t, t 4 ) will aue the hybridization of all probe of {p, p, p, p 6 }; thu, we an determine the preene of target t for intane, but annot determine whih other target i preent. The olution to the multiple target problem i to ue aggregated target that repreent the preene or abene of ubet of target in the ample [1]. An aggregated target t a i thu a ubet of target, and the ingle target eparation problem i a peial ae of the multiple target eparation problem. Stated formally, given an m n target-probe inidene matrix H with a target et T = {t 1,..., t m } and a probe et P = {p 1,..., p n }, and a minimum overage parameter min, a minimum eparation parameter min and a parameter d max 1, the aim of the non-unique probe eletion problem i to determine a ubet P min = {q 1, q,, q } P of probe uh that: 1) = P min n i minimal. ) Eah target t T i min -overed by ome probe in P min. ) Eah pair of aggregated target (t a x, t a y) T T, with t a x, t a y d max, i min -eparated by ome probe in P min. In Point ) above, we eek to guarantee min -eparation not only between pair of ingle target but alo between pair of mall ubet of target t a x, t a y T ( T i the power et of T ), eah ubet up to ardinality d max. Thee requirement are alled the group eparation ontraint [1]. Thi problem wa proved to be NP-hard, in [1], by performing a redution from the et overing problem. It i NP-hard even for ingle target a well a for min = 1 or min = 1. The work of [] wa the firt to formulate the non-unique probe eletion problem for ingle target a an integer linear programming (ILP) problem. Let C = {(i, k) 1 i < k m} be the et of all ombination of target indie. Aign x j = 1 if probe p j i hoen and otherwie. Then, we have the following ILP formulation: Subjet to: Minimize: n x j. (1) j=1 x j {, 1} 1 j n, () n h ij x j min 1 i m, () j=1 n h ij h kj x j min 1 i < k m. (4) j=1 Funtion (1) minimize the number of probe. The probe eletion variable are binary-valued in Retrition (). Contraint () and (4) are the overage and eparation ontraint, repetively. Note that Contraint (4) are for ingle target only. A oppoed to thi, in [1], an ILP formulation wa propoed, whih inlude the group eparation ontraint for aggregated target. In thi paper, we olve the original ILP formulation of [], that i for ingle target eparation only, uing determiniti greedy heuriti ombined with an evolution trategy. Note that one an eaily hek if the original et of andidate probe atify all the ontraint. If not, then there are no feaible olution. In thi ae, we an inert unique virtual probe in the original probe et only for thoe target or target-pair that are not min -overed or min -eparated. Thi will enure the exitene of feaible olution. III. PREVIOUS WORK Shliep et al. [] firt introdued the non-unique probe eletion problem and deribed a imple but fat greedy heuriti, whih ompute an approximate olution that guarantee min -eparation for pair of mall aggregated target. Klau et al. [1] [] propoed two ILP formulation for thi problem, repetively for ingle target and aggregated target, then olved it uing the ILP olver CPLEX on pre-redued problem intane. They alo proved that the non-unique probe eletion problem i NP-hard. Menee et al. [] propoed a determiniti greedy heuriti, for ingle target only, that firt ontrut an initial feaible olution through loal earh, and then applie a redution method to further redue thi olution. Ragle et al. [4] developed an optimal utting-plane ILP heuriti, for ingle target only, to find optimal olution within pratial omputational

3 limit. Wang et al. [6] [8] propoed determiniti greedy heuriti that elet probe baed on their ability to help atify the ontraint. Reently, Wang et al. [7] ombined the probe eletion funtion of [8] with a geneti algorithm, and produed reult that are at leat omparable to (and in mot ae, better than) thoe obtained by the method of [4], whih i onidered the bet approah for thi problem. IV. PROBE SELECTION FUNCTION We want to elet a minimum number of probe uh that eah target i min -overed and eah target-pair i min - eparated. Conider a target-probe inidene matrix, H, the parameter min and min, the initial feaible andidate probe et, P = {p 1,..., p n }, and the target et T = {t 1,..., t m }. Let P ti be the et of probe hybridizing to target t i, and P tik be the et of probe eparating the target-pair t ik. A probe p P ti i an eential overing probe if and only if P ti = min. In Table I, for intane, the probe in P t = {p 1, p, p 6 } are eential overing probe if min =. Eential eparating probe are defined imilarly. Eential probe mut be ontained in any minimal olution; that i, removing any uh probe will make the olution unfeaible. A redundant probe i the one for whih a feaible olution remain feaible when the probe i removed. Note that a probe may be redundant for ome andidate olution but non-redundant for other. There i a degree of redundany between probe uh that highly redundant probe are in very few or no minimal olution. A. Coverage Funtion We want to hooe the minimum number of probe uh that eah target i min -overed. Given the matrix H, the parameter min, the andidate probe et P = {p 1,..., p n } and the target et T = {t 1,..., t m }, we defined the funtion ov dr : P T [, 1] in [8] a follow: ov dr (p j, t i ) = h ij min P ti, () where p j P ti, t i T, and P ti i the et of probe hybridizing to target t i ; ov dr (p j, t i ) i the amount that p j ontribute to atify the overage ontraint for target t i. For target t i, p j i likely to be redundant for a larger value of P ti and likely to be non-redundant for a maller value of P ti. We defined the overage funtion C dr : P [, 1] in [8] a follow: C dr (p j ) = max t i T pj {ov dr (p j, t i ) 1 j n}, (6) where T pj i the et of target overed by p j. C dr (p j ) i the maximum amount that p j an ontribute to atify the minimum overage ontraint. Table II how the overage funtion table produed from Table I. Funtion C dr favor the eletion of probe that min - over target t i that have the mallet ubet P ti ; thee are the eential or near-eential overing probe. In Table II, for example, target t ha the minimal value P t =, and hene TABLE II COVERAGE FUNCTION TABLE OBTAINED FROM TABLE I. p 1 p p p 4 p p 6 t min min t min t min min min min t 4 min min C min min min min min min dr 4 any probe that over it an be eleted firt. In partiular, funtion C dr guarantee the eletion of near-eential overing probe that min -over dominated target; t i dominate t k if P tk P ti. In Table II, for example, t dominate t 4 ine P t4 = {p, p 4, p } {p, p, p 4, p, p 6 } = P t. Any min -over of the dominated target t k will alo min -over all it dominant target, and therefore, more target are min - overed. Probe overing the dominated target t k have larger ov dr value than probe overing it dominant target t i, ine P tk < P ti, and hene they will be eleted firt. We would alo like to favor the eletion of dominant probe; p j dominate p l if T pl T pj. In Table II, for intane, p 6 dominate p 1 ine T p1 = {t 1, t } {t 1, t, t } = T p6. Seleting dominant probe intead of dominated probe over more target. In the example, however, we have C dr (p 1 ) = C dr (p 6 ), and hene p 1 ould be eleted for target overage rather than p 6, depending on a partiular order of the probe. On the other hand, p 6 dominate p and C dr (p 6 ) > C dr (p ), and hene p 6 will be eleted firt. To favor the eletion of a dominant probe that ha the ame value, C dr, a ome of it dominated probe, we define a new ov dr funtion, renamed ov dp, a follow: ov dp (p j, t i ) = h ij min P ti T p j m, (7) where p j P ti, t i T, and ov dp (p j, t i ) [, 1]; P ti i the et of probe hybridizing to target t i, T pj in the penalty term i the et of target overed by p j, and m i the number of target. In Equation (7), probe that over fewer target are penalized more than probe that over more target. B. Separation Funtion We want to hooe the minimum number of probe uh that eah target-pair i min -eparated. We defined the funtion ep dr : P T [, 1] in [8] a follow: ep dr (p j, t ik ) = h ij h kj min P tik, (8) where p j P tik, t ik T, and P tik i the et of probe eparating target-pair t ik ; ep dr (p j, t ik ) i what p j an ontribute to atify the eparation ontraint for target-pair t ik. We defined the eparation funtion S dr : P [, 1] in [8] a follow: S dr (p j ) = max t ik T p j {ep dr (p j, t ik ) 1 j n}, (9) 6

4 where Tp j i the et of target-pair eparated by p j. S dr (p j ) i the maximum amount that p j an ontribute to atify the minimum eparation ontraint. Table III how the eparation funtion table produed from Table I. TABLE III SEPARATION FUNCTION TABLE OBTAINED FROM TABLE I. p 1 p p p 4 p p 6 t 1 min min t min 1 t min min min 14 min t min min min t min 4 min min t 4 S min min min min min min dr Funtion S dr alo favor the eletion of probe that min - eparate target-pair t ik that have the mallet ubet P tik and further favor the eletion of near-eential eparating probe that min -eparate dominated target pair. Similarly to the overage funtion, to favor the eletion of a dominant probe that ha the ame value, S dr, a ome of it dominated probe, we define a new ep dr funtion, renamed ep dp, a follow: ep dp (p j, t ik ) = h ij h kj min P tik T p j m(m 1), (1) where p j P tik, t ik T, and ep dp (p j, t ik ) [, 1]; P tik i the et of probe eparating target-pair t ik and T p j i the et of target-pair eparated by p j. In Equation (1), probe that eparate fewer target-pair are penalized more than probe that eparate more target-pair. C. Seletion Funtion We want to elet the minimum number of probe uh that all overage and eparation ontraint are atified; that i, we mut elet a probe aording to it ability to help atify both overage and eparation ontraint. In [8], we ombined funtion C dr and S dr into a ingle probe eletion funtion, D dr : P [, 1] a follow: D dr (p j ) = max{(c dr (p j ), S dr (p j )) 1 j n}. (11) D dr (p j ) i the degree of ontribution of p j, that i, the maximum amount required for p j to atify all ontraint. D dr enure that all eential probe p j will be eleted for inluion in the ubequent andidate olution, ine C dr (p j ) = 1 or S dr (p j ) = 1. With our definition of D dr, probe p that over dominated target or eparate dominated target-pair have the highet D dr (p) value. By eleting a probe p to over a dominated target t i or to eparate a dominated target-pair t ik, we are alo eleting p to over a many a many target a poible (all target that dominate t i ) or to eparate a many target-pair a poible (all target-pair that dominate t ik ). Thi i the main greedy trategy in our heuriti in Setion V. We define another probe eletion funtion, D dp : P [, 1], that ombine the funtion C dp and S dp a follow: D dp (p j ) = max{(c dp (p j ), S dp (p j )) 1 j n}. (1) In D dp, dominant probe among all probe that have equal value in D dp will be eleted firt; thi i the eondary greedy eletion priniple. Thee two greedy priniple together allow a larger overage and eparation when uing D dp than D dr in a greedy earh method. V. GREEDY SELECTION WITH EVOLUTION STRATEGY In thi etion, we propoe two determiniti greedy heuriti that filter out bad probe a in Menee et al. []. That work ued no eletion funtion to deide whih probe to filter out; probe are removed a long a the feaibility of a given andidate olution i not ompromied: thee are the bad probe in []. They ort the probe in inreaing order of the number of target they hybridize, then elet probe in thi order to obtain a feaible olution, and finally remove redundant probe to obtain a near minimal olution. In the data et, many probe hybridize to the ame number of target and thi i not enough to identify the bet probe. In our heuriti, the value of D dr or D dp tore muh more information about the urrent probe et, in uh a way that the algorithm an deide whih probe to onider bet for eletion. A. Dynami Dominated Row Covering Heuriti The Dynami Dominated Row Covering (DDRC) heuriti introdued in [6] deide at any given moment whih probe i the bet to inlude in or exlude from a andidate olution. Thi deiion depend on the urrent probe et to elet from and on the urrent olution. DDRC ue the D dr funtion to elet the bet probe. Let P ol be the urrent olution and let p l P P ol be the newly eleted probe. We re-ompute the ov dr (p j, t i ) and ep dr (p j, t ik ) entrie only for thoe row t i and t ik overed by p l, a follow: ov dr (p j l, t i ) = h ij min C ti P ti C ti, (1) where p j P ti C ti, t i T, ov dr (p j l, t i ) [, 1], and C ti i the et of eleted probe (inluding the new eleted probe p l ) that already over row t i ; and ep dr (p j l, t ik ) = h ij h kj min S tik P tik S tik, (14) where p j P tik S tik, t ik T, ep dr (p j l, t ik ) [, 1], and S tik i the et of eleted probe (inluding the new eleted probe p l ) that already over row t ik. We then retrit the olumn of H to the probe in P {p l }. The update are uh that ome dominant row may beome dominated, and therefore, the algorithm an onentrate it effort to elet probe for overing thee new dominated row along with the urrent dominated row. 7

5 Likewie, one a row i already min -overed or min - overed, the algorithm hould fou it effort on eleting probe for the remaining row only. We illutrate thi in the example given in Table IV. If we elet p 1, then only the entrie for row t 1 and t will be updated. Alo, the value of C dr (indiated by - ), S dr and D dr are updated only for probe p, p, p 4, p 6 after removing p 1. TABLE IV COVERAGE MATRIX FOR DDRC BEFORE AND AFTER SELECTING p 1. p 1 p p p 4 p p 6 t min min t min t min min min min t 4 min min C min min min min min min dr 4 p 1 p p p 4 p p 6 t min 1 1 min 1 1 t min 1 1 t min min min min min t 4 min min min C dr min - Note that one the eleted probe already atify the ontraint on a target or target-pair, the row aoiated with that target or target-pair beome all-zero. In the table, if min = and p i eleted next after p 1, then all the ov dr value for target t 1 will beome zero. DDRC diard row that are already atified by the urrently eleted probe et o that it onentrate it effort only on the remaining row. Note alo that t i now a dominated target, and hene it will ompete with other dominated target for overage. DDRC i dynami, in the ene that the overage and eparation matrie hange after eah eletion of a probe. The DDRC heuriti onit of the Initialization, the Contrution and the Redution phae and i hown in Algorithm 1 4. Algorithm 1 Dynami Dominated Row Covering heuriti Input: T = {t 1,..., t m }, P = {p 1,..., p n }, and H = [h ij ] Output: Near-minimal olution P min P ini Initialization(P, T, H) P on Contrution(P ini, P, T, H) P red Redution(P on, P, T, H) Return P min P red. In the Initialization phae, we ompute the initial value of D dr (p) for eah probe p P, and reate an initial but poibly non-feaible olution P ini whih ontain eential probe only. In the Contrution phae, we edly inert high-degree probe into P ini until an initial feaible olution P on i obtained. In the Redution phae, we redue P on by edly removing low-degree probe o a to obtain a final near minimal feaible olution P red. DDRC i greedy a it a earhe the pae P (the power et of P ) by eleting Algorithm Initialization Phae Input: P, T, H Output: Initial unfeaible olution P ini G H, /* ave initial H */ P ini {p P p i eential} for all t a (1 a m) and t ab (1 a < b m) overed by eah q P ini do Compute D dr (p) for all p {P ta C ta } {P tab S tab }. H H P Pini, /* retrition of H to in P P ini */ P P P ini Return P ini. Algorithm Contrution Phae Input: P ini, P, T, H Output: Feaible olution P on P on P ini for eah target t i not min -overed by P on do n i #probe needed to omplete min -overage of t i P on P on {highet-degree probe q P P on that over t i } for all t a (1 a m) and t ab (1 a < b m) overed by q do Update D dr (p) for all p {P ta C ta } {P tab S tab } H = H P {q} P = P {q} until n i probe are inerted for eah target-pair t ik not min -eparated by P on do n ik #probe to omplete min -eparation of t ik P on P on {highet-degree probe q P P on that eparate t ik } for all t a (1 a m) and t ab (1 a < b m) overed by q do Update D dr (p) for all p {P ta C ta } {P tab S tab } H H P {q} P P {q} until n ik probe are inerted Return P on. only the high-degree probe to atify the ontraint and by removing only the low-degree probe to maintain the feaibility of a andidate olution. Solution P min i not neearily optimal. 1) Computational Complexity of DDRC: In the wort ae, the time omplexity of the DDRC i dominated by the eparation ontraint atifation in the Contrution phae. 8

6 Algorithm 4 Redution Phae Input: P on, P, T, H Output: Redued olution P red /* Redution Phae */ P red P on H G Pred, /* retore initial H and retrit to P red */ Compute D dr (p) for all p P red Sort P del {p P red D dr (p) < 1} in inreaing D dr (p) if P red {p} i feaible for eah p P del then P red P red {p} end if Return P red. Let M = m m. The inner for loop ha at mot m + M iteration; there are at mot n probe to update and eah update take m + M tep; thu, the loop take O(nM ) time. The loop iterate at mot n ik min n time; thu, the loop take O(n M ) time. The outer for loop ha at mot M iteration. Therefore, DDRC run in O(n M ) = O(n m 6 ). In pratie, DDRC run muh fater than O(n m 6 ) beaue n ik min < min m n. B. Dynami Dominant Probe Seletion Heuriti The Dynami Dominant Probe Seletion (DDPS) heuriti, introdued in [6], i imilar to DDRC exept that we ue the funtion D dp, intead. After eah probe eletion, the overage and eparation matrie are update uing the following equation: ov dp (p j l, t i ) = h ij min C ti P ti C ti T p j U pj, m (1) and ep dp (p j l, t ik ) = (16) h ij h kj min S tik P tik S tik T p j U p j m(m 1) where ov dp (p j l, t i ) [, 1] and ep dp (p j l, t ik ) [, 1]. U pj T pj and U p j T p j are, repetively, the et of target in T pj and target-pair in T p j that are already min -overed and min -eparated by the urrently eleted probe et. A explained before, the row aoiated with thee target or target-pair will be all-zero, and therefore, they hould be diarded from T pj or T p j for probe p j. 1) Computational Complexity of DDPS: In the wort ae, the time omplexity of the DDPS i dominated by the eparation ontraint atifation in the Contrution phae. The differene between DDRC i that eah update in the inner for loop take (m + M)M tep, ine omputing the T p for eah probe p take M tep; thu, the loop take O(nM ) time. Thu, DDPS run in O(n m 8 ), and hene, i O(m ) time lower than DDRC., C. Evolution Strategy In thi etion, we deribe an Evolution Strategy (ES) [] that optimize the olution obtained by our determiniti greedy method. Our ES i hown in Algorithm with the DDRC heuriti. Algorithm Evolution Strategy with DDRC Heuriti Input: T = {t 1,..., t m }, P = {p 1,..., p n }, and H = [h ij ] Output: Near-minimal olution P min P min DDRC(P, T, H) P mut Mutation(P min, P ) P on Contrution(P mut, P, T, H) P red Redution(P on, P, T, H) if P red < P min then P min P red end if until n gen generation are performed until n ite iteration are performed Return final P min. Our ES tart with the initial parent olution obtained from DDRC, P min, and maintain a ingleton population in eah generation. A hild olution, P red, i obtained in eah generation after applying mutation, ontrution and redution on P min. P red replae P min only if it i maller. Thi proe ontinue for n gen generation. A parent P min may not be optimized after n gen generation; thu, we iterate ES n ite time to eape loal optima and to further optimize P min. In Algorithm, P min i alway the bet olution o far. However, mutation i diruptive and the mutant, P mut, may no be feaible; hene, ontrution and redution are applied in order to generate a feaible near-minimal olution, P red. Mutation operation i hown in Algorithm 6. Algorithm 6 Mutation in DDRC ES Input: P min, P = {p 1,..., p n } Output: New olution P mut P mut P min Generate a random number r [1, P ] Randomly elet a probe p P if p P mut then With probability 1 D dr (p): P mut P mut {p}, ele With probability D dr (p): P mut P mut {p} end if until r probe are proeed Return final P mut. Mutation proeed by randomly eleting a random number r [1, P ] of probe from the initial total probe et P. Depending on it preene or abene in P min, a eleted probe p i then randomly exluded from P min with 9

7 probability 1 D dr (p), or randomly inluded into P min with probability D dr (p). Thi mutation tend to remove bad probe and inert good probe. Mutant P mut, though poibly not feaible, give potentially a good olution, P red, after ontrution and redution. With a random value of r, our ES perform both: a loal earh around P min with a mall value of r (to exploit P min ), and, a global earh in the pae P with a large value of r (to eape a loal optimum trap). VI. COMPUTATIONAL EXPERIMENTS We performed experiment to how the minimization ability of DDRC ES and DDPS ES and that they produe approximate reult that are loe to thoe of the urrent tate-of-the-art advaned heuriti for thi problem. The program were written in C and all tet ran on two Intel Xeon TM CPU.6GHz with GB of RAM under Ubuntu 6.6 i86. We onduted experiment on ten artifiial data et and three real data et, that were kindly provided by Dr. Ragle and Dr. Pardalo [4]. Thee data et were ued in all previou tudie mentioned in Setion III exept for HIV-1 and HIV- et whih were ued only in [] [4] [6] [7] [8]. Table V how, in the eond and third olumn, the dimenion T P (number of target number of probe) of the inidene matrix for eah et (M for Meiobentho i the larget et). Column A i the number of required virtual probe inerted into P to maintain the feaibility of the initial probe et P. Due to pae ontraint, we refer the reader to [] [] [] for the full detail on the ontrution of thee data et. TABLE V DATA SETS AND DIMENSIONS. Set T P A a a 6 81 a a a b b b b4 4 6 b 4 68 M HIV HIV All experiment were performed with parameter min = 1 and min =, a in all previou tudie. For ES, the value for parameter (n ite, n gen ) were, repetively, (1, 1) for DDRC ES and (1, 1) for DDPS ES. DDRC ES terminated in two week given all the thirteen data et altogether. The parameter value for DDPS ES were determined uh that it terminate in two week; we tried imilar ombination uh a (1, 1) but with reult omparable to (1, 1). DDRC and DDPS, alone, terminated in about hour given all data et together. Table VI how, for all data et, the minimum ize P min attained by the determiniti greedy method, GrdM, DDRC, DDPS, and the random hill-limbing method, DDRC ES and DDPS ES. GrdM i the bet publihed greedy method [] for the non-unique probe eletion. Table VII ompare DDRC ES and DDPS ES with the urrent bet method for the non-unique probe eletion problem: the Integer Linear Programming (ILP) of [1] [], the Optimal Cutting- Plane Algorithm (OCP) of [4] and the Geneti Algorithm (DRC GA) of [7]. In both table, the final P min inlude the virtual probe inerted into P. TABLE VI DDRC ES AND DDPS ES VERSUS DDRC AND DDPS. Set P GrdM DDRC DDPS DDRC ES DDPS ES a a a a a b b b b b M HIV HIV TABLE VII DDRC ES AND DDPS ES VERSUS ILP, OCP AND DRC GA. Set P DDRC ES DDPS ES ILP OCP DRC GA a a a a a b b b b b M HIV HIV Given two heuriti X and Y, we ay that X < Y in term of their overall performane on the data et, if X produe larger olution than Y in the majority of the data et; for intane, in Table VI, we have DDRC < DDPS ine DDRC give a larger P min in 8 out of 1 data et, while DDPS give a larger P min in out of 1 data et. From Table VI and VII, we have the following order: GrdM < DDRC < ILP < DDPS < DDPS ES < DDRC ES < OCP < DRC GA. DDRC ES and DDPS ES ubtantially 6

8 outperformed the greedy DDRC and DDPS heuriti in all intane, whih in turn greatly outperformed GrdM in all intane. A explained in the firt paragraph of Setion V, GrdM ue no eletion funtion and thu, doe not know whih probe are good or bad. Reall that both ES method loally and globally optimize the olution produed by their orreponding determiniti greedy method. The fat that DDRC ES performed better than DDPS ES i due to their value in parameter n ite : a larger value allowed DDRC ES to eape loal optima, however, DDPS ES i expeted to perform better given the ame parameter value of DDRC ES. All our heuriti ahieved greater redution on the M et than ILP. In partiular, DDRC ES and DDPS ES ubtantially outperformed ILP in at leat 6% of the intane and ahieved reult very loe to thoe of OCP. The author of [1] [], firt applied the greedy heuriti of [] to redue the initial probe et (and to redue the ILP running time), and then further optimized the redued probe et with the ILP olver CPLEX (CPLEX i one of the leading mathematial programming oftware pakage available and few heuriti, if any, are able to ompete with it reult). CPLEX wa retrited to earh only a mall portion of the olution pae, hene ILP wa not aware of the full initial probe et. Our heuriti had no uh retrition and run fater. Table VIII report the improvement, Imp, of DDRC ES over ILP, OCP and DRC GA omputed a in Equation 17. We hoe DDRC ES beaue it performed bet among our heuriti. Imp = P DDRC ES min P Heu min P Heu min 1, (17) where Heu i either ILP, OCP or DRC GA. A negative (poitive) value of Imp mean that a DDRC ES reult i Imp% better (wore) than an Heu reult. Conequently, Imp i negative when DDRC ES return a probe et maller than P Heu min. Therefore the maller the value of Imp, the better DDRC ES. TABLE VIII IMPROVEMENTS OF DDRC ES OVER ILP, OCP AND DRC GA. Set ILP OCP DRC GA a a a a a b b b b b M HIV HIV The mean improvement of DDRC ES relative to ILP, OCP and DRC GA are 4.%, +1.% and +.98%, repetively; whih are very low and hene very good. Thi ugget that a better hoie of parameter, a well a a better ES trategy, would give a better overall performane than both OCP and DRC GA. In our previou work [7], the geneti algorithm DRC GA ued the eletion funtion D dr without, however, re-omputing D dr aording to Equation (1) and (14) after eah probe eletion; that i the overage and eparation matrie were tati. Among our greedy method introdued in [6], the tati D dr funtion performed muh wore than the dynami D dr funtion ued here. Through it impliit parallel earh ability, DRC GA i more able to eape loal optima than DDRC ES. Finally, we note that OCP ue the integer linear programming priniple and wa propoed in [4] a an improvement (in time and minimization) of the ILP method of [1] []. Both, OCP and DRC GA are not retrited to earh a portion of the initial probe et a in ILP. VII. CONCLUSIONS AND FUTURE RESEARCH In thi paper, we introdued two evolution trategy approahe to the non-unique probe eletion problem. We onidered the ae for ingle target eparation only. The experiment howed that thee heuriti are able to obtain reult that are very loe to the reult of the tate-of-theart heuriti for thi problem. Further improvement on our DDRC ES and DDPS ES heuriti ould potentially yield better reult than OCP and GA in fater time. Our eletion funtion an be modified to be ued in any (tandard) et overing problem uh a thoe that appear in bioinformati. REFERENCES [1] G.W. Klau, S. Rahmann, A. Shliep, M. Vingron, and K. Reinert, Integer linear programming approahe for non-unique probe eletion, Direte Applied Mathemati, vol. 1, pp , 7. [] G.W. Klau, S. Rahmann, A. Shliep, M. Vingron, and K. Reinert, Optimal robut non-unique probe eletion uing integer linear programming, Bioinformati, vol., pp. i186 i19, 4. [] C.N. Menee, P.M. Pardalo, and M.A. Ragle, A new approah to the non-unique probe eletion problem, Annal of Biomedial Engineering, vol., no. 4, pp , 7. [4] M.A. Ragle, J.C. Smith, and P.M. Pardalo, An optimal utting-plane algorithm for olving the non-unique probe eletion problem, Annal of Biomedial Engineering, vol., no. 11, pp., 7. [] A. Shliep, D.C. Torney, and S. Rahmann, Group teting with DNA hip: generating deign and deoding experiment, Pro. IEEE Computer Soiety Bioinformati Conferene (CSB ), p ,. [6] L. Wang, A. Ngom, L. Rueda and R. Gra Seletion Baed Heuriti for the Non-Unique Oligonuleotide Probe Seletion Problem in Miroarray Deign, IEEE/ACM Tranation on Computational Biology and Bioinformati, 8, under review. [7] L. Wang, A. Ngom, and R. Gra Non-Unique Oligonuleotide Miroarray Probe Seletion Method Baed on Geneti Algorithm, Pro. 8 IEEE Congre on Evolutionary Computation, June 1-6, Hong Kong, China, 8, to appear. [8] L. Wang, and A. Ngom, A model-baed approah to the nonunique oligonuleotide probe eletion problem, Seond International Conferene on Bio-Inpired Model of Network, Information, and Computing Sytem (Bioneti 7), Deember 1-1, Budapet, Hungary, ISBN: , 7. 61