Solving Large Double Digestion Problems for DNA Restriction Mapping by Using Branch-and-Bound Integer Linear Programming

Transcription

1 The First Internationa Symposium on Optimization and Systems Bioogy (OSB 07) Beijing, China, August 8 10, 2007 Copyright 2007 ORSC & APORC pp Soving Large Doube Digestion Probems for DNA Restriction Mapping by Using Branch-and-Bound Integer Linear Programming Zhijun Wu 1 Yin Zhang 2 1 Department of Mathematics, Iowa State University, Ames, Iowa, U.S.A. 2 Department of Computationa and Appied Mathematics, Rice University, Houston, Texas, U.S.A. Abstract The doube digestion probem for DNA restriction mapping has been proved to be NPcompete, and is intractabe if the numbers of the DNA fragments generated by the two restriction enzymes are arge. Severa approaches to the probem have been used, incuding exhaustive search and simuated anneaing, and have proved to be effective ony for sma probems, as the probem size or in other words, the size of the search space for the probem grows as a factoria function (n!)(m!) of the numbers n and m of the DNA fragments generated by the two restriction enzymes, respectivey. In this paper, we formuate the doube digestion probem as a mixed-integer inear program by foowing Waterman 1995 in a sighty different form. We show that with this formuation and by using some state-of-the-art integer programming techniques, we can actuay sove doube digestion probems for fairy arge sizes. In particuar, we can sove a set of randomy generated probems of sizes up to (242!)(250!), which are many magnitude increases from previousy reported (16!)(16!) testing sizes. 1 Introduction A DNA sequence can be broken down (digested) into a set of sma fragments by using some specia proteins caed restriction enzymes. The sequences of the fragments usuay become easier to determine and the engths of the fragments can aso be measured by using, for exampe, the eectrophoretic ge. Once determined, the fragments can be put back in the origina order to obtain the whoe sequence. The atter process is caed the DNA restriction mapping and is an important step in genomic sequencing [14]. A critica part of DNA restriction mapping is to find the right order of the fragments given ony the engths of the fragments. This probem is caed the digestion probem and in particuar, the doube digestion probem if two different restriction enzymes are used in the experiments [17]. Let the two restriction enzymes be denoted by A and B. Let a = {a 1,a 2,...,a n } Z n be the engths of the fragments obtained after appying enzyme A to a given sequence, and b = {b 1,b 2,...,b m } Z m the engths of the fragments obtained after

2 268 The First Internationa Symposium on Optimization and Systems Bioogy appying enzyme B, where the engths are measured in numbers of DNA base pairs. Let c = {c 1,c 2,...,c } Z be the engths of the fragments obtained after first appying enzyme A and then enzyme B to the same sequence. The fragments in a, b, and c are given in arbitrary orders. However, as shown in Figure 1, if we can put the fragments in a and b in their origina orders, we can immediatey obtain a the fragments in c by simpy combining a the restriction sites in a and b as if they were digested by a singe enzyme. On the other hand, if we put a and b in other orders, we wi in genera obtain a set of fragments different from c. The doube digestion probem is to find the orders for a and b, given c as the resut of the sequence being digested by A and B combined. A B C Figure 1: Doube digestion A sequence of 19 base pairs is digested by a restriction enzyme A at sites 1, 4, 16, and is cut into 4 fragments of engths 1, 3, 12, 3, and by B at sites 2, 6, 12, 15, 18, and is cut into 6 fragments of engths 2, 4, 6, 3, 3, 1. If both A and B are appied, the sequence wi be digested at both A and B sites, and is cut into 9 fragments of engths 1, 1, 2, 2, 6, 3, 1, 2, 1 in C. The doube digestion probem is soved routiney in moecuar bioogy abs for constructing the restriction maps for newy coned DNA sequences [17]. The soution of a generaized version of the probem may aso be appied to protein determination in proteomics using mass spectrometry [11]. The probem can be difficut to sove, however, if the numbers of the generated fragments are arge. In genera, it is intractabe as shown in Godstein and Waterman 1987 [4]. Severa approaches to the probem have been proposed, such as exhaustive search [12, 15], simuated anneaing [4, 5, 18], and fragment matching [3, 8, 16], but they a have proved to be effective ony for sma probems, as the probem size or in other words, the size of the search space for the probem, grows as a factoria function ( a!)( b!) of a and b, the numbers of the fragments in a and b [4]. In this paper, we formuate the doube digestion probem as a mixed-integer inear program by foowing Waterman 1995 [17] in a sighty different form. We

3 Soving Large Doube Digestion Probems for DNA Restriction Mapping 269 show that with this formuation and by using some state-of-the-art integer programming techniques, we can actuay sove doube digestion probems for fairy arge sizes. In particuar, we can sove a set of randomy generated probems of sizes up to (242!)(250!), which are many magnitude increases from previousy reported (16!)(16!) testing sizes [4]. More specificay, the doube digestion probem can be soved in two steps. First, based ony on their engths, find the fragments in c that can fit in each of the fragments in a and b (ike soving some one-dimensiona puzzes). This is caed the fragment matching. Second, based on the obtained matching reationships of a and b with c, find the orders of the fragments in a and b. The first step is the computationay most difficut part of the probem, because the second step can be accompished easiy in order of a + b time (Chapter 2 in [17]). We therefore focus on fragment matching and formuate the probem as a mixed-integer inear program that minimizes the matching errors in either 1 or norm. We then use the CPLEX mixed-integer inear programming software [7] to sove the probem. In this way, we have been abe to sove a set of randomy generated test probems with the numbers of the generated fragments in a and b ranging from 18 to 242 and 20 to 250, respectivey. Since the tota number of possibe orderings for c is proportiona to ( a!)( b!) [4], assuming most eements in a (and b) are different, the probem sizes or the search spaces for our test probems are proportiona to (18!)(20!) to (242!)(250!). We describe the mixed-integer inear programming formuation in greater detai in Section 2 and introduce the CPLEX mixed-integer inear programming software in Section 3. Computationa resuts are presented and discussed in Section 4. Comments and remarks are given in Section 5. 2 MIP Formuation Let A and B be two restriction enzymes. Let a = {a 1,a 2,...,a n } Z n be the engths of the fragments obtained after appying enzyme A to a given DNA sequence and b = {b 1,b 2,...,b m } Z m the engths of the fragments obtained after appying enzyme B. Let c = {c 1,c 2,...,c } Z be the engths of the fragments obtained after appying enzyme A foowed by enzyme B, or in the other way around, appying enzyme B foowed by enzyme A. In order to find correct orders for a and b, we need to find for every eement in c the eement in a it comes from, and the same thing in b. We ca this probem the fragment matching probem. It is an important part of the doube digestion probem. Once this probem is soved, a soution to the doube digestion probem can be constructed easiy using for exampe a fragment ordering agorithm as described in Chapter 2 of Waterman 1995 [17]. Let X be a n matrix and x i, j the i, j-eement of X, and { 1, sequence o f c x i, j = j sequence o f a i, 0, otherwise.

4 270 The First Internationa Symposium on Optimization and Systems Bioogy Let Y be a m matrix and y i, j the i, j-eement of Y, and { 1, sequence o f c y i, j = j sequence o f b i, 0, otherwise. Since the eements c j in c that come from the ith fragment of a add to a i and that from the ith fragment of b add to b i, a i = j=1 x i, j c j, i = 1,...,n, and simiary, b i = j=1 y i, j c j, i = 1,...,m. In matrix form the equations are equivaent to a = Xc, b = Y c. Note that each eement in c may come from ony one eement in a or b. Therefore, n i=1 x i, j = 1, m i=1 y i, j = 1, j = 1,...,, j = 1,...,. Ceary the fragment matching probem for a tripe sets a, b, and c is basicay to find X and Y so that the above equations hod. As stated in Waterman 1995 [17], the probem can be formuated as a inear integer program, min α,β,x,y sub ject to α + β α β n i=1 m i=1 j=1 j=1 x i, j = 1, y i, j = 1, x i, j c j a i α, y i, j c j b i β, x i, j, y i, j {0,1}. j = 1,..., j = 1,..., i = 1,...,n i = 1,...,m Here, we recognize that the program is separabe and equivaent to two separate sub-

5 Soving Large Doube Digestion Probems for DNA Restriction Mapping 271 programs, min α,x sub ject to α α n i=1 j=1 x i, j = 1, x i, j {0,1}. x i, j c j a i α, j = 1,..., i = 1,...,n and min β,y sub ject to β β m i=1 j=1 y i, j = 1, y i, j {0,1}. y i, j c j b i β, j = 1,..., i = 1,...,m The two sub-programs have the same mathematica structure. It suffices to consider ony one of them. We put it in the foowing genera form, (P ) min t,x t sub ject to t n i j x i, j c j a i t, x i, j = 1, x i, j {0,1}. j = 1,..., i = 1,...,n This program is to find an appropriate assignment matrix X such that the errors for the equations a i = j x i, j c j, i = 1,...,n are minimized in norm. We therefore name the probem P. Of course, the errors

6 272 The First Internationa Symposium on Optimization and Systems Bioogy can aso be minimized in 1 norm, and the program then becomes (P 1 ) min t,x n sub ject to i t i t i n i j x i, j = 1, x i, j {0,1}. x i, j c j a i t i, j = 1,..., i = 1,...,n Here we name the probem P 1. Minimizing 1 or norm may have different effects on the soution to the probem. If the errors are not equa to zero at the soution, minimizing 1 norm reduces tota errors even if there are a few big errors, whie minimizing norm keeps the biggest error as sma as possibe. If an exact soution is found where the errors are a equa to zero, there are no differences in the soution quaities with 1 or norms. However, the soutions may sti be different if there are mutipe soutions, and the agorithm may perform quite differenty as we. In the foowing sections, we show how the probems P 1 and P can be soved by using the CPLEX mixed-integer inear programming software. 3 CPLEX MIP Sover CPLEX is a widey-used software package for soving integer, inear and quadratic programming probems. It was originay deveoped by Bixby [2] and ater commerciaized. A inear programming probem usuay is referred to as an optimization probem with a inear objective function and inear constraints. The foowing is a typica inear program: min x1,x 2,...,x n c 1 x 1 + c 2 x 2 + c n x n sub ject to a 11 x 1 + a 12 x a 1n x n b 1 a 21 x 1 + a 22 x a 2n x n b 2 a m1 x 1 + a m2 x a mn x n b m 1 x 1 u 1 2 x 2 u 2... n x n u n where x 1,x 2,...,x n are continuous variabes in genera. The probem becomes a mixed-integer inear program if some variabes are required to be integers. CPLEX

7 Soving Large Doube Digestion Probems for DNA Restriction Mapping 273 has sovers for inear programs and in particuar, a mixed-integer sover for mixedinteger inear programs (MIP). Linear programming probems with continuous variabes (LP) can be soved in poynomia time, whie mixed-integer inear programs can be hard, incuding many NP-compete probems. The CPLEX MIP sover is buit upon its LP sovers and uses a branch-and-bound strategy, with the aid of cutting pane techniques, to find integer soutions. Two types of cuts, based on maxima ciques and minima covers, are generated when necessary to cut off non-integra soutions. In the branch-and-bound agorithm for soving a MIP probem, a series of LP sub-probems is soved. A tree of subprobems is generated with each subprobem being a node of the tree. The root node is the LP reaxation of the origina MIP. If the soution to the reaxation has integer variabes taking fractiona vaues, then one of such variabes is chosen for branching, and two new subprobems are generated, each with more restrictive bounds for the branching variabe. For exampe, for binary variabes, one node wi have the variabe fixed at zero, and the other at one. The subprobem can resut in an a-integer soution, an infeasibe probem, or another fractiona soution. If the soution is fractiona, the process is repeated. A node is cut off when the objective function vaue of the subprobem associated with the node becomes higher than a known upper bound on the optima vaue of the origina MIP. Note that every a-integer soution gives such an upper bound. Aso note that the node is cut off because further branching from the node wi aways generate subprobems with the same or even higher objective function vaues, and therefore, the whoe branch can be removed. For more detaied descriptions on the branch-andbound agorithm, the readers are referred to [9]. The branch-and-bound agorithm searches for an integer soution for the MIP probem in the subprobem tree. In the worst case, it may need to examine the entire tree, which can be computationay prohibitive since there are exponentiay many nodes in the tree with respect to the integer variabes. In practice, the performance of the agorithm depends on specific search strategies such as how to seect nodes for branching, how to obtain tighter bounds, etc. The CPLEX MIP sover provides severa ways for users to choose among best possibe search strategies. Readers are referred to the CPLEX Users Manua [7] for more detaied descriptions. For mixed-integer programming, in addition to branch-and-bound, cutting pane is another important soution technique. It is often used in combining with branchand-bound. A cutting pane, or a cut, is an inequaity added to a probem that restricts or cuts off non-integra soutions. Adding cuts usuay reduces the number of branches that are needed to sove a mixed-integer program. The CPLEX MIP sover generates two types of cuts derived from maxima ciques and minima covers: Cique inequaities are generated by ooking at the reationships between binary variabes before optimization starts. The reationships are given in the a-binary inequaities of the origina probem. A graph representing the reationships is constructed, and maxima ciques are found. At most one variabe from each cique can be positive in a feasibe soution. Inequaities that describe these restrictions are gen-

8 274 The First Internationa Symposium on Optimization and Systems Bioogy erated. If the soution to a subprobem vioates one of these cique inequaities, the inequaity is added to the probem. Cover inequaities are derived by ooking at each a-binary inequaity with nonunit coefficients. For a constraint with a-negative coefficients, a minima cover is a subset of the variabes in the inequaity such that if a variabes were set to 1, the inequaity woud be vioated, but if any variabe were excuded, the inequaity woud be satisfied. If a cover inequaity is vioated by the soution to the current subprobem, it is added to the probem. For more detaied descriptions on cique and cover cuts for inear integer programming, the readers are referred to [6]. 4 Computationa Resuts We used the CPLEX MIP sover (CPLEX 6.5.2) to obtain the soutions for probems P 1 and P. The program was run on a muti-processor SGI Origin 2000 computer with sixteen 300MHZ R12000 processors and 10GB main memory. However, ony one processor was used at a time. In order to see how doube digestion probems can be soved by our approach, we generated a set of random test probems by foowing a procedure suggested by Godstein and Waterman in [4]. We assumed a set of DNA sequences of 100, 200, 300, 400, and 500 units. The unit here may correspond to one or more than one base pairs. It does not matter what the actua unit is since the difficuty of the probem is correated to the numbers and engths of the fragments, which is independent of the size of the unit. For each sequence, we simuate the first enzyme restriction by cutting the sequence at every unit with a probabiity p 1, and the second enzyme restriction with a probabiity p 2. After combining the two, we obtain a tripe set of fragments a, b, and c. In order to cover certain amount of possibiities, we generated different sets of probems with p 1 = p 2 = 0.2, p 1 = p 2 = 0.3, p 1 = p 2 = 0.4, and p 1 = p 2 = 0.5. Obviousy, more fragments are generated with increasing probabiities p 1 and p 2, and the difficuties of the probems may vary. We ist the sizes of the fragment sets for a generated probems in Tabe 1. Note that since the orders of the fragments in a and b determine the order of the fragments in c and there are tota ( a!)( b!) possibe orderings for a and b, the probem size, or the search space, for the doube digestion probem is proportiona to ( a!)( b!). Since the sizes of the generated fragment sets range from a = 18 and b = 20 to a = 242 and b = 250, the search spaces for the generated probems can be as arge as (242!)(250!), which are many magnitude increases from previousy reported (16!)(16!) testing sizes using simuated anneaing [4]. Note aso that if the fragments in c is to be matched to those in a, the corresponding integer program has n binary variabes, where n is the number of fragments in a and the number of fragments in c. In our test probems, the number of fragments in a or b ranges from 18 to 250, whie that in c from 35 to 376. This impies that the corresponding integer programming probems can have up to 94,000 binary variabes. An integer program with this many binary variabes is often considered too arge to be sovabe even on powerfu computers. However, as we can see in Tabe 2 to 5, the integer programs for our test probems have been soved very we by using the CPLEX MIP sover.

9 Soving Large Doube Digestion Probems for DNA Restriction Mapping 275 Tabe 1: Randomy Generated Probems # units p 1 = p 2 = 0.2 p 1 = p 2 = 0.3 p 1 = p 2 = 0.4 p 1 = p 2 = 0.5 a = 18 a = 27 a = 36 a = : b = 22 b = 33 b = 42 b = 55 c = 35 c = 49 c = 62 c = 75 a = 33 a = 45 a = 66 a = : b = 39 b = 59 b = 77 b = 102 c = 61 c = 86 c = 115 c = 146 a = 55 a = 79 a = 109 a = : b = 58 b = 89 b = 114 b = 149 c = 104 c = 148 c = 181 c = 227 a = 71 a = 103 a = 142 a = : b = 84 b = 124 b = 157 b = 197 c = 137 c = 191 c = 241 c = 298 a = 90 a = 135 a = 184 a = : b = 115 b = 165 b = 211 b = 250 c = 189 c = 256 c = 319 c = 376 Tabes 2 to 5 contain the resuts obtained from using the CPLEX MIP sover on the test probems we have generated in Tabe 1. Note that for every tripe sets a, b, and c, there are two integer programs to sove, one for matching c to a and another for c to b. We denote the two programs by c a and c b. For each of these probems, we ist in the tabes the number of branch-and-bound nodes used in the CPLEX MIP sover and the time to obtain a soution to the probem. Tabe 2 contains the resuts for the probems soved as P 1 type programs. Tabe 3 contains the resuts for the probems soved as P type programs. These two sets of resuts were obtained with the given fragment sets in the origina orders. However, we have aso tested the probems with the given fragment sets sorted in an ascending order. The resuts are isted in Tabes 4 and 5. From Tabes 2 to 5 we can see that the CPLEX MIP sover soved the probems very we athough many of their corresponding integer programs have arge numbers of integer variabes. The probems were soved within reasonabe amount of time except for one that took about 10 hours. Comparing the resuts for P 1 and P types probems, the timings are comparabe. The choice between soving P 1 or P shoud depend on practica needs. On the other hand, the resuts for probems with or without sorting the fragment data seem quite different. Most of the probems were soved faster when the fragment sets were sorted except for a coupe of cases. In genera, the more nodes are there in the branch-and-bound procedure, the onger time the program wi run. However, as we can see in Tabes 2 to 5, the number of branch-and-bound nodes does not aways correate with the running time. Some probems required fewer nodes but onger time. The reason is that the cost of soving inear programs corresponding to different nodes can vary greaty.

10 276 The First Internationa Symposium on Optimization and Systems Bioogy Tabe 2: Test Resuts for P 1 Type Probems # units program p 1 = p 2 = 0.2 p 1 = p 2 = 0.3 p 1 = p 2 = 0.4 p 1 = p 2 = c a 45/1s 23/1s 21/2s 21/2s c b 88/1s 33/1s 38/2s 20/3s 200 c a 430/8s 206/11s 73/13s 145/28s c b 718/12s 762/27s 191/31s 76/24s 300 c a 529/32s 400/1m10s 463/2m5s 193/2m35s c b 4156/2m4s 602/1m56s 2369/7m0s 65/1m9s 400 c a 2390/3m13s 1097/6m23s 541/6m49s 160/5m7s c b 2669/3m36s 1228/6m24s 845/9m49s 324/9m19s 500 c a 1235/3m33s 7648/46m36s 662/15m47s 144/7m43s c b 6616/17m14s 18862/1h30m42s 14661/1h43m13s 1238/38m5s Tabe 3: Test Resuts for P Type Probems # units program p 1 = p 2 = 0.2 p 1 = p 2 = 0.3 p 1 = p 2 = 0.4 p 1 = p 2 = c a 11/1s 43/1s 28/2s 21/2s c b 97/2s 50/1s 37/2s 21/3s 200 c a 226/4s 225/9s 182/37s 87/35s c b 4070/38s 1086/39s 148/44s 81/33s 300 c a 545/25s 177/1m20s 138/1m16s 97/1m50s c b 198/18s 237/2m4s 391/4m17s 78/1m16s 400 c a 2293/2m27s 269/2m42s 268/5m1s 280/18m35s c b 3751/4m13s 1727/24m6s 1071/19m36s 324/11m34s 500 c a 208/53s 334/10m13s 401/40m31s 144/10m46s c b 2193/12m10s 1520/1h24m42s 414/17m11s 821/2h44m25s 5 Concuding Remarks In this paper, we have considered a mixed-integer inear programming formuation for the doube digestion probem and showed that with this formuation, the probem can be soved efficienty by using state-of-the-art mixed-integer programming techniques. In particuar, we used the CPLEX mixed-integer inear programming software and obtained the exact soutions for a set of randomy-generated test probems. We have focused on the probem of matching the doube-digested fragments to the singe-digested ones. The probem is formuated as a mixed-integer inear program minimizing the matching errors in either 1 or norm. The program is separated into two sub-programs. We have used the CPLEX MIP sover to sove both 1 and types of probems. The test probems were generated randomy to cover possibe sizes of probems, engths of fragments, and distributions of random fragments. We have tested probems with hundreds of fragments and hence tens of thousands of binary variabes in the corresponding integer programs. Many of the

11 Soving Large Doube Digestion Probems for DNA Restriction Mapping 277 Tabe 4: Test Resuts for P 1 Type Probems with Sorted Fragments # units program p 1 = p 2 = 0.2 p 1 = p 2 = 0.3 p 1 = p 2 = 0.4 p 1 = p 2 = c a 232/2s 52/1s 26/2s 33/3s c b 1198/5s 33/1s 41/3s 19/2s 200 c a 446/7s 380/15s 196/30s 42/20s c b 198/5s 461/21s 162/25s 21/18s 300 c a 668/37s 169/50s 83/58s 57/1m12s c b 1251/1m5s 1115/2m58s 104/1m5s 91/1m c a /9h59m31s 1345/6m28s 415/5m13s 94/4m14s c b 4000/4m57s 2341/15m48s 437/5m50s 199/7m0s 500 c a 1120/4m8s 1694/1m21s 159/5m24s 119/7m19s c b 992/4m10s 846/13m32s 342/10m18s 289/15m9s Tabe 5: Test Resuts for P Type Probems with Sorted Fragments # units program p 1 = p 2 = 0.2 p 1 = p 2 = 0.3 p 1 = p 2 = 0.4 p 1 = p 2 = c a 25/1s 51/1s 21/2s 21/2s c b 102/1s 77/2s 41/2s 27/3s 200 c a 76/2s 191/8s 87/18s 54/19s c b 152/4s 584/29s 52/13s 50/21s 300 c a 212/12s 154/33s 83/51s 70/1m14s c b 453/27s 254/1m27s 107/51s 106/1m27s 400 c a 308/40s 572/5m15s 193/5m39s 69/4m8s c b 741/1m36s 198/2m47s 175/6m13s 133/6m42s 500 c a 222/1m16s 321/11m56s 197/13m33s 124/13m5s c b 1342/6m32s 261/15m42s 182/9m4s 131/17m8s probems were soved in ony a few minutes on a singe SGI R12000 processor. In terms of the sizes or the search spaces of the probems, they range from (18!)(20!) to (242!)(250!) possibe orderings, which are many magnitudes arger than previousy reported (16!)(16!) testing sizes. The reason for such an advancement in soving an NP-hard probem ike the doube digestion probem is not just because of the improvement on the computer speed over the past ten or twenty years, but is rather because of the deveopment of many new integer programming techniques such as the cutting pane methods and the improved inear programming sovers as impemented in modern software ike CPLEX that can expoit the probem structures more effectivey. Simiar resuts were obtained for other types of NP-hard probems as we. For instance, in 1987, Padberg and Rinadi soved a traveing saesman probem with 532 American cities [10], whie in 1998, by introducing various integer programming techniques, Appegate et a. were abe to sove a probem of same type with American cities [1], which was considered as a major advance in modern computer science other than just a

12 278 The First Internationa Symposium on Optimization and Systems Bioogy simpe ceebration on the computer hardware improvement. Severa issues yet to be resoved in future work. One of them is the handing of the errors in the experimenta data. If the measured engths for the fragments are sti integers, the mixed-integer programs wi essentiay be at the same eve of difficuty as those we have tested. If the engths are rea numbers, however, the soutions to the mixed-integer programs may take onger time to find. We pan to conduct more computationa experiments to study this issue. Another issue is the mutipe soutions to the mixed-integer programs and hence the doube digestion probem. As discussed in the iterature, this probem cannot be soved uness additiona knowedge of the sequence is given. In any case, the soutions to the two sub-programs for fragment matching show ony how the singe and doube-digested fragments match but not how they shoud be ordered. The fragments need to be examined together and put into sensibe orders. Fortunatey, this can be done in an efficient and systematic fashion [17]. When using the CPLEX MIP sover for the test probems, we have so far not taken advantages that CPLEX provides for improving the performance of the sover, such as using user-suppied initia soutions, heuristic upper bounds, and better node seection orders, etc. We have ony used defaut settings for our test. It is highy probabe that by taking these advantages, one shoud be abe to expoit the probem structure more effectivey and achieve even better computationa resuts than what we have just observed. References [1] D. A. Appegate, R. E. Bixby, V. Chvta, and W. Cook, On the Soution of Traveing Saesman Probems Documenta Mathematica Journa der Deutschen Mathematiker-Vereinigung, Internationa Congress of Mathematicians, 1998, [2] R. E. Bixby, Impementing the Simpex Method: The Initia Basis, Technica Report TR90-32, Department of Computationa and Appied Mathematics, Rice University, Houston, TX, [3] W. M. Fitch, T. F. Smith, and W. W. Raph, Mapping the Order of DNA Restriction Fragments, Gene, 22, 1983, pp [4] L. Godstein and M. S. Waterman, Mapping DNA by Stochastic Reaxation, Adv. App. Math., 8, 1987, pp [5] A. V. Grigorjev and A. A. Mironov, Mapping DNA by Stochastic Reaxation: A New Approach to Fragment Sizes, Appic. Biosci.,, pp [6] M. Grötsche, L. Lovász, and A. Schrijver, Geometric Agorithm and Combinatoria Optimization, Springer, [7] ILOG Inc., Using the CPLEX Caabe Library, ILOG Inc., [8] M. Krawczak, Agorithms for the Restriction-Site Mapping of DNA Moecues, Proc. Nat. Acad. Sci. USA, 85, 1988, pp

13 Soving Large Doube Digestion Probems for DNA Restriction Mapping 279 [9] G. L. Nemhauser and L. A. Wosey, Integer and Combinatoria Optimization, John Wiey & Sons, Inc., [10] M. Padberg and G Rinadi, Optimization of a 532-City Symmetric Traveing Saesman Probem by Branch and Cut, Operations Research Letters, 6, 1987, 1-7. [11] C. Pandurangan and H. Ramesh, The Restriction Mapping Probem Revisited, Department of Computer Science, Brown University, Providence, RI, [12] W. R. Pearson, Automatic Construction of Restriction Site Maps, Nuceic Acids Res., 10, 1984, [13] W. Schmitt and M. S. Waterman, Mutipe Soutions of DNA Restriction Mapping Probems, Adv. App. Math., 12, 1991, pp [14] D. P. Snustad and M. J. Simmons, Principes of Genetics, John Wiey & Sons, Inc., [15] M. Stefik, Inferring DNA Structures from Segmentation Data, Artif. Inte., 11, 1978, pp [16] P. Tuffery, P. Dessen, C. Mugnier, and S. Hazout, Restriction Map Construction Using Compete Sentences Compatibiity Agorithm, Comput. Appic. Biosci., 4, pp [17] M. S. Waterman, Introduction to Computationa Bioogy: Sequences, Maps, and Genomes, Chapman Ha, [18] L. W. Wright, J. B. Lichter, J. Reinitz, M. A. Shifman, K. K. Kidd, and P. L. Mier, Computer-Assisted Restriction Mapping: An Integrated Approach to Handing Experimenta Uncertainty, CABIOS, Vo. 10, No. 4, 1994, pp