Solving the large scale knapsack feasibility problem using a distributed computation approach to integer programming

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1 RESEARCH Open Access Solvng the large scale napsac feasblty problem usng a dstrbuted computaton approach to nteger programmng Zhengtan Wu 1,2*, Fuyuan Hu 1 and Baochuan Fu 1 *Correspondence: wzht8@mal.usts.edu.cn 1 School of Electronc and Informaton Engneerng, Suzhou Unversty of Scence and Technology, Keru Road, Suzhou, Chna Full lst of author nformaton s avalable at the end of the artcle Abstract The napsac feasblty problems have been ntensvely studed both because of ther mmedate applcatons n ndustry and fnancal management, but more pronounced for theoretcal reasons, as napsac problems freuently occur by relaxaton of varous nteger programmng problems. In ths wor, the large-scale napsac feasblty problem s dvded nto two subproblems. The frst subproblem s transformng of the napsac feasblty problem nto a polytope udgement problem whch s based on lattce bass reducton. In the next subproblem, a dstrbuted mplementaton of Dang and Ye s fxed-pont teratve algorthm s ntroduced to solve the polytope udgement problem generated n the former subproblem. Compared wth the branch and bound method, numercal results show that ths dstrbuted fxed-pont method s effectve. Keywords: The napsac feasblty problems, Integer programmng, Dstrbuted mplementaton, Fxed-pont teratve algorthm Introducton The napsac problems, whch have been ntensvely studed snce the poneerng wor of Dantzg (1957), have played an mportant role n ndustry, fnancal management, and so on. What s more, varous nteger programmng problems can be relaxed to the napsac problems. Therefore, the computng of the napsac problems has been becomng the mar level of the computaton n nteger problems. To tacle ths problem, many new contrbutons have been made n the followng lterature. A heurstc based upon genetc algorthms has been developed for multdmensonal napsac problem n paper (Chu and Beasley 1998). Based on the harmony search method, a new bnary-coded verson of harmony search (Kong et al. 2015) s presented to solve large-scale multdmensonal napsac problem. In ths proposed algorthm, attenton s pad to the probablty dstrbuton rather than the exact value of each decson varable, and the concept of mean harmony s developed n the memory consderaton. Inspred by regon partton of tems, an effectve hybrd algorthm based on greedy degree and expectaton effcency s constructed n the paper (Lv et al. 2016). Combnng advanced features both from the path relnng method and the responsve threshold search algorthm, the frst evolutonary path relnng approach s ntroduced n paper (Chen et al. 2016) for solvng the uadratc multple napsac problem approxmately. The Author(s) Ths artcle s dstrbuted under the terms of the Creatve Commons Attrbuton 4.0 Internatonal Lcense ( whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded you gve approprate credt to the orgnal author(s) and the source, provde a ln to the Creatve Commons lcense, and ndcate f changes were made.

2 Page 2 of 7 In ths paper, the dstrbuted Dang and Ye s fxed-pont teratve method (Dang and Ye 2015) s mplemented to solve large-scale napsac feasblty problem. Ths fxedponted algorthm has been extended to arlne dsrupton problem and other problems. The dea of solvng multple fleet arlne dsrupton problems usng a dstrbuted computaton approach to nteger programmng has been developed n the prevous wor (Wu et al. 2017a). The paper (Wu et al. 2017b) uses the Dang and Ye s teratve fxed-pont method for nteger programmng to generate feasble flght routes whch are used to construct an arcraft reassgnment n response to the groundng of one arcraft. Some computng of Nash eulbra methods are derved from ths Dang and Ye s algorthm, such as the method from the papers (Wu et al. 2014, 2015). Some other engneerng problems (Zhang et al. 2015, 2016) can also use ths fxed -pont method to solve. The Dang and Ye s fxed-ponted method can be explaned as follows. Let P ={x R n Ax + Gw b, for some w R p }, where A R m n s an m n nteger matrx wth n 2, G R m p an m p matrx, and b a vector of R m. Let and x max = (x1 max, x2 max,..., xn max ) T wth x max = max x P x, = 1, 2,..., n x mn = (x mn 1, x mn 2,..., x mn n ) T wth x mn = mn x P x, = 1, 2,..., n. Let D(P) = { x Z n x l x x u}, where x l = x mn and x u = x max. For z R n and N 0, let P(z, ) ={x P x = z,1, and x z, + 1 n}. Gven an nteger pont y D(P) wth y 1 > x l, Dang and Ye (2015) developed a fxedpont teratve algorthm whch s presented n Fg. 1. In Dang and Ye s algorthm for nteger problem, a defnton of an ncreasng mappng, whch s from a fnte lattce nto tself, s developed. Any nteger pont, whch s outsde the P, s mapped nto the frst pont n P that s smaller than them n the lexcographcal y 0 y, n 1, 0. Output y. y P y l x. solve ln ear program: max n 1 x s.t. x P ( y, ), 1, 2,, n. l N, y x. 1. P ( y, ) l Let y x, 1. 1, y x ( y ). Let y ( y, y, y ) wth y T 1 2 n y f 1, 1 x ( y ) f 1 n, 1, 2,, n, and 1. denote the optmum of LP by x ( y ), 1, 2,, n. Let y ( y, y, y ) T 1 2 n y f 1 1, 1 wth y y 1 f, u x f 1 n, 1, 2,, n, 1, and 1. Fg. 1 Flow dagram of the teratve method

3 Page 3 of 7 order of x l. All the nteger ponts, whch are nsde the polytope, are the fxed ponts through ths ncreasng mappng. Gven an ntal nteger pont, the method ether proves there s no such pont by a lmted number of teratons or generates an nteger pont n the polytope. For more proofs and detals about ths teratve algorthm, one can consult the paper (Dang and Ye 2015). An appeal feature of ths fxed-pont teratve method, whch wll be used n ths paper, s that t can be easly mplemented n a dstrbuted way. The rest of the paper s organzed as follows. Some detals of transformaton of the napsac feasblty problem nto a polytope udgement problem, whch s based on a LLL bass reducton, wll be presented n Transformaton of the napsac feasblty problem nto apolytope udgement problem secton. The computaton detals and numercal results wll be gven n Dstrbuted computaton and numercal results secton. Some conclusons and future wor wll be dscussed n the last Conclusons and future wor secton. Transformaton of the napsac feasblty problem nto a polytope udgement problem In order to analyze the napsac feasblty problem easly, ths problem s defned as follows (Dantzg 1957). Defnton 1 Fnd a 0 1 nteger soluton of p T x = d, where p = (p 1, p 2,..., p n+1 ) T > 0 and p = p for all = After analyss, one can see ths napsac problem s one specal example of the maret splt problem. To convert ths problem nto an euvalent problem of determnng whether there s an nteger pont n a full-dmensonal polytope gven by P ={x R n Ax b}, we can apply the LLL bass reducton algorthm (Khorramzadeh 2012) whch has been descrbed n paper (Wu et al. 2013). Therefore, ths napsac feasblty problem can be formulated euvalently as follows: Does there exst a vector λ Z n m s.t. x d X 0 λ e (n m) 1 x d. (1) The solvng of the Defnton 1 can be transformed to udge whether there exsts an nteger pont n the polytope (1). The dstrbuted teratve algorthm, developed by Dang and Ye (2015), has a good performance to udge whether the polytope (1) has an nteger pont or not. Dstrbuted computaton and numercal results One master computer combned wth several slave computers s eupped to mplement the dstrbuted computaton. The master computer taes charge of solvng the soluton space of the polytope, dvdng the soluton space to segments, sendng the segments to the slave computers, recevng the computaton result from the slave computers, and exportng the computaton result. Each slave computer receves the segment, udges whether there exts an nteger pont n ts segment usng Dang and Ye s fxed-pont

4 Page 4 of 7 teratve method, and sends ts result to the master computer. The outlne of the dstrbuted computaton process n ths paper s llustrated n Fg. 2 In present study, the dstrbuted computaton system conssts of three computers whch are OptPlex 330 wth two processors. The programs are wrtten n C++ language. Subsegments whch are dstrbuted by the master computer are mutually ndependent. Therefore, subsegments can be solved smultaneously. Luc Laponte (1994) has been adopted to establsh a communcaton networ among the computers. The other settngs and parameters are taen the same as the research (Wu et al. 2013). Message Passng Interface (MPI) s a tool to establsh a stable communcaton networ between the master computer and the slave computers n the dstrbuted computaton. The pseudocode of buldng the dstrbuted networ by MPI s descrbed n Fg. 3. In the followng case study, two methods have been employed to solve the same example, respectvely. One s dstrbuted fxed-pont method, whle the other s the branch and cut method (Padberg and Rnald 1991) whch s a classcal algorthm for nteger programmng. The numercal results are shown n the followng Table 1. For smplfcaton, some symbols are defned as follows. NumLPs: The number of teratons for a certan algorthm BC: The branch and cut method From Table 1, we can see that the dstrbuted algorthm s superor to the branch and cut method regardng the number of teratons. Fg. 2 Flow dagram of the dstrbuted computaton process

5 Page 5 of 7 Fg. 3 Dstrbuted computaton pseudocode Table 1 Two method to solve the napsac feasblty problems Prob. Dmenson n The method BC NumLPs F NumLPs F Feasble 1428 Feasble Feasble 2023 Feasble Feasble 1360 Feasble Feasble 1117 Feasble Feasble 1122 Feasble Infeasble 1315 Infeasble Feasble 978 Feasble Feasble 1486 Feasble Feasble 2017 Feasble Feasble 2587 Feasble Feasble 879 Feasble Feasble 724 Feasble Conclusons and future wor In ths paper, a new method has been used to solve the napsac feasblty problem. Ths method s dvded nto two steps. In the frst step, the napsac feasblty problem s transformed nto a polytope udgement problem based on a LLL bass reducton. In the other step, a dstrbuted fxed-pont method for nteger programmng s mplemented to solve the polytope udgement problem. Compared wth the branch and cut method

6 Page 6 of 7 whch s consdered to be the best algorthm for the problem of ths nd, numercal results show that ths dstrbuted fxed-pont method s promsng. However, the dmenson of nstances s low and the number of slave computers n the numercal experment s only three. The potental ablty of ths method has not been fully expressed. In the next step, these two shortcomngs wll be settled. Wth large number of slave computers, one can be confdent to beleve that the numercal results wll be more better. Addtonally, the dstrbuted method s easy to be extended to solvng other problems. Authors contrbutons The authors dscussed the problem and the solutons were proposed all together. All authors partcpated n draftng and revsng the fnal manuscrpt. All authors read and approved the fnal manuscrpt. Author detals 1 School of Electronc and Informaton Engneerng, Suzhou Unversty of Scence and Technology, Keru Road, Suzhou, Chna. 2 Department of Systems Engneerng and Engneerng Management, Cty Unversty of Hong Kong, Tat Chee Avenue, Hong Kong, Chna. Competng nterests The authors declare that they have no competng nterests. Avalablty of data and materals Data sharng s not applcable to ths artcle as no datasets were generated or analyzed durng the current study. Fundng Ths wor was partly supported by Natonal Nature Scence Foundaton of Chna under Grants , , and ; Research Foundaton of USTS under Grants. XKQ201517; Jangsu Provncal Department of Housng and Urban-Rural Development under Grants. 2017ZD253; Natural scence fund for colleges and unverstes n Jangsu Provnce under Grants.17KJD110008; Fundamental Research Funds for the Central Unverstes under Grant ; and Natural Scence Foundaton of Jangsu Provnce under Grant. BK Publsher s te Sprnger Nature remans neutral wth regard to ursdctonal clams n publshed maps and nsttutonal afflatons. Receved: 28 August 2017 Accepted: 14 December 2017 References Chen Y, Hao JK, Glover F (2016) An evolutonary path relnng approach for the uadratc multple napsac problem. Knowl Based Syst 92:23 34 Chu PC, Beasley JE (1998) A genetc algorthm for the multdmensonal napsac problem. J Heurstcs 4(1):63 86 Dang C, Ye Y (2015) A fxed pont teratve approach to nteger programmng and ts dstrbuted computaton. Fxed Pont Theory Appl 2015(1):1 15 Dantzg GB (1957) Dscrete-varable extremum problems. Oper Res 5(2): Khorramzadeh M (2012) Numercal experments wth the lll-based hermte normal form algorthm for solvng lnear dophantne systems. Int J Contemp Math Sc 7(13): Kong X, Gao L, Ouyang H, L S (2015) Solvng large-scale multdmensonal napsac problems wth a new bnary harmony search algorthm. Comput Oper Res 63:7 22 Lv J, Wang X, Huang M, Cheng H, L F (2016) Solvng 0 1 napsac problem by greedy degree and expectaton effcency. Appl Soft Comput 41: Luc Laponte JM (1994) Message Passng Interface Forum. MPI: A Message Passng Interface Standard. J combn theory Ser A 112:44 81 Padberg M, Rnald G (1991) A branch-and-cut algorthm for the resoluton of large-scale symmetrc travelng salesman problems. SIAM Rev 33(1): Wu Z, Dang C, Zhu C (2013) Solvng the maret splt problem usng a dstrbuted computaton approach. In: 2013 IEEE nternatonal conference on nformaton and automaton (ICIA), IEEE, pp Wu Z, Dang C, Karm HR, Zhu C, Gao Q (2014) A mxed 0-1 lnear programmng approach to the computaton of all pure-strategy nash eulbra of a fnte n-person game n normal form. Math Probl Eng 2014:8. org/ /2014/ Wu Z, Dang C, Hu F, Fu B (2015) A new method to fndng all Nash eulbra. In: Internatonal conference on ntellgent scence and bg data engneerng, pp Wu Z, L B, Dang C (2017a) Solvng multple fleet arlne dsrupton problems usng a dstrbuted-computaton approach to nteger programmng. IEEE Access 5:

7 Page 7 of 7 Wu Z, L B, Dang C, Hu F, Zhu Q, Fu B (2017b) Solvng long haul arlne dsrupton problem caused by groundngs usng a dstrbuted fxed-pont computatonal approach to nteger programmng. Neurocomputng 269: do.org/ /.neucom Zhang Y, Le ZX, Zhang LW, Lew KM, Yu JL (2015) nlocal contnuum model for vbraton of sngle-layered graphene sheets based on the element-free p-rtz method. Eng Anal Bound Elem 56:90 97 Zhang Y, Zhang LW, Lew KM, Yu JL (2016) Free vbraton analyss of blayer graphene sheets subected to n-plane magnetc felds. Compos Struct 144:86 95

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