SEMI-DEFINITE RELAXATION ALGORITHM FOR SINGLE MACHINE SCHEDULING WITH CONTROLLABLE PROCESSING TIMES

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【出版日期】2005-01-25

【摘要】<正>The authors present a semi-definite relaxation algorithm for the scheduling problem with controllable times on a single machine. Their approach shows how to relate this problem with the maximum vertex-cover problem with kernel constraints (MKVC). The established relationship enables to transfer the approximate solutions of MKVC into the approximate solutions for the scheduling problem. Then, they show how to obtain an integer approximate solution for MKVC based on the semi-definite relaxation and randomized rounding technique.

【刊名】Chinese Annals of Mathematics

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6 1 .Introduetion Scheduling Problems with eontrollable times on a single maehine ean be stated as follows. Given a set of 0 jobsJ={1,2,…,n}.Eaeh job夕任J has a weight码任Z+and a normal proeessing time马〔Z+.The proeessing times of jobs are eontrollable in the following manner.The normal proeessing time of job j ean be redueed by up tou,units(。,〔z+ and。,三Pj)if its proeessing 15 speeded up·Each unit reduetion of proeessing time of job J requires a eost of勺due to the faet that some additional resourees are neeessary for the speedup·In a given sehedule,艺J denotes the reduetion of proeessing time and乃=Pj一云, the actual proeessing time of job了.Let几denote the eompletiont而e ofjob J.Our goal 15 to find a schedule of the jobs and a proeessing time reduetiont,(云j三。J)for each job J sueh that the total eost ineluding the total weight eompletion time of jobs and the total eost of speedup,i.e.,艺。,q+E乌t7,15 minimum,whieh ean be denoted asl/印t/兄。,几+勺t,, J任Jj任J where the notion“ePt,,stands for“eontrollable Proeessing times,,. The probleml/叩t/艺哟马+勺t,15 Np一hard(see【1」).viekson【21 shows that for the probleml/印t/E。,马+勺亡,,there 15 an optimal sehedule that satisfies the following all- or一one ProPerty:the Proeessing time of eaeh jobj〔J 15 either fully redueed or not redueed at all,i.e.,t,〔{0,。,},and its aetual proeessing time几任{Pj,马一。,}.Huang and Zhang [3]give a polynomial time algorithm forl/cpt/E、J几+勺艺j with。,三。and勺三e· 】5魂 CHEN,F.&ZHANG,L.及 Sinee the seminar work of Goemans and Williamson!4}for MAX一CUT,semidefinit(、 I)rograr,lrning(SDP)relaxations have reeently proved useful in obtaining irnproved appr(〕xi- 。lation algorithms for several eombinatorial problems,ineluding MAXZSAT(see【4」),MAX k一CUT and MAX BISECTION(see{5}),non一eonvex quadratie(see【6{),quadratic prograT‘卜 。ling(see【7」)and 50 on.A review 15 given in{8」. Semi一definite programming(SDP)ean be stated as follows: 】1lln 5 .t. C·X. A、·X二认, X卜0. where,C,A,(乞=1, linear indePendent, are n x n real symmetrie matriees,andA,(乞=l,2, A·刀=冗A‘,B、,=Tr(ATB)denotes the inner 二,7n)ar。 Prodllet of 乞,J matrixA,B:X匕0 rePresents X 15 a semi一definite matrix.Semi一definite Programnling can be solved by ellipsoid algorithm(see!9」)and interior一point polynomial一time methods(se。 【10!). It 15 known that Skutella 15 the first researeher who has sueeessfully used the eonvexa一zd semidefinite programming techniqlles to design the apProximation algorithms for seheduling, In[11],Skutella presentsa号一即proximation forR//艺二,Cj and 1.2752一approximation for RZ//艺二,cj·Inspired by the paper,Zhang,Tang and Chen{12」give a3/2 approxirna- tion algorithrn for a general Problem based on eonvex quadratie Programming relaxation. l/甲t/艺二,q+cjt,,as a speeial ease,ean also be solved by the methods.Reeently,X,l !13}also used the same teehnique oeeurring in!11}to get an improvement algorithnit() 1 .27一approximation forl/叩‘/Etv,Cj+马艺,· Although it 15 diffieult for us to solve minimum eombinatorial oPtimization Problems di- r‘,Ctly by th‘之te‘:hnique Of semi一definite programming relaxation,Skutella!11}gets a 1.122- 叩proxinlation for几//艺二,马by translating the problemPZ//又竹q into a Max eut Problem.In this PaPer,we use the similar idea to develoP a semi一definite relaxation algo- rithn、 forl/叩t/艺二,Cj+勺t,· This paper 15 organized as foll0Ws·In Section 2 we show that the probleml/叩艺/艺‘,q +勺t:15 equivalent to a maximum vertex一eover problern with kernel eonstraints(MKVC)·In Seetion3,we point out that the algorithm for(MKVC)ean be emplo”d to solve the problem 1/印忿/艺二,Cj+e,t了and has a better performanee under some reason曲le assumptions. 9 2 .Equivalenee of Two Problems In the seetion,we will show that the probleml/叩t/艺、,q+e了t,15 equivalent to the maximum vertex一eover Problem with kernel eonstraints whieh ean be stated as follows. I;etG=(认E)be an undireeted gr即h whereV={1,2,…,n}15 the set of vertexes,E i、the set of edges.Git.en kernel setsKI,凡,二,万m〔V。eaeh kernel set has justt确 elements.There are weights二,,=二j:任Z+on edge(乞,j)任E and eaeh vertex has a weight ,刀,〔Z+The problem 15 to determine a subsets〔V sueh that the sum of total weight of the edges eovered by 5 and the total weight of the vertex in 5 15 maximized with kernel eonstraints,i·e.,just one element ofK‘will be eovered by S.W七will denote this problenl by MKVC.Without 1055 of generality,we assume that no two kernel sets have eomrnorl element,otherwise there 15 not a feasible solution for MKVC.Let if该任S, if艺杯5. 1工n曰 f夕1、 一一 X SEMI一DEFINITE RELAXA〕,ION ALGORITHM 155 Then the Problem MKVC ean be formulated as follows: MKVC: 黔 艺 二*,+艺、‘ orj任S,乞(J 云任S s.t.x。十xf二1, e,f任凡, xj〔{0,1}, i=1,…,m, J=1,…,n. Based on the property of all一or一none,we ean see that thel/叩t/艺二j马+乌t,15 equiv- alent to the following Problem with non一eontrollable Proeessing times.Given a set ofZ几 jobsH={1,2,…,Zn}.Eaeh jobj任H has a weight巧and a non一controllable process- ing time Tj if it 15 proeessed on maehine,where vj=二,,乃=勿一科,ifj三n and ”,=二,一。,兀=Pj一。ifj>n·If job了〔H 15 proeessed on the single maehine,it ineursa processing eost内wi

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