decompositions, network flows, and a precedence constrained single-machine scheduling problem
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Decompositions, Network Flows, and a PrecedenceConstrained Single-Machine Scheduling ProblemFrançois Margot, Maurice Queyranne, Yaoguang Wang,
To cite this article:François Margot, Maurice Queyranne, Yaoguang Wang, (2003) Decompositions, Network Flows, and a Precedence ConstrainedSingle-Machine Scheduling Problem. Operations Research 51(6):981-992. http://dx.doi.org/10.1287/opre.51.6.981.24912
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DECOMPOSITIONS, NETWORK FLOWS, AND A PRECEDENCECONSTRAINED SINGLE-MACHINE SCHEDULING PROBLEM
FRANÇOIS MARGOTGSIA, Carnegie Mellon University, 5000 Forbes Avenue, Pittsburgh, Pennsylvania 15213, [email protected]
MAURICE QUEYRANNEFaculty of Commerce, University of British Columbia, Vancouver, British Columbia, Canada, [email protected]
YAOGUANG WANGPeopleSoft, Inc., Pleasanton, California 94566, [email protected]
We present an in-depth theoretical, algorithmic, and computational study of a linear programming (LP) relaxation to the precedence con-strained single-machine scheduling problem 1�prec�∑j wjCj to minimize a weighted sum of job completion times. On the theoretical side,we study the structure of tight parallel inequalities in the LP relaxation and show that every permutation schedule that is consistent withSidney’s decomposition has total cost no more than twice the optimum. On the algorithmic side, we provide a parametric extension toSidney’s decomposition and show that a finest decomposition can be obtained by essentially solving a parametric minimum-cut problem.Finally, we report results obtained by an algorithm based on these developments on randomly generated instances with up to 2,000 jobs.
(Network/graphs, flow algorithms: parametric flows and Sidney decompositions. Production/scheduling, approximations: 2-approximationalgorithm. Programming, integer, algorithms, relaxation/subgradient: integer formulation.)
Received August 2000; revisions received July 2002, January 2003; accepted February 2003.Area of review: Optimization.
1. INTRODUCTION
We consider the following single-machine scheduling prob-lem, denoted 1�prec�∑j wjCj in the scheduling literature(see, e.g., the extensive survey by Lawler et al. 1993): A setN of n jobs is to be processed nonpreemptively on a singlemachine, which can process only one job at a time. Associ-ated with each job j are a positive processing time pj and anonnegative weight wj . A feasible job schedule must obeya partial order specified by an acyclic graph G = �N �.The objective is to find a feasible sequence (or schedule)of jobs which minimizes the weighted sum
∑j∈N wjCj of
completion times. (The basic definitions and notations canbe found in §2.)
This scheduling problem is NP-complete (Lawler 1978)and has a long history; see Lawler et al. (1993) for refer-ences. It is a very basic problem in scheduling theory, andit appears as a subproblem in more elaborate settings. Anattractive idea for solving this problem is to use a decom-position technique introduced by Sidney (1975): Identifyjob subsets such that an optimal schedule can be obtainedfrom pasting together optimal schedules for the job subsets.One of the goals of this paper is to give an efficient way tocompute the finest such decomposition. We shall providea detailed exposition of and extensions to Sidney’s resultsin §3.
In this paper we also study structural, algorithmic, andcomputational properties of a linear programming (LP)relaxation that uses as its only decision variables the job
completion times Cj . This type of formulation was firstintroduced by Balas (1985) and studied by several authors.In particular, Queyranne and Wang (1991a) presented anextensive polyhedral study of the LP relaxation determinedmainly by two families of valid inequalities, called paral-lel inequalities and series inequalities. Wolsey (1990) laterextended the formulation with O�n2 sequence-determiningbinary variables and proved that this formulation is at leastas tight as the one given by parallel and series inequali-ties. See the survey by Queyranne and Schulz (1996) foran exposition and references.
The linear programming formulation considered in thispaper, introduced by Queyranne and Wang (1991b), wasused by Schulz (1996) and Hall et al. (1997) to designa polynomial-time 2-approximation algorithm for thisscheduling problem. In §4, we study the structure of tightparallel inequalities in this LP formulation. We show that,if the problem is not amenable to Sidney’s decomposition,then a single, global parallel inequality suffices in the LPformulation. As a consequence, we also get that every per-mutation schedule that is consistent with Sidney’s decom-position has total cost no more than twice the optimum, asurprising extension to Schulz’s 2-approximation result.
In §3, we provide a parametric extension to Sidney’sdecomposition and show that a finest decomposition can beobtained by essentially solving a parametric minimum-cutproblem, which by Gallo et al. (1989) (see also McCormick1998) requires about the time of a single maximum-flow
0030-364X/03/5106-09811526-5463 electronic ISSN 981
Operations Research © 2003 INFORMSVol. 51, No. 6, November–December 2003, pp. 981–992
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calculation. In §5 we show that, after this Sidney decom-position, the LP formulation can be solved as essentiallya single dual minimum cost flow problem. Both networkflow problems are on networks with the jobs as nodes (plusone source and one sink) and the nonredundant precedenceconstraints as arcs (plus one source or sink arc per job).
In §6, we report results obtained by an implementa-tion taking advantage of these theoretical developmentson randomly generated instances with up to 2,000 jobs.By repeatedly applying the extended Sidney decompositionprocedure and using series decompositions, most of theseproblems end up as a collection of subproblems, the largestof which contains approximatively 60% of the jobs of theinitial problem. In short computing time (determined pri-marily by the performance of the network flow algorithms)we obtain feasible solutions and lower bounds with an aver-age optimality gap of about 1%, which never exceeded3.5%.
2. BASIC NOTATIONS AND DEFINITIONS
Throughout this paper, we use the following notation:
w ∈ �N+ for the row vector of the nonnegative job weights,p ∈ �N+ for the row vector of the positive job processing
times,C ∈ �N for a column vector of job completion times,
where w and p are given and C are the decision variables.A subset J of the job set N is proper if J �= N ; it is non-trivial if it is proper and nonempty.
Let G= �N � be an acyclic digraph that represents theprecedence relations among the jobs in N . (See Figure 1.)For a job subset J ⊆ N , let
+�J = �ij ∈ � � i ∈ J j ∈ N\J � and
−�J = �ji ∈ � � i ∈ J j ∈ N\J ��A subset I ⊆ N is initial (in N ) if −�I = ; equiv-
alently, if ij ∈ � and j ∈ I imply i ∈ I . Similarly, a sub-set T ⊆ K is terminal if +�T =; equivalently, if i ∈ Tand ij ∈ � imply j ∈ T . In Figure 1, the set �1345�is initial and the set �67� is terminal. Initial subsets arealso known, among others, as (order) ideals (Davey and
Figure 1. A precedence graph on seven jobs; numbersa�b next to node i correspond to the pairwi�pi.
1;1 41;1
3;1
0;1
1;4
1
2 50;2 6
7
3 1;1
Priestley 1990), selections (Lawler 1976), and closures(Picard 1976), whereas terminal sets are also known as fil-ters (Davey and Priestley 1990). Let � denote the collec-tion of all initial sets (in N ), and � that of all terminal sets.From these definitions, it follows immediately that
N and are each an initial and a terminal set;a subset I ⊆ N is initial if and only if its complement N\I
is terminal;� and � are a sublattices of 2N , that is, each is closed
under set union and intersection; i.e., the union U ∪Vand intersection U ∩V of any initial (resp., terminal) setsU and V are also initial (resp., terminal) sets.
We assume throughout that G= �N � does not contain adirected Hamiltonian path and that wj > 0 for some j ∈ N ,for otherwise the scheduling problem would be trivial.
The problem 1�prec�∑wjCj can now be stated as a dis-junctive linear program as follows:
z∗=min wC
s.t. Cj−Ci�pj ∀ij ∈�Cj−Ci�pj ∨ Ci−Cj�pi ∀i �= j ij �∈�C�p�
(1)
This problem is known to be strongly NP-hard. In thesequel, we denote the problem (1) by P�wpN �. Asw � 0, the optimal schedule is obtained as a permutationof the jobs, processed with no idle time.
For a set J ⊆ N , we use w�J for∑j∈J wj and p�J is
defined similarly. The weight vector w is tailing off if thereexists some terminal set T ∈� such that w�T = 0; in sucha case, the scheduling problem reduces to one defined onthe job set N\T , and we may schedule the jobs in T in anarbitrary feasible sequence after all those in N\T withoutaffecting the objective function value. In Figure 1, w istailing off as the terminal set T = �27� has w�T = 0.
For any nonempty job subset J , define ��J = w�J /p�J . The problem P�wpN � is stiff if ��I < ��Nfor all proper initial subsets I of N ; otherwise, it is Sidneydecomposable. In Figure 1, for the initial set I = �345�we have ��I= 5/3> ��N= 7/11, and thus this instanceis not stiff.
A simple result that will be used several times is thefollowing.
Lemma 2.1. Let ab � 0 and cd > 0. Then �a+b/�c+d < b/d if and only if a/c < b/d. (A similar state-ment holds for > and = operators.)
Proof. By cross multiplication. �
A reader interested in Sidney decompositions may nowcontinue with §3, whereas one more immediately interestedin properties of the LP relaxation of the scheduling problemmay now proceed directly to §4.
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3. SIDNEY DECOMPOSITIONS
In this section, we study extensions, properties, and algo-rithmic aspects of Sidney decompositions. Finding a Sidneydecomposition will allow us to decompose an instanceof the scheduling problem into smaller, more manageableinstances.
3.1. Parametric Sidney Decomposition
We begin with a parametric extension to Sidney’s decom-position theorem in Sidney (1975). In addition to its algo-rithmic implications, this extension also provides a shortproof of Sidney’s main result. For nonnegative real number# and job set H , define f#�H by
f#�H= w�H−#p�H�Theorem 3.1. Consider any fixed # � 0 and any initialsubset J in N such that ��J = max�f#�I � I ∈ ��� Thenthere exists an optimal schedule in which J precedes N\J �Proof. As in the proof of Lemma 3 in Sidney (1975),for �′ = �\ +�J , we consider the related problemP�wpN �′; that is, we eliminate all precedence rela-tions from J to N\J � Thus, P�wpN �′ is a relaxationof P�wpN �.
Lemma 3.2. J ∈ arg max�f#�I � I is initial in �N �′��
Proof. First, note that J is initial in �N �′. Now, to get acontradiction, assume that there is an initial set I in �N �′with f#�I > f#�J . Let U = J ∪ I and V = J ∩ I , so U andV are initial sets in �N �′ and f#�U+ f#�V = f#�J +f#�I > 2f#�J � But note that U and V are also initial in�N �: this is trivial for V as it is contained in J , and �and �′ coincide on J ; regarding U , if jk ∈ � and k ∈ U ,then either jk ∈�′ implying j ∈ U , or jk ∈�\�′ implyingj ∈ J ⊂U . Because U and V are initial in �N �, we havef#�U� f#�J and f#�V � f#�J , a contradiction. �
Lemma 3.3. There exists an optimal schedule for �N �′in which J precedes N\J .Proof. As in Sidney (1975), consider an optimal permu-tation ' on �N �′ and write '= �'�G1'�H1'�G2 � � � ,'�Hr'�Gr+1, where '�K denotes the restriction of ' toa subset K, such that J = ⋃
i Hi, and where G1 or Gr+1
may be empty. By the Adjacent String Interchange Lemma(Lemma 2 in Sidney 1975), we have ��H1���G2� · · ·���Hr, for otherwise we could exchange a subset with itspredecessor in the list, obtaining a feasible permutation for�N �′ with lower total cost. Note that ��Hr � #, forotherwise f#�J\Hr = f#�J − �w�Hr−#p�Hr > f#�J a contradiction with Lemma 3.2, because J\Hr is ini-tial in �N �′. On the other hand, because H1 is ini-tial in �N �′, we have ��H1 � #. In addition, if G1 �=, we also have ��G1 � # for otherwise f#�J ∪G1 =f#�J + �w�G1 − #p�G1 > f#�J again contradictingLemma 3.2, because J ∪G1 is initial in �N �′. Therefore��G1= ��H1= ��G2= · · · = ��Hr= #, where the first
equality holds provided G1 is nonempty. This implies thatwe can exchange each Hi with all preceding Gj ’s, obtain-ing an optimal permutation for �N �′ as stated in thelemma. �
Theorem 3.1 now follows immediately, because an opti-mal schedule as in Lemma 3.3 is feasible for �N �, andproblem P�wpN �′ is a relaxation of P�wpN �.
�
Let �∗ = max���I � I ∈ ��. Note that Sidney’s resultfollows as a special case, using #= �∗: As f�∗�J � 0 forall initial sets J , and is equal to 0 if and only if ��J = �∗,we have:
Corollary 3.4. If J ⊂ N is an initial set with ��J = �∗then there exists an optimal schedule for �N � in whichJ precedes N\J .
3.2. An Algorithm for ParametricSidney Decomposition
We now discuss an algorithmic implication of Theorem 3.1.As observed by Picard (1976) and Lawler (1976), Lawler(1978), finding an initial set J that maximizes f#�J isequivalent to finding a minimum s t-cut in the network�# = �N ∗�∗ c#, where N ∗ = N ∪ �s t� and s t � Nare a source and a sink; �∗ = � ∪ �sj jt � j ∈ N� andc# � �
∗ �→ �+ is defined by c#�ij=+� for all ij ∈ �,
c#�si= max�−wi+#pi0� and
c#�it= max�wi−#pi0�for all i ∈N . Indeed, by construction of the graph �N ∗�∗,every subset J ⊂N ∗ with s � J , t ∈ J and finite cut capacityc#�
−N ∗�J <+� defines an initial set I = J\�t� in N with
f#�I= c#� −N ∗��t�−c#� −N ∗�J . That is, the sink side ofthe minimum cut identified by the flow defines an initialset in N maximizing f#. Note that (because w � 0) N isoptimal for #= 0 and (because p > 0) is optimal for all#� max�wj/pj � j ∈ N�.
As the parameter # � 0 increases, we obtain a paramet-ric minimum cut problem where arcs adjacent to the sourcehave monotonically nondecreasing capacity, those ajacentto the sink have monotonically nonincreasing capacity, andother arcs have constant capacity. This is precisely the set-ting for the parametric maximum flow algorithm, hereaftercalled the GGT algorithm, due to Gallo et al. (1989). Indeed,the GGT algorithm produces, in about the time needed tocompute a single maximum flow on a network �#, a nestedfamily of subsets = H0 ⊂ H1 ⊂ · · · ⊂ Hk = N and asequence of breakpoints+�= #0>#1> · · ·>#k � #k+1 =0 such that for all i= 0 � � � k, Hi maximizes f# for all val-ues of # in the interval +#i+1#i,. (Note that we may have#k = #k+1 = 0 if w is tailing off, as defined at the end of §2.)
Proposition 3.5. Let = H0 ⊂ H1 ⊂ · · · ⊂ Hk = N bea nested sequence of initial sets constructed by the GGTalgorithm. For i= 1 � � � k define Ji =Hi\Hi−1. Then there
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exists an optimal schedule for P�wpN � in which Jiprecedes Ji+1 for all i = 1 � � � k−1.
This proposition shows that we can use the GGT algo-rithm to identify a decomposition of the job set N , suchthat an optimal schedule is obtained from optimal sched-ules for the job subsets in the decomposition. In the restof this subsection we prove this proposition, and in thenext subsection, we show how it relates to Sidney’s originaldecomposition. In particular, we shall need to consider spe-cial cases where there are several optimal initial sets. Thereader not interested in such details may now skip directlyto §4.
For the proof of Proposition 3.5 and subsequent devel-opments, we need to extend the notations and definitionsof §2 to subsets of an arbitrary job set K ⊆ N . For anysubsets J ⊆ K, let
+K�J = �ij ∈ � � i ∈ J j ∈ K\J � and
−K�J = �ji ∈ � � i ∈ J j ∈ K\J ��
A subset I ⊆ K is initial in K if −K�I = ; similarly, asubset T ⊆ K is terminal in K if +K�T = . Let ��Kdenote the collection of all initial sets in K, and � �K thatof all terminal sets in K.
Proof of Proposition 3.5. The proposition vacuouslyholds for k = 1, the case where no decomposition can befound, i.e., the problem is stiff; so, assume k � 2. Fori = 1 � � � k, define Ki = N\Hi−1 (see Figure 2). Observethat each Ki is a terminal set in N and each Ji is an initialset in Ki. We prove by induction on i that there exists anoptimal schedule for P�wpN � in which Ji precedesKi. Because J1 = H1 is an initial set in K1 = N that maxi-mizes f#1
, Theorem 3.1 implies that it is optimal to sched-ule all jobs in J1 before all jobs in K2 = K1\J1. Therefore,we may from now on restrict attention to optimal schedulesfor P�wpK2�2 where �2 is the restriction of � to K2.So, by induction, assume we have proved the propositionfor all i � h, and now restrict attention to optimal sched-ules for P�wpKh�h, where �h is the restriction of �to Kh. We claim that Jh is an initial set in Kh that max-imizes f# for # = #h. By contradiction, assume that thereexists an initial set I in Kh with f#�I > f#�Jh. BecauseHh−1 ∪ I is an initial set in N we have
f#�Hh−1� f#�Hh−1 ∪ I= f#�I+ f#�Hh−1
> f#�Hh−1+ f#�Jh= f#�Hh= f#�Hh−1
where the last equality follows from Hh−1 and Hh beingboth optimal for # = #h. This produces a contradiction.Thus Jh is an initial set in Kh that maximizes f#, and The-orem 3.1 implies that it is optimal to schedule all jobsin Jh before all jobs in Kh+1 =Kh\Jh, and therefore beforeJh+1. �
Figure 2. Illustration of Proposition 3.5; ��Hi−1 =#i−1 and ��Hi = #i; Ki = N −Hi−1 is notshown.
HiHi−1
Ji
N
3.3. Sidney Decompositions andthe Semilattice of Sidney Partitions
We now consider in greater detail the structure of thevarious decompositions one may obtain by using theparametric method described above or Sidney’s originaldecomposition method, as well as through different choiceswithin these methods. We shall start with a few definitions.
For job subset K ⊆ N , let
�∗�K= max���I � I is an initial subset in the subgraphof G induced by K��
Let �J1 � � � Jk denote an ordered partition of K, that is, asequence of nonempty subsets of K satisfying
⋃ki=1 Ji = K
and Jh∩Ji = whenever h �= i. We say that �J1 � � � Jk isa Sidney decomposition of K if, for all i = 1 � � � k, subsetJi is an initial set in Ki
�= Ji ∪· · ·∪ Jk and ��Ji= �∗�Ki.Lemma 3.6. The GGT algorithm constructs a Sidney decom-position of the job set N .
Proof. Letting, for h = 1 � � � k, subset Ji be as definedin Proposition 3.5 and, as above, Ki = N\Hi−1, we needonly to prove that ��Ji= �∗�Ki. Assume by contradictionthat there exists an initial set I in Ki with ��I > ��Ji.Note that, as in the proof of Proposition 3.5, we havef#i �Ji= f#i �Hi−1−f#i �Hi= 0, so #i = ��Ji < ��I. Butthen Hi−1 ∪ I is an initial set such that
f#i �Hi−1 ∪ I= f#i �Hi−1+ f#i �I > f#i �Hi−1
a contradiction. Thus we must have ��Ji= �∗�Ki. �
Sidney (1975) constructs a decomposition of the jobset N by the following simple process: First, let K1 = N ;then for i = 1 � � � and while Ki �= , choose a nonemptyinitial subset Ji of Ki with ��Ji = �∗�Ki and let Ki+1 =Ki\Ji. Clearly, this is a Sidney decomposition as definedabove, but it need not coincide with that obtained from theGGT algorithm. We now investigate the collection of allSidney decompositions of a job set N ; we will show inparticular that there is a finest such decomposition and thatit can be obtained from the GGT algorithm with modestadditional work.
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The �-profile of a Sidney decomposition � = �J1 � � � Jlis the sequence �#1 > · · ·> #q of the distinct values ��Jifor all members Ji of � . The proof of the following lemmais immediate, and is left to the interested reader.
Lemma 3.7. Let � = �J1 � � � Jl denote a Sidney decom-position of K. If ��Ji > ��Jk then i < k.
A consequence of this lemma is that in a Sidney decom-position � , the subsets Ji must be in nonincreasing orderof their � values. The reduction of a Sidney decompo-sition � with �-profile �#1 > · · · > #q is the orderedpartition � = �R1 � � � Rq, where for i = 1 � � � q, Ri =⋃�Jj � ��Jj = #i�. The proof of the following lemma is
immediate, and is also left to the interested reader.
Lemma 3.8. The reduction of a Sidney decomposition is aSidney decomposition.
Theorem 3.9. All Sidney decompositions of a giveninstance have the same �-profile and the same reduction.
Proof. Let � = �J1 � � � Jk and � ′ = �J ′1 � � � J ′l be twoSidney decompositions of the same job set N , with �-profiles �#1 > · · · > #q and �#′1 > · · · > #′r , respectively,and reductions � = �R1 � � � Rq and �′ = �R′
1 � � � R′r ,
respectively. We will show by induction on i that, for all i,#i = #′i and Ri = R′
i, thus implying that q = r and yield-ing the result. By definition, we have #1 = �∗�N = #′1.If R1 �= R′
1 then let U = R1 ∪R′1 and assume, w log, that
R1 ⊂ U . Because U is a union of initial sets in N , itis itself an initial set in N . Therefore, U\R1 is an ini-tial set in N\R1, with ��U\R1 = �∗�N by Lemma 2.1.As �∗�N > #2 = �∗�N\R1, we get a contradiction. So,we must have R1 = R′
1. The general case proceeds simi-larly: If #h = #′h and Rh =R′
h for all h� i, then we observethat �Ri+1 � � � Rq and �R′
i+1 � � � R′r are Sidney decom-
position of N\⋃h�i Rh with profiles �#i+1 > · · ·> #q and�#′i+1 > · · · > #′r , respectively; and the same argumentapplies to show that #i+1 = #′i+1 and Ri+1 = R′
i+1, complet-ing the proof. �
Accordingly, call the common �-profile of all Sidneydecompositions of N the �-profile of N , and call thecommon reduction of all Sidney decompositions of Nthe reduced Sidney decomposition of N . Thus all Sidneydecompositions of N differ only in the way the subsets Riin the reduced Sidney decomposition are partitioned andordered. For our scheduling problem, we will prefer to usea finer decomposition than the reduced Sidney decomposi-tion, whenever one exists; indeed, we prefer to decomposethe problem into as many (small) pieces as possible. Wewill show that this is indeed possible.
A Sidney decomposition � = �J1 � � � Jk of a job set Ndefines a partition �� = �J1 � � � Jk� of N consisting of thesame subsets, by simply disregarding their order. Call sucha partition a Sidney partition of N . The subsets Ji are calledthe blocks of the partition. We now characterize all the Sid-ney decompositions that define a same Sidney partition �� .
Theorem 3.10. Let �� = �J1 � � � Jk� be a Sidney parti-tion. An ordered sequence �J1 � � � Jk of its blocks is aSidney decomposition if and only if, for all distinct i j ∈�1 � � � k�,
(i) if ��Ji > ��Jj then i < j; and(ii) if uv ∈ � with u ∈ Ji and v ∈ Jj then i < j .
Proof. The conditions are clearly necessary: (i) byLemma 3.7, and (ii) because its violation would imply thatJj is not an initial set in
⋃nh=j Jh. Therefore, assume that
� = �J1 � � � Jk is an ordered sequence satisfying condi-tions (i) and (ii), and consider any index i ∈ �1 � � � k�. Weprove by induction on i that Ji is a initial set in Ki =
⋃nh=i Jh
with ��Ji= �∗�Ki. This follows immediately from (i) and(ii) when i = 1. So assume that this holds for all h < iwhere i � 2. This implies that the blocks J1 � � � Ji−1 canbe identified by Sidney’s algorithm as the first i−1 blocksin a Sidney decomposition � ′ = �J1 � � � Ji−1 J
′i � � � J
′r .
Condition (ii) implies that Ji is an initial set in Ki. Let�#1 � � � #q denote the �-profile of N . Since �� is definedby a Sidney decomposition, by Theorem 3.9, ��Ji = #jfor some index j . Therefore, letting � denote the Sidneyreduction of N , we have Ji ⊆ Rj . This implies �∗�Ki ���Ji = #j . By condition (i), we have Rh ∩Ki = for allh < j . By Theorem 3.9 applied to � ′, this implies that�∗�Ki= ��J ′i � #j . Therefore �∗�Ki= #j = ��Ji. �
For our scheduling problem, we will be indifferent be-tween using any of several Sidney decompositions thatdefine a same partition; although the resulting optimalsequences will differ, they will remain (globally) optimal,and the computational effort will be identical because wewill need to solve identical subproblems. We will showthat the Sidney partitions of a job set N form a (finite)meet semilattice, and thus that there exists a “finest” Sidneypartition.
Recall that the set Part�N of all partitions of a set N isa lattice (Grätzer 1978). The associated partial order “�”is defined as follows: A partition � = �Q1 � � � Qq� of Nis finer than partition �′ = �Q′
1 � � � Q′r �, denoted �� �′, if
for all i= 1 � � � q there exists an index j such that Qi⊆Q′j .
Note that this implies q � r . A collection �⊆ Part�N is ameet sub-semilattice of Part�N if it is closed for the (par-tition lattice) meet operation, that is, if ��′ ∈� then theirmeet �∧�′ is also in �, where �∧�′ ∈� is the coarsestpartition �′′ ∈ Part�N such that �′′ � � and �′′ � �′.
Theorem 3.11. The set of all Sidney partitions of a set Nis a meet subsemilattice of the partition lattice Part�N .
Proof. Given any two Sidney decompositions � = �J1 � � � ,Jk and � ′ = �J ′1 � � � J ′l of N , we need to prove that themeet (largest lower bound) �� ∧ �� ′ of their associated Sid-ney partitions is also a Sidney partition of N . Recall (e.g.,Lemma IV.4.1 in Grätzer 1978) that �� ∧ �� ′ is the partitionconsisting of all the nonempty intersections Hij = Ji ∩ J ′j .Order these nonempty intersections Hij in lexicographicorder of the indices i and j (that is, if Hij is before Huvthen either i < u or i = u and j < v). It follows that the
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sequence of all nonempty intersections Hij thus orderedsatisfies condition (ii) of Theorem 3.10. By Theorem 3.9,we have Hij �= only if ��Ji = ��J ′j . Therefore thisordered sequence of nonempty Hijs also satisfies condi-tion (i) of Theorem 3.10, implying that �� ∧ �� ′ is a Sidneypartition. �
Corollary 3.12. There exists a finest Sidney decomposi-tion of a set N , unique up to permutations of some of itsblocks.
The following example shows that the set of all Sidneypartitions is not closed for the join operation.
Example. let N = �a b cd e� and � = �ab cd�; letall pi = wi = 1, so all nonempty subsets J ⊆ N have��J = �∗�N = 1; then � = ��a e� �b c� �d� and � ′ =��a� �b� �c� �d e� are Sidney decompositions, and yetthe join �� ∨�� ′ = ��ad e� �b c�� of their associated Sid-ney partitions is not a Sidney partition, as neither of itsblocks is an initial set in N .
Note, however, that there is a coarsest, i.e., least fine,Sidney partition, �� defined by the reduced Sidney decom-position � of N . In contrast to the finest Sidney partition,the coarsest Sidney partition �� is unique.
3.4. Constructing a FinestSidney Decomposition
We now show how to construct a finest Sidney decompo-sition, using the GGT algorithm described in §3.2.
First recall (see, e.g., Hu 1970) that, given a network�# = �N ∗�∗ c# with source s and sink t in N ∗, the col-lection of sink sets T of all minimum capacity s t-cutsforms a sublattice of 2N
∗. This sublattice is isomorphic
to the sublattice of ideals I ⊆ N with maximum weightf#�I=w�I−#p�I, by simply letting I = T \�t�, as seenin §3.2. Recall also that the largest and smallest such sinksets T∨ and T∧, respectively, can be obtained by simplelabeling. For T∨, start from the source s and apply theFord-Fulkerson labeling procedure: then T∨ is the set of allunlabeled nodes at the end of the procedure (i.e., all nodesnot reachable from the source s in the augmenting net-work associated with the current maximum flow). For T∧,take all nodes from which one can reach the sink t inthe augmenting network (see Hu (1970) for details). Notethat each procedure requires O�m+ n operations, wheren = �N � = �N ∗� − 2 and m = ��� (so the number of arcsin �# is m+2n).
Because by Theorem 3.9, R1 = T∨\�t� defines the largestideal with maximum weight f#�I, it follows that we imme-diately obtain the reduced Sidney decomposition by usingFord-Fulkerson’s original labeling procedure at each stepof the GGT algorithm.
For our scheduling purposes, however, we want to con-struct a finest Sidney decomposition. Recall that the GGTalgorithm identifies a sequence of breakpoints +� =#0 > #1 > · · · > #k � #k+1 = 0, where, as follows from
§§3.2 and 3.3, �#1 � � � #k is the �-profile of N . Let-ting, as above, � = �R1 � � � Rk denote the reduced Sid-ney decomposition of N , each set Ii =
⋃h�i Rh is, for
i= 0 � � � n, the largest ideal I in N with maximum weightf#i �I. Consider a step of the GGT algorithm where wehave determined a maximum flow for #= #i. We now seekto determine a finest Sidney decomposition, which amountsto finding a maximal nested family of minimum s t-cuts in�#. All minimum s t-cuts in �# can be obtained as follows(details can be found in Picard and Queyranne 1980).
Let AG��# f be the augmentation graph associatedwith the optimal flow f on �# and let A1 � � � Au be itsstrongly connected components. Let AG′ be the directedgraph obtained from AG by contracting to a single vertexaj the component Aj for j = 1 � � � u. Observe that AG′
is a directed acyclic graph, thus inducing a natural partialorder on its nodes. Any minimum s t-cut in �# is the setof arcs entering the union of components corresponding toan initial set in AG′. A maximal nested family of such cutscan be obtained by ordering the nodes of AG′ according toany order compatible with the partial order represented byAG′, say �b1 � � � bu� and considering the cuts generatedby the sets b1 ∪· · ·∪bj for j = 1 � � � u−1.
Because all the above operations can be carried out intime O�m+n, we have shown the following.
Theorem 3.13. Let MF(va) denote the time required tosolve a maximum flow problem in a network with v nodesand a arcs, using an algorithm compatible with the Gallo-Grigoriadis-Tarjan parametric maximum flow approach(Gallo et al. 1989). Then a finest Sidney decomposi-tion of a scheduling problem with n jobs and m prece-dence constraints can be constructed in MF�nm+2n+O�n�m+n time. The coarsest Sidney decomposition canbe constructed in MF�nm+2n+O��m+n time.
4. PROPERTIES OF THE LP RELAXATION: TIGHTSETS AND DECOMPOSITIONS
In this section, we study an LP relaxation of the problem.We explore some interesting properties of the LP optimalsolutions. We derive a necessary and sufficient conditionfor the Sidney decomposition of the problem. Finally, weshow that any feasible solution of a stiff instance has objec-tive value at most twice the optimal value. (This has beenproved independently by Chekuri and Motwani 1999.)
For convenience, we introduce some notation. For anyvectors u v ∈ RN and J ⊆ N , let u�J =∑
j∈J uj , u2�J =∑j∈J u2
j , and u∗v�J =∑j∈J ujvj . The following set func-
tion plays a fundamental role in the study of schedulingpolyhedra (see, e.g., Queyranne 1993 or Queyranne andSchulz 1996): For J ⊆ N , let
g�J = 12 +p�J
2 +p2�J ,�
The following identity (Queyranne 1993) will be useful:For any J H ⊆ N ,
g�J ∪H+g�J ∩H= g�J +g�H+p�J\Hp�H\J � (2)
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For any J ⊆ N , inequality p ∗C�J � g�J is knownto be valid for (1), and it is called a parallel inequality(Queyranne and Wang 1991a).
Without loss of generality we may assume that the prece-dence graph G is transitively reduced, that is, � is the min-imum collection of arcs representing the partial order. (Thisis also called the Hasse diagram of the partial order.)
Let A denote the ���×�N � arc-job incidence matrix of G,where rows of A are indexed by arcs ij ∈ �, columns byjobs h ∈ N , and
Aijh =
−1 if h= i and ij ∈ �
1 if h= j and ij ∈ �
0 otherwise�
Let Ah denote the column of A indexed by job h. Alsodefine the column vector b ∈ R� as bij = pj for all ij ∈ �.Let 0 denote a vector of all 0s, and 1 a column vector ofall 1s.
Replacing the disjunctive constraints and C � p in (1)with all parallel inequalities, we obtain the following LPrelaxation:
wC∗ = min wC
s.t. p ∗C�J � g�J ∀ J ⊆ NAC � b�
(3)
Using a variable 8J for each J ⊆ N and a variable yij foreach arc ij ∈ �, the LP dual formulation of (3) is
D�8∗ y∗= max∑�g�J 8J � J ⊆ N+yb
s.t.∑�ph8J � J � h J ⊆ N+yAh = wh
∀h ∈ N8� 0 y � 0 (4)
where �8∗ y∗ denote an optimal dual solution.Now, for any optimal solution C to (3), define
;�C= �J � J ⊆ N p ∗C�J = g�J ��That is, ;�C is a family of tight sets. By convention, ∈;�C.
Lemma 4.1. Let J ∈ ;�C be any nontrivial tight set. Then�i ∀ i ∈ J Ci � p�J
and it holds with equality iff J\�i� ∈ ;�C;
�ii ∀ j �∈ J Cj � p�J +pjand it holds with equality iff J ∪ �j� ∈ ;�C.
(5)
Proof. Note that by (2), g�J − g�J\�i�= pip�J . Thus,(i) follows from
g�J = piCi+p ∗C�J\�i�� piCi+g�J\�i��Similarly, because g�J ∪ �j�− g�J = pj�p�J +pj, (ii)follows from
g�J ∪ �j�� pjCj +p ∗C�J = pjCj +g�J �Note that we have used the fact that all parallel inequalitiesare valid. �
Lemma 4.2. Let J ∈ ;�C be any tight subset. If one of thefollowing conditions holds:
(i) there exists no tight set �J with �J ⊂ J and � �J � = �J �−1,(ii) there exists no tight set �J with J ⊂ �J and � �J � =
�J �+1,then Cj −Ci > pj for all i ∈ J and j ∈ N\J .Proof. Suppose that for some i ∈ J and j ∈ N\J , Cj −Ci � pj . Then by (i) and (ii) of (5) we have Cj −p�J �Cj −Ci � pj � Cj −p�J . This implies
(a) Ci = p�J and (b) Cj = p�J +pj .(1) If �J � � 1 and Cj −Ci � pj for some i ∈ J and j ∈
N\J then, by (a) above and (i) of (5), J\�i� ∈ ;�C, acontradiction to the minimality of J .
(2) Observe that (b) above and (ii) of (5) implythat �J ∪ �j� ∈ ;�C with � �J � = �J � + 1, again acontradiction. �
Note that it is not necessary that � contain the pair ijor ji.
The following proposition establishes some importantproperties of tight sets.
Proposition 4.3. For any optimal solution C to (3),(i) ;�C contains a nonempty set;(ii) for any pair J H ∈ ;�C, either J ⊂ H or H ⊂ J
holds;(iii) each J ∈ ;�C is initial;(iv) N ∈ ;�C if w is not tailing off.
Proof. (i) If (i) does not hold, then by complemen-tary slackness, 8∗ = 0. So w = y∗A, and w�N = w1 =y∗A1= 0, implying w = 0, a contradiction.
(ii) If (ii) is false, then there exists a pair J H ∈ ;�Cwith p�J\Hp�H\J > 0. Using two parallel inequalitiesinduced by J ∪H and J ∩H to obtain the first inequalitybelow and the identity (2) for the second line, we obtain acontradiction because
g�J ∪H+g�J ∩H� p ∗C�J ∪H+p ∗C�J ∩H= p ∗C�J +p ∗C�H= g�J +g�H= g�J ∪H+g�J ∩H
−p�J\Hp�H\J �(iii) If some J ∈ ;�C is not initial, then there exists
some j �∈ J but ji ∈ � for some i ∈ J . Then Cj � Ci−pi,and Ci � p�J by (i) of (5). This implies that Cj < p�J , acontradiction to inequality (ii) of (5).
(iv) Suppose that N �∈ ;�C. Then by (i) and (ii), thereexists a unique maximal tight set J ∗ with J ∗ �=N . Let H =N\J ∗, and note that H is terminal because J ∗ is initialby (iii). Define �C by �Ch = Ch if h ∈ J ∗ and �Ch = Ch− <if h ∈ H . By maximality of J ∗ and (ii) of Lemma 4.2,we have Cj −Ci > pj for all ij ∈ +�J ∗. Furthermore, themaximality of J ∗ ensures that no tight set contains a nodein H . Therefore, for sufficiently small < > 0, �C is primalfeasible. But wC � w �C = wC− <w�H, implying wj = 0for all j ∈H and w is tailing off. �
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Theorem 4.4. Assume that G= �N � does not contain adirected Hamiltonian path. The problem P�wpN � isSidney decomposable if and only if there exists an optimalLP solution C such that p ∗C�J = g�J holds for somenontrivial subset J ⊂ N .Proof. Necessity. Let I ∈ � be a nontrivial initialset such that ��I = w�I/p�I = �∗. Let H = N\I .Because P�wpN � is Sidney decomposable and usingLemma 2.1, we have
w�I
p�I�w�N
p�N= w�I+w�Hp�I+p�H ⇐⇒ w�I
p�I�w�H
p�H� (6)
Now, consider any optimal LP solution C. If there existssome nontrivial subset J ⊂N with p∗C�J = g�J , we aredone. Otherwise, for < > 0, define C<i = Ci−<p�H for alli ∈ I and C<j = Cj + <p�I for all j ∈ H . Observe that forany < > 0, the following inequality,
p ∗C<�N= p ∗C�I− <p�Hp�I+p ∗C�H+ <p�Ip�H� g�N (7)
and all precedence constraints are still satisfied by C<
(because I is initial). Furthermore, by (6)
wC � wC< = wC− <w�Ip�H+ <w�Hp�I� wC�
So C< is also an optimal solution if no parallel inequalityp ∗C<�J � g�J induced by nontrivial subset J ⊂ N isviolated as < increases. Clearly, we can increase < untilp ∗C<�J < � g�J < becomes binding for some nontrivialsubset J <. The resulting C< is the required solution.Sufficiency. Let C be the optimal solution with ;�C
containing some nontrivial subset. We need to find somenontrivial initial subset J ∗ with ��J ∗� ��N.
First, observe that if N �∈ ;�C, then by (iv) ofProposition 4.3, w is tailing off, which implies that thereexists some nontrivial terminal subset H with w�H= 0. SoJ ∗ =N\H is the required initial set, and we are done. Thus,for the rest of the proof assume that �;�C� � 3 (because;�C contains the empty set, a nontrivial tight set and N ).
By (ii) and (iii) of Proposition 4.3, we know that ;�Ccontains k nested nonempty tight sets J1 ⊂ · · · ⊂ Jk where2 � k� n, and that all these sets are initial. We distinguishthe following two cases.
Case 1� 2 � k < n.(I) We first claim that, for some q with 1 � q < n, Cj−
Ci > pj for all ij ∈ +�Jq.If �J1�� 2, then (i) of Lemma 4.2 implies that the claim
holds; Otherwise, since k < n, there exist tight sets Jq andJq+1 with �Jq+1�� �Jq�+2. Then the claim follows from (ii)of Lemma 4.2. This proves the claim.
(II) Let I = Jq and T = N\Jq . Define C<i = Ci+ <p�T for all i ∈ I and C<j = Cj − <p�I for all j ∈ T . For suffi-ciently small < > 0, C< violates no precedence constraints
by the above claim, and moreover, using 4.3(ii), it isstraightforward to verify that
p ∗C<�Q > g�Q for all Q �= N ,
and that p ∗C<�N= g�N�So C< is a feasible solution, and
0 � wC<−wC = <w�Ip�T − <w�T p�I=⇒ w�I
p�I�w�T
p�T �
It follows that ��I� ��N as required.
Case 2� k = n. Without loss of generality assume thatC1 < C2 < · · · < Cn. It follows from repeated applicationof Lemma 4.1 that C forms a schedule, that is, C1 = p1,Cj+1 = Cj + pj for j = 1 � � � n− 1. First observe thatCj −Ci > pj for any ij ∈ � with j � i+ 2. (Otherwise,by Lemma 4.1, Ji ∪ �j� �∈ ;�C is also tight, a contradic-tion.) If there exists some proper tight subset Jq such that�q q+ 1 �∈ �, then using the above observation and theconstruction of C< as in (II) of CASE 1, we show thatI = Jq is the required subset and we are done. Otherwise,we must have �q q+ 1 ∈ � for q = 1 � � � n− 1. So �forms a chain, again a contradiction. �
When P�wpN � is stiff (i.e., not Sidney decompos-able), some interesting consequences follow from the abovetheorem. First, one obtains a new (trivially computable)lower bound for the optimal objective value z∗. Second, anyfeasible schedule with no idle times has its objective valuewithin a factor of 2 of the optimal objective value.
Theorem 4.5. If P�wpN � is stiff, then the optimal LPvalue wC∗ �w�Np�N/2. Moreover, there exists a familyof instances such that this inequality is asymptotically tight.
Proof. Because P�wpN � is stiff, by Theorem 4.4,;�C= �N� for any optimal LP solution C (since w �= 0).By duality, w = 8∗
Np+ y∗A, and the optimal LP valueis wC∗ = 8∗
N g�N+ y∗b. Since w�Np�N = w1p�N =�8∗
Np�N+0p�N= 8∗Np�N
2,
wC∗ = 8∗N g�N+y∗b � 8∗
N
p�N2
2= w�Np�N
2
as required. The required family of instances can be con-structed as follows. Let N = �1 � � � n� and � = �jn � j =1 � � � n−1�. Let wj = 0 for j = 1 � � � n−1 and wn = n,and let pj = 1, for j = 1 � � � n. The optimal LP solutionC∗ is given by C∗
j = �n2 +n− 2/2n for j = 1 � � � n− 1and C∗
n = �n2 + 3n− 2/2n with wC∗ = �n2 + 3n− 2/2nand w�Np�N/2 = n/2. �
Corollary 4.6. If P�wpN � is stiff, then any feasibleschedule with no idle times has its objective value within afactor of 2 of the optimal value.
Using the same LP formulation, Schulz (1996) and Hallet al. (1997) obtained 2-approximation algorithms by con-structing a feasible solution with value at most twice thevalue of the linear relaxation. Corollary 4.6 has the sameflavor, but shows that for a stiff instance, even the worstfeasible solution is within a factor of 2 of the optimal value.
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5. SOLVING THE LP FOR A STIFF INSTANCE
In this section, we show how to solve the LP (3) for a stiffinstance P�wpN � by network flow techniques. ByTheorem 4.4, the LP (3) for a stiff instance reduces to theinequalities generated by the precedence constraints and tothe equality p ∗C�N= g�N. In the reminder of this sec-tion, we find convenient to work with variables associatedwith starting times instead of completion times. For J ⊆N ,we define g′�J = 1
2
[p�J 2 −p2�J
]and for j ∈ N , we use
Sj = Cj −pj to denote the start time for job j . Hence,
pjSj = pjCj −p2j and Cj −Ci � pj ⇐⇒ Sj −Si � pi�
Rewriting the LP (3) with these variables (see Queyranneand Schulz 1996), we obtain:
min z= wSs�t� pS = g′�N
Sj −Si � pi for all ij ∈ �
S � 0�
(8)
Let � = w�N/p�N and w̄ = w − �p. Consider theLagrangian relaxation of this problem obtained by dualiz-ing the equality constraint with multiplier �, i.e.:
min z′ = wS−��pS−g′�N = w̄S+�g′�N s�t� Sj −Si � pi for all ij ∈ � (9)
S � 0�
We say that an arc ij ∈ � is tight for a solution S ifSj −Si = pi. The next lemma implies that z= z′.Lemma 5.1. There exists a solution S∗ optimal for both LP(8) and LP (9).
Proof. Because the problem P�wpN � is stiff,for each initial set I ∈ � we have w̄�I = w�I −�w�N/p�Np�I < 0, and thus for each terminal setT ∈ � we have w̄�T > 0. Because the extreme rays of LP(9) are generated by the characteristic vectors of the termi-nal sets, the optimal value of LP (9) is bounded. As LP (9)is contained in the positive orthant, there exists an optimalextreme point S. Let K be a connected component of thesubgraph of G induced by the tight arcs for S. Note that Kcontains a node i with Si = 0, for otherwise, adding (resp.subtracting) a small < > 0 to Sj for all j in K would yielda feasible point S+ (resp. S−). But then S = �S+ +S−/2,a contradiction. Hence, for each node j ∈ N there exists apath of tight arcs joining j to a node i�j with Si�j = 0(ignoring the orientation of the arcs).
Let t = arg max�Si � i ∈ N� and t0 � � � tk t be a sim-ple path of tight arcs joining t0 to t with St0 = 0. Thenwe have St � pt0 + · · ·+ptk and, by choice of t, St′ � Stfor all t′ ∈ N . Consider now the schedule obtained bystarting with the jobs t0 � � � tk t in that order and thenputting the remaining jobs in arbitrary order. (This sched-ule is probably not feasible for LP (9), but this is of no
importance.) Denote by y the starting times of that sched-ule. Note that S � y and that py = g′�N . It follows thatpS � g′�N and thus, for q = �g′�N −pS/p�N, we haveq � 0. Hence S∗ = �S+q1 is feasible for LP (9) and satis-fies p S∗ = g′�N , i.e., S∗ is feasible for LP (8). Moreover,since w̄�N = 0, we have w̄S∗ = w̄S, implying that S∗ isan optimum solution to LP (9) feasible for LP (8), i.e., anoptimum solution to LP (8). �
The LP (9) is, in fact, a problem on node potentials withnonnegativity constraints. Hence, if we forget about thenonnegativity constraints, it is the dual of a max-cost flowproblem with supply-demand vector w̄ and with arc capac-ities all infinity. By solving this max-cost flow problemand constructing a dual feasible solution by complemen-tary slackness, we can find an optimal solution of LP (8),as outlined in the proof of the previous lemma. The com-plexity of the whole procedure is dominated by the time tocompute a max-cost flow, e.g., O����� logn����+n logn(Ahuja et al. 1993).
Depending on the algorithm chosen for solving the max-cost flow problem, the dual variables may be readily avail-able. Otherwise, starting from an optimal solution f ∗ ofthe flow problem, we consider the connected componentsK1 � � � Ku induced by the arcs ij with f ∗
ij > 0. For each1 � v� u, one can set potential Si � 0 for the nodes i ∈Kvsuch that all arcs of Kv are tight for S and each com-ponent contains a node with potential 0. This solution Smay violate some precedence constraints associated witharc ij ∈� with i and j in different connected components.But because the potential associated to the nodes of oneconnected component is defined up to an additive constant,it remains to determine additive constants >1 � � � >u � 0such that adding >v to the potential of the nodes in Kv forall 1 � v � u yield a feasible optimal solution to LP (9).
These >s can be found by solving a longest path problemin the digraph G′ obtained from G by contracting eachconnected component to a single node, and replacing arcij joining i in component Ku with j in component Kv byan arc joining Ku to Kv with weight Si+pi−Sj . Add onenode s joined to all nodes in G′ by an arc of length 0.Because LP (9) is feasible, G′ does not contain a directedcycle with positive weight and we can use Bellman-Fordalgorithm to compute the longest path from s to the othernodes in G′ in O�nm operations. It is straightforward tocheck that using the value of the longest path from s tothe node Ku for >u yield a feasible solution to LP (9) thatsatisfies the complementary slackness condition with f ∗,implying the optimality of this solution.
6. COMPUTATIONAL RESULTS
We implemented the algorithms described in §§3, 4, and 5and tested its performance on random instances. We firstgive a high-level description of the algorithm before detail-ing its steps and explicating some of the terms it uses inthe remainder of the section.
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1. Initialize the ordered list L of subproblems as con-taining only the initial problem;
2. For each subproblem P in the list L, repeat
2.1. if P is series decomposable, replace P in L bythe ordered subproblems of its series decompo-sition; break;
2.2. If P is Sidney decomposable, replace P in Lby the ordered subproblems of a finest Sidneydecomposition of P ; break;/* Now P is non series decomposable and stiff */
2.3. Remove P from L;2.4. Compute the value of the LP (3) for P ; Try to
improve the value of this lower bound;2.5. Compute a heuristic solution for P ;
3. Paste the heuristic solutions and LP values of the nondecomposable subproblems to obtain an upper boundand lower bound on the optimal value for the initialproblem.
Decompositions. Steps 2.1 and 2.2. We use two types ofdecompositions: series decompositions and Sidney decom-positions. Series decompositions occur when there existsan initial set I such that each node in N\I may be reachedfrom each node in I by a directed path in G. In otherwords, the jobs in I have to be completed before any job inN\I can start. Thus, jobs in I appear before jobs in N\I inany feasible schedule. Series decompositions can be foundin O�n+m for a graph with n nodes and m arcs. Thealgorithm to find the Sidney decompositions outlined in §3uses a parametric network flow algorithm. Unfortunately,we could not find an implementation of the the parametricnetwork flow algorithm of Gallo et al. and thus we find theSidney decompositions by O�n calls to a max-flow code(we use the code Maxflow written by Goldfarb and Grigo-riadis 1988).
Solving the LP. Step 2.4. To find the optimal solutionof the LP (3) for a stiff instance, we follow the discussionof §5 to formulate the problem in the form of LP (9). Wesolve its dual, a max-cost flow problem, with the networksimplex solver of the linear optimizer CPLEX4.0. We foundthat, for our problems, it is at least as efficient as the max-cost flow codes at our disposal. From the optimal flow andas discussed in §5, we construct an optimal solution of LP(9) from which an optimal solution of the LP (8) and thenof the LP (3) are easily obtained. The optimum value ofthe LP (3) is denoted by LP.
Lower bound improvement. Improving the lowerbound given by the LP (9) is possible, as observed byHoogeveen and Van de Velde (1996). They give a strength-ening for the Lagrangian relaxation of the precedence con-straints of the LP (9). We did not implement an ascentmethod to find the best Lagrange multipliers as suggestedby Hoogeveen and Van de Velde, but merely used the max-cost flow solution. The value of this improved lower boundis denoted by LPL.
Heuristics. Step 2.5. A simple heuristic to find a rela-tively good solution of the scheduling problem is to order
the jobs as indicated by the optimal solution C of the LP (3),starting with the job with the smallest entry in C. This order-ing is feasible since if ij ∈ � then Cj −Ci � pj is a con-straint of the LP and thus Ci < Cj . The solution obtained inthis way is denoted by OUB for original upper bound.
Note that it is possible to replace C by a point C ′ obtainedby subtracting 'ipi to Ci for all i ∈N , where 'i is any num-ber in +01. Ordering the jobs according to C ′ will alsogive a feasible solution. This idea, introduced by Phillipset al. (1998) for converting preemptive schedules to non-preemptive ones for problems with release dates, is partof several approximation algorithms for scheduling prob-lem (Goemans et al. 2002). We generated 10 such feasibleschedule O1 � � � O10 for each subproblem by randomizingthe 'is.
Most of the time, substantial local improvements of afeasible solution are possible by permuting two consecutivejobs, or more generally, by permuting two adjacent groupsof at most d jobs, for a fixed d. Applying this improve-ment heuristic to several feasible schedules usually yieldsa very good feasible solution. However, when the numberof jobs increases above a few hundred, this procedure maybecome quite time consuming, depending on the value ofd (we used d = 10). In the results reported below, the timespent for this heuristic may be as high as 50% of the totaltime for problems with 1,000 jobs or more. Obviously, it ispossible to devise a much faster heuristic (using a smallervalue for d, or looking for improvements only for a fixedamount of time) that would return almost the same results.But because we use the best known feasible solution toapproximate the gap, and we want this gap to be as smallas possible, we don’t mind spending a large fraction of thetime in the heuristic procedure. We denote by UB the bestsolution found by this improvement procedure on the 10feasible schedules O1 � � � O10.
Pasting the solutions. Step 3. In Step 2 of the algorithm,we construct an ordered list of subproblems �J1 � � � Jkforming a partition of N and for each of these subproblems(also called piece) Jr , we find an heuristic solution UBr andcompute the lower bound LPr corresponding to the LP (3).Pasting the solutions of the subproblems together, we get anupper bound UB and a lower bound LP on the value of theoptimal solution of the original problem using
UB =k∑i=1
[UBi+w�Ji
i−1∑j=1
p�Ji
]and
LP =k∑i=1
[LPi+w�Ji
i−1∑j=1
p�Ji
]�
We also compute the simple lower bound WP derivedfrom Theorem 4.4, i.e.,
WP =k∑i=1
[w�Jip�Ji/2+w�Ji
i−1∑j=1
p�Ji
]�
Instances. Problems are generated as proposed by Potts(1985). The precedence graph is a random acyclic directed
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graph on n nodes with density B, i.e., a directed graphobtained by selecting independently and uniformly arc ij ,with i < j , with probability B. For a given value of n, 20problems are generated, 2 for each value of
B ∈ �0�0010�020�040�060�080�10�150�20�30�5��
The processing times are drawn from the discrete uniformdistribution on +1100,, and the weights are drawn fromthe discrete uniform distribution on +110,. The graphs arethen replaced by their transitive reduction. The number ofarcs in the resulting digraphs varies, of course, with B. Forexample, for problems with 1000 jobs, we get roughly 430arcs for B = 0�001, 4000 arcs for B = 0�02, 3000 arcs forB = 0�08, and 2000 arcs for B = 0�3.
We report results for problems generated in this way for100, 500, 1000, 1500, and 2000 jobs. We partition the 20problems generated for a value of n into four groups: prob-lems 1–2 (B = 0�001), problems 3–10 (B ∈ �0�020�040�060�08�), problems 11–18 (B ∈ �0�10�150�20�3�),and problems 19–20 (B = 0�5). The problems generatedwith B = 0�001 (problems 1 and 2) or B = 0�5 (problems19 and 20) are usually easy instances because the formerare close to a digraph with no arcs and the latter resemblea complete acyclic digraph. The other problems are harderbut of relatively homogeneous difficulty inside a group.
Results. To assess the efficiency of the local improve-ment heuristic and of the Lagrangian relaxation procedure,we report the percentage of the gap that is closed bythe corresponding procedure, gapim for the improvementheuristic and gapl for the Lagrangian procedure. Ideally,if the optimal value of a problem is OS, the best we canhope is that the local improvement heuristic close the gapbetween OUB and OS. Unfortunately, for most of the prob-lems we consider, the value of the optimal solution is not
Table 1. Aggregated results.
n inst. a_gap m_gap a_gapim a_gapl a_r_wp a_mpc m_mpc a_cpu m_cpu
100 1–2 0�00 0�00 0�00 0�00 100�00 3�00 3 0�28 0�283–10 0�74 2�27 72�86 31�38 98�02 29�00 59 0�26 0�33
11–18 2�16 3�51 47�97 13�82 94�82 55�12 79 0�17 0�2019–20 0�36 0�63 33�53 35�86 98�48 22�50 28 0�17 0�20
500 1–2 0�00 0�00 50�00 50�00 100�00 3�50 4 1�60 1�633–10 1�48 2�35 50�77 5�57 97�87 248�50 408 1�53 1�83
11–18 1�20 2�46 32�97 8�64 97�76 323�00 460 1�13 1�4819–20 0�09 0�15 41�23 24�48 99�64 53�00 76 1�00 1�03
1,000 1–2 0�00 0�00 74�01 43�24 99�99 6�50 7 3�67 3�703–10 1�31 2�44 41�12 3�78 98�28 618�62 881 4�87 6�83
11–18 1�42 3�48 20�48 5�51 97�80 579�12 950 3�32 4�8019–20 0�03 0�04 46�26 28�83 99�86 38�50 55 2�40 2�42
1,500 1–2 0�00 0�00 83�05 64�73 99�98 13�00 15 6�36 6�373–10 1�00 1�33 40�20 3�36 98�72 991�88 1422 9�66 15�95
11–18 0�77 1�49 20�50 5�15 98�74 802�25 1091 5�18 6�3519–20 0�02 0�03 44�27 24�34 99�88 48�00 57 4�02 4�07
2,000 1–2 0�00 0�00 78�41 45�86 99�98 13�50 16 10�40 10�683–10 0�93 1�30 36�73 2�76 98�84 1027�12 1381 14�05 26�93
11–18 1�13 1�92 13�23 3�03 98�27 1280�00 1933 9�79 13�3719–20 0�01 0�02 49�21 28�40 99�93 54�50 63 6�76 6�85
known. Hence we use LPL instead of OS to obtain a pes-simistic estimator of the performance of the heuristic. Sim-ilarly, the Lagrangian procedure may be able to close thegap between LP and OS, and we use UB instead of OSto get an underestimation of the performance of the proce-dure. More precisely, we report
gapim = 100OUB−UBOUB−LPL and gapl = 100
LPL−LPUB−LP �
In addition, we also report the gap between the bestheuristic solution UB and the best lower bound LPL andthe ratio r_wp between the lower bounds WP and LP:
gap = 100UB−LPLLPL
and r_wp = 100WP
LP�
For a group of instances, we report the average valueof gap, gapim, gapl, r_wp, the size of the largest piecein the decomposition mpc and the cpu time in secondscpu with names prefixed with “a_”. We also report themaximum of gap, mpc, and cpu with names prefixed with“m_”. The machine used was a SUN Sparcstation 5 with a400-Mz processor running SunOS5 and using the compilergcc2.6.3.
Several conclusions may be drawn from Table 1. First,the gap between the best known feasible solution UB andthe lower bound LPL is quite small: the maximum valueof gap is 3.48%, attained for a single problem with 1,000jobs. The average gap is between 1% and 1.5%. Interest-ingly, the gap seems to decrease as the number of jobsincreases above 1,000. The local improvement procedurefor a feasible solution closes in average 20%–40% ofthe gap between OUB and LPL, whereas the Lagrangianrelaxation closes in average 3%–10% of the gap between
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UB and LP. Here, it seems that the latter procedure is moreefficient for small values of n, closing up to 31% of the gapbetween the LP solution and the best known feasible solu-tion for problems with 100 jobs. We also note that the sim-ple bound derived from Theorem 4.4 is a very good bound,in particular for problems with 1,500 jobs or more, consid-ering the simplicity of its computation. The decompositionsare able to reduce in average the size of the largest pieceto roughly 60% of the initial size. The cpu times are hereas an indication that the decomposition procedure does nottake a huge amount of time. Interestingly enough, the timeneeded to apply our randomized heuristic on a problemwith �500 jobs (i.e., no decomposition is performed and10 calls to the heuristic are made) takes far more time thanthe time needed to decompose the problem, apply the ran-domized heuristic on the pieces of the decomposition, andpaste the solutions. The value of the solutions obtained arealso slightly better when decompositions are in use.
Note that the LP formulation we use on a stiff pieceis one of the simplest formulations, i.e., it reduces to theprecedence constraints and the parallel inequality on thejobs in the piece. This formulation is generally very weakfor random instances, but our results show that it is quitestrong for the nonseries decomposable stiff instances weget by decompositions.
ACKNOWLEDGMENTS
The authors thank two anonymous referees for their helpfulcomments and suggestions.
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