BULLETIN OF THE POLISH ACADEMY OF SCIENCES TECHNICAL SCIENCES, Vol. 67, No. 1, 2019 DOI: 10.24425/bpas.2019.127335 Better polynomial algorithms for scheduling unit-length jobs with bipartite incompatibility graphs on uniform machines T. PIKIES* and M. KUBALE Department of Algorithms and System Modelling, Gdańsk University of Technology, ETI Faculty, Gabriela Narutowicza 11/12, 80-233 Gdańsk, Poland Abstract. The goal of this paper is to explore and to provide tools for the investigation of the problems of unit-length scheduling of incompat- ible jobs on uniform machines. We present two new algorithms that are a significant improvement over the known algorithms. The first one is Algorithm 2 which is 2-approximate for the problem Qm pj = 1, G = bisubquartic Cmax. The second one is Algorithm 3 which is 4-approximate for the problem Qm p = 1, G = bisubquartic ΣC , wherej m {2, 3, 4}. The theoryj behind the proposed algorithms is based on the properties j j 2 of 2-coloring with maximalj coloring width, andj on the properties of ideal machine, an abstract machine that we introduce in this paper. Key words: approximation algorithm, graph coloring, incompatible job, polynomial algorithm, scheduling, uniform machine, unit-time job. 1. Introduction partite subcubic (subquartic) graph is said to be bisubcubic (bi- subquartic). The symbol α(G) stands for the size of a maximum Suppose we have to process n bins of chemical substances independent set in G. We say that a graph is k-colorable if its and we have m parallel uniform machines, i.e. machines that vertices can be partitioned into k independent sets, called color have identical functionality but may have different processing classes, and by a k-coloring of G we mean such a partition of its speeds. On no machine can we process two substances that may vertices. The width of a coloring is the difference between the react with each other in order to guarantee machines safety or sizes of the largest and the smallest color class of the coloring. due to spacial reasons. The aim is such an assignment of the For other definitions the reader is referred to [10]. Conflicts bins to the machines that the processing time of all bins is as between jobs can be modeled by an incompatibility graph, i.e. short as possible or the mean flow time of a bin is as short as by a graph which has exactly one unique vertex assigned to possible. We assume here that each of the bins has to be pro- each of the jobs and which has an edge between a pair of ver- cessed on exactly one machine without interruptions. tices if the corresponding jobs cannot be processed on the same The problem can be expressed as the following scheduling machine. We say that jobs in a given set are compatible if the problem. Let us assume that we have n identical jobs with unit vertices corresponding to them form an independent set in the execution times. We also have m parallel uniform machines. incompatibility graph. They are uniform in the sense that the execution of a job on Now let us introduce some definitions concerning sched- a machine takes time inversely proportional to the speed of uling theory. Let us denote the set of jobs by J = { j1, …, jn}. Let the machine to be completed. There may be incompatibilities us also denote the set of the machines by M = {M1, …, Mm}. By between some jobs, that is, jobs belonging to incompatible s(M ) we denote the speed of machine M. Without loss of gener- pairs cannot be scheduled on the same machine. We consider ality we assume that s(M ) … s(M ), i.e. that the machines 1 ¸ ¸ m two criteria of optimality: the length of schedule and the total are ordered according to their speeds. By stotal = ∑ M M s(M ) completion time. we denote the sum of the speeds of all machines. A 2schedule Let us recall definitions of graph theory used further in this is for each job an allocation of a time interval to a machine paper. For a graph G = (V, E) by n(G) = V we denote the [2]. A subschedule is a schedule for a restricted set of jobs number of its vertices (nodes) and by e(G) =j Ej the number of J J on a restricted set of machines M M . By n (M ) 0 µ 0 µ S its edges. We denote by ∆(G) the maximum degreej j of a vertex we denote the number of jobs scheduled on machine M in in G. By V0(G) we mean the set of all vertices of a graph G a schedule S. By CS (M ) we denote the latest completion time with degree 0. A graph G is said to be a cubic (quartic) graph of the jobs scheduled on machine M in a schedule S. We have if it is 3-regular (4-regular). If, however, G fulfills ∆(G) 3 C (M ) = n (M )/s(M ). Thus the schedule length is equivalent ∙ S S (∆(G) 4) then it is called subcubic (subquartic). A bipartite to maxC (M ). Symbol Σ (N ) stands for the total completion ∙ S S cubic (quartic) graph is said to be bicubic (biquartic), and a bi- time of jobs from a set N J in a schedule S, therefore the µ total completion time is denoted by ΣS (J). By a greedy as- signment of compatible jobs to machines we understand an *e-mail: [email protected], [email protected] assignment in which the jobs are scheduled one by one and Manuscript submitted 2018-01-04, revised 2018-04-16, initially accepted each job is assigned to the machine that guarantees the shortest for publication 2018-05-08, published in February 2019. completion time at the moment of scheduling this job. In the Bull. Pol. Ac.: Tech. 67(1) 2019 31 T. Pikies and M. Kubale following algorithms all the assignments of jobs to machines 2. 2-coloring with maximal coloring width should be performed greedily, e.g. using an Algorithm given in [3]. We present a simple Algorithm that constructs a 2-coloring of For a set of the machines K M by an ideal machine M we a bipartite graph with maximal coloring width. µ id mean a machine with the speed equal to ∑M K s(M )Tytus, thatPikies, can Marek Kubale ignore incompatibilities between jobs. By C (2N) we denote the id Algorithm 1 Non-Equitable Coloring latestN completionJ scheduled time on of the the ideal jobs machine from setM Nid , hence J scheduledCid(N)= Algorithm 1 Non-Equitable Coloring ⊆ µ Input: A bipartite graph G. onN the/∑ idealM K smachine(M). By MΣidid, (henceN) we C denoteid(N ) = the N total/∑M completion K s(M ). Input: A bipartite graph G. | | ∈ j j 2 Bytime Σ of(N jobs) we from denote a set theN totalJ completionscheduled on time the idealof jobs machine from Output:Output: AA two-coloring two-coloring with with maximal coloring width id ⊆ a Msetid ,N hence JΣ idscheduled(N)=0.5 Non( theN + ideal1)/∑ machineM K s(M ).M , hence 1. Col1 = Col2 = /0 µ | | | | ∈ Tytusid Pikies, Marek Kubale1 2 Σid(ThereN) = 0.5 areN several( N papers+ 1)/∑ addressingM K s(M ) the. problem of schedul- 2. Let G1, …, ,...,GGc be beconnected connected components components of ofG G. j j j j 2 1 c ingThere with are incompatible several papers job constraintsaddressing the on identicalproblemTytus of machines. sched Pikies,- Marek3. Kubalei = 1 c N J scheduled on the ideal machine Mid, hence Cid(N)= Algorithm3. for i = 1 1 Non-Equitabletoto c dodo Coloring ulingHowever,⊆ with incompatible to the best of job our constraints knowledge, on thereidentical are fewmachines. results 4. Split V(G ) into color classes Col (G ) and Col (G ), N /∑ s(M). By Σid(N) we denote the total completion A bipartiteV( i) graph G. (1 )i (2 ) i N JM scheduledK on the ideal machine Mid, hence Cid(N)= Input:Algorithm4. Split 1 Non-EquitableGi into color Coloringclasses Col1 Gi and Col2 Gi , However,|involving⊆| ∈ to uniform the best machines. of our knowledge, The problem there isare hard few even results for where Col1(Gi) Col2(Gi) . timeN / of jobss( fromM). a By setΣN (NJ) wescheduled denote theon the total ideal completion machine Output:whereA two-coloring |Col (G )|≥| with Col maximal(G )| . coloring width identical∑M K machines. Theid authors⊆ of [1] have proved that even if Input:5. ColA bipartite= Col1 graphColi G(G. ). 2 i involvingM| id| , hence∈ uniformΣid(N)= machines.0.5 N ( NThe+ 1problem)/∑ sis( Mhard). even for 1. =1 j =1 j ¸1 j i j time of jobs from a set N J scheduledM onK the ideal machine Output:5. Col Col1 A Col= two-coloring Col2 ∪/0 Col with(G ) maximal. coloring width identicalthere are machines. only three The identical authors|⊆| | machines| of [1] have and∈ proved the incompatibili- that even 6. Col21 = Col21 [Col21(Gii). M There, hence areΣ several(N)= papers0.5 N addressing( N + 1)/ the problems(M). of schedul- 2. Let G1,...,Gc be∪ connected components of G. tiesid betweenid jobs can be modeled by a∑ bipartiteM K graph then the 1.6.7. Col Col1 =2 Col= Col2 =2 /0 Col2(Gi). ifing there with are incompatible only three identical job| | constraints| |machines on ∈and identical the incompati machines.- 3. endi for= c [ There are several papers addressing the problem of schedul- 2. forLet G ,...,1 toG dobe connected components of G.