Scheduling with Contact Restrictions -- A Problem Arising in Pandemics
Moritz Buchem, Linda Kleist, Daniel Schmidt genannt Waldschmidt
SScheduling with Contact Restrictions –A Problem Arising in Pandemics
Moritz Buchem ! School of Business and Economics, Maastricht University, The Netherlands
Linda Kleist ! Department of Computer Science, TU Braunschweig, Germany
Daniel Schmidt genannt Waldschmidt ! Institut für Mathematik, TU Berlin, Germany
Abstract
We study a scheduling problem arising in pandemic times where jobs should keep sufficient distanceduring transit times to machines. To model this, each job has a transit time before and afterits processing time, i.e., three parameters. We seek conflict-free schedules in which the transittimes of no two jobs on machines in close proximity intersect. More formally, for the problem
SchedulingWithContactRestrictions ( scr ), we are given a set of jobs, a set of machines, anda conflict graph on the machines. The goal is to find a conflict-free schedule of minimum makespan.We show that, unless P = NP, the problem does not allow for a constant factor approximationeven for identical jobs and every choice of fixed positive parameters, including the unit case. However,given an oracle for a maximum collection of disjoint independent sets , we provide a 2-approximationalgorithm for identical jobs.Moreover, we present optimal and near optimal schedules for unit jobs on several graph classes.For bipartite graphs, we give a / -approximation and an asymptotic / -approximation. Furthermore,we solve the problem on complete graphs, complete bipartite graphs, and traceable bipartite graphsin polynomial time. For trees, we compute schedules with highest load density yielding near optimalsolutions. Theory of computation → Scheduling algorithms; Mathematics ofcomputing → Combinatorial optimization; Theory of computation → Packing and covering problems
Keywords and phrases
Scheduling, conflict graph, conflicting machines, maximum independent set,maximum induced c-colorable subgraph, approximation algorithm, NP-hard, inapproximability,pandemic, bipartite graph, tree, load density
Funding
Daniel Schmidt genannt Waldschmidt was funded by the Deutsche Forschungsgemeinschaft(DFG, German Research Foundation) under Germany’s Excellence Strategy – The Berlin MathematicsResearch Center MATH+ (EXC-2046/1, project ID: 390685689).
Acknowledgements
We thank Tjark Vredeveld for helpful comments concerning the presentationof this manuscript.
In pandemic times, prohibiting close proximity is crucial to control the spread of the virus.This raises various interesting questions of scheduling under contact restrictions.To study these questions, we generalize the problem of makespan minimization on identicalparallel machines. The scheduler is faced with potentially infectious jobs which should keepsufficient distance to each other, because its usually unknown whether a job is infectious.We are particularly interested in the situation when the actual processing of jobs on themachines may be achieved in isolation, whereas, before and after being processed, the jobscould potentially be in contact during their transits. Here, the proximity of the machinescome into play. To give a concrete example, consider a testing center in which patients a r X i v : . [ c s . D M ] F e b Scheduling with Contact Restrictions (a)
Test center with path-ways and three test rooms. (b)
Conflict graph (c)
Schedule for path P with seven unit jobs;transit and processing times are displayed indark and light gray, respectively. Figure 1
A scheduling problem under contact restrictions. need to move towards their designated room, receive the test and leave again. During thesetransits, patients may potentially meet, whereas, during the test itself, they are isolated fromeach other in their respective rooms. Figure 1(a) depicts a testing center with three testrooms; the proximity relations are captured by a path on three vertices, see Figure 1(b). Weassume that each patient needs (at most) a fixed amount of transit time and testing time; inthis example one time unit each. Figure 1(c) presents a schedule for seven patients such thatno two of them are in close proximity.In the context of infectious jobs , similar problems arise in vaccination centers or whenscheduling shifts in factories, warehouses, and office buildings. Additionally, optimizing undercontact restrictions plays a crucial role when jobs may have private information or data thatshould not be shared; e.g., the interrogation of suspects in multiple rooms.
The problem
To model these situations we introduce the concept of scheduling with contactrestrictions. The problem
SchedulingWithContactRestrictions ( scr ) is a schedulingproblem in which jobs on conflicting machines are processed such that certain blockingintervals of their processing time do not overlap. The objective is to minimize the makespan.To the best of our knowledge the concept of conflicting machines has not been considered inthe literature.An instance of scr is defined by a set of jobs J and a conflict graph G = ( M , E ) ona set of machines M where two machines i and i ′ are in conflict if and only if ii ′ ∈ E . Incontrast to classical scheduling problems, each job j has three parameters ( ↼ b j , p j , ⇀ b j ), where ↼ b j and ⇀ b j denote the first and second blocking time of j , respectively, and p j denotes its processing time . Together they constitute the system time q j = ↼ b j + p j + ⇀ b j ; note that theorder ↼ b j , p j , ⇀ b j must be maintained. We seek schedules in which the blocking times of no twojobs on conflicting machines intersect. Formally, a (conflict-free) schedule Π is an assignmentof jobs to machines and starting times such thatfor each point in time, every machine executes at most one job, i.e., the intervals of thesystem times of two jobs assigned to the same machine do not overlap interiorly,for every edge ii ′ ∈ E and two distinct jobs j, j ′ ∈ J assigned to machines i and i ′ ,respectively, the intervals of the blocking times of j and j ′ do not overlap interiorly.In particular, every job j has a starting time S Π j , and a completion time C Π j = S Π j + q j . Wesay a job is running at every point in time in the open interval ( S Π j , C Π j ). We define the makespan of Π as ∥ Π ∥ := max j ∈J C Π j and seek for a schedule with minimum makespan, i.e.,the objective is min Π ∥ Π ∥ . For an example of an optimal schedule on the path P with sevenunit jobs, consider Figure 1(c).Throughout this paper, we use n := |J | and m := |M| to refer to the number of jobs andmachines, respectively. We highlight that, by definition, all schedules are non-preemptive. . Buchem, L. Kleist and D. Schmidt genannt Waldschmidt 3 We provide a spectrum of new results for questions arising from scheduling under contactrestrictions.The problem scr generalizes the classical problem of makespan minimization on parallelidentical machines P || C max .In Section 2, we consider instances with long blocking times and show that scr reducesto P || C max on a maximum independent set of the conflict graph. This relationship to themaximum independent set problem implies that no constant factor approximation exists.In Section 3, we strengthen this insight by showing that there exists no constant factorapproximation, even in the case of identical jobs for all positive fixed parameters ( ↼ b , p, ⇀ b )(Theorem 3.3); this includes the unit case (1 , , c -colorable subgraph (max c -IS), we present a 2-approximation foridentical jobs (Theorem 3.4); similarly, given a / γ -approximate max c -IS-oracle, weprovide a 4 γ -approximation (Theorem 3.5).In Section 4, we investigate unit jobs. For bipartite graphs, the above mentionedapproximation has a performance guarantee of / ; we additionally provide an asymptotic / -approximation (Theorem 4.3). Moreover, we present optimal and near optimalschedules for several graph classes. We show that optimal schedules for complete bipartitegraphs and traceable bipartite graphs consist of two simple patterns and can, therefore,be computed in polynomial time (Theorem 4.4). In contrast, we show that optimalschedules of trees are more intricate and compute schedules with highest load densityyielding near optimal solutions for all number of jobs (Corollary 4.12).Adding to polynomial algorithms for bipartite graphs, we present a polynomial timealgorithm to compute optimal schedules for identical and symmetric jobs with longprocessing time (Theorem 5.1). Our problem scr generalizes the classical scheduling problem of makespan minimization onparallel identical machines , also denoted by P || C max . There are two possible options to viewthis generalization because P || C max is equivalent to scr if the blocking times of all jobs vanish, i.e., ↼ b j = ⇀ b j = 0 for all j ∈ J , orif the edge set of the conflict graph is the empty set.For a constant number of machines, the problem P || C max is weakly NP-hard, while it isstrongly NP-hard when m is part of the input [13]. Graham [14, 15] introduced list schedulingalgorithms to obtain the first constant approximation algorithms for this problem. Improvedapproximation guarantees have been achieved by a fully polynomial time approximationscheme (FPTAS) when m is constant [27] and a polynomial time approximation scheme(PTAS) when m is part of the input [19]. In subsequent work, the latter has been improvedto efficient polynomial time approximation schemes (EPTAS) in subsequent work, we referto [1, 8, 18, 20, 21]. Scheduling with conflict graphs has been investigated when conflicts between jobs arespecified. To this end, one distinguishes two types of restrictions: conflicting jobs cannot be scheduled on the same machine ( jc-intra ), or conflicting jobs cannot be processed concurrently on different machines ( jc-inter ).For jc-intra , Bodlaender and Jansen [5] show NP-hardness even for unit time jobs onbipartite graphs and co-graphs. Bodlaender et al. [7] present approximation algorithms for Scheduling with Contact Restrictions bipartite graphs and graphs with bounded treewidth. Das and Wiese [9] consider jc-intra where the conflict graph is a collection of cliques and introduce a PTAS for identical machines. jc-inter has been studied under various names: mutual exclusion scheduling problem,scheduling with conflicts and scheduling with agreements where an agreement graph is thecomplement of a conflict graph. For jc-inter for jobs with unit processing time (MES),Baker and Coffman [2] showed that MES is NP-hard for general graphs and a fixed numberof m ≥ G with the minimum number of colorssuch that each color appears at most m times. The computational complexity of MES hasbeen investigated for various graph classes, see [2, 4, 6, 10, 12, 16]. For bipartite graphs,MES remains NP-hard [6], while it becomes polynomial time solvable for a fixed number ofmachines [2, 6, 16]. Even et al. [11] consider jc-inter for non unit jobs on two machineswith few job types and show that it can be solved (approximately) when p j ∈ { , , } .When p j ∈ { , , , } , jc-inter is APX-hard even for bipartite conflict graphs [11]. Forcomplements of bipartite graphs, Bendraouche and Boudhar [3] showed that jc-inter isNP-hard even with a restricted number of job types. Mohabeddine and Boudhar [26] showedthat jc-inter is NP-hard on complements of trees, while it is solvable in polynomial timefor complements of caterpillars or cycles. In this section, we identify several special cases of scr which reduce to the problem of(classical) makespan minimization on a maximum independent set of identical machines.The common denominator of these cases is the property that each job has a "long blockingtime". These long blocking times lead to so-called basic schedules in which jobs on conflictingmachines do not run in parallel. To this end, we introduce some notation in a general formin view of the next sections. All omitted proofs of this section are presented in Appendix B.
Maximum c -colorable subgraph For a grah G = ( V, E ) be a graph and c ∈ N ≥ , a(maximum) induced c -colorable subgraph , or short (maximum) c -IS , of G is a set of c disjointindependent sets I , . . . , I c ⊆ V (whose union has maximum cardinality). We denote thecardinality of a maximum c -IS of G by α c ( G ); we also write α c if G is clear from the context.Clearly, a 1-IS is an independent set. For c = 1, it is well known that, unless P = N P ,there exists no constant factor approximation algorithm for maximum 1-IS [13, 17]. Lundand Yannakakis [25] show that the inapproximability of maximum 1-IS translates directly tothe inapproximability of finding c -ISs. Basic schedule
A schedule Π is basic , if for every edge ii ′ ∈ E and for every pair of jobs j and j ′ assigned to i and i ′ , respectively, have non-overlapping system times, i.e., S Π j ≤ S Π j ′ implies C Π j ≤ S Π j ′ .We identify three types of instances, in which all feasible schedules are basic. ▶ Lemma 2.1.
For an instance of scr (G, J ), every schedule Π is basic if any of the threefollowing properties are fulfilled: (i) all jobs j are identical ( ↼ b j = ↼ b , p j = p, ⇀ b j = ⇀ b ) and ( ↼ b > p or ⇀ b > p ) , ( prop-i ) (ii) all jobs j have equal system time ( q j = q ) and ↼ b j > p j and ⇀ b j > p j , or ( prop-ii ) (iii) all jobs j, k fulfill ↼ b j > p k and ⇀ b j > p k . ( prop-iii ) . Buchem, L. Kleist and D. Schmidt genannt Waldschmidt 5 Proof.
Suppose for the sake of a contradiction that there exists two jobs j and j ′ assigned tomachines i and i ′ , respectively, with ii ′ ∈ E such that j ′ starts while j is processed. Becausetheir blocking times cannot overlap, we also know that j ′ starts after the first blocking timeof j and the first blocking time of j ′ ends before the second blocking time of j starts, seeFigure 2. Consequently, ↼ b j ′ ≤ p j . Moreover, p j ′ ≥ ⇀ b j if j ′ ends after j ; in particular, thisholds for equal system times as in (i) and (ii). jj (cid:48) p j p j (cid:48) Figure 2
Illustration of the proof of Lemma 2.1. (i)
Because all jobs have the same parameters, we obtain the contradiction ↼ b = ↼ b j ′ ≤ p j = p and p = p j ′ ≥ ⇀ b j = ⇀ b . (ii) Using the assumptions ↼ b j ′ > p j ′ and ⇀ b j > p j , we obtain ↼ b j ′ > p j ′ ≥ ⇀ b j > p j ≥ ↼ b j ′ , acontradiction. (iii) The assumption ↼ b j ′ > p j yields an immediate contradiction to ↼ b j ′ ≤ p j . ◀ We denote scr restricted to instances fulfilling one of the above three properties by scr-prop-i , scr-prop-ii and scr-prop-iii , respectively. As it turns out, even in the case ofidentical jobs, these problems are hard to solve. ▶ Theorem 2.2.
Unless P = NP, scr-prop-i , scr-prop-ii and scr-prop-iii do not admitconstant factor approximations, even in the case of identical jobs and when ↼ b , p, ⇀ b are fixed. Proof.
By containment, we restrict our attention to instances of scr-prop-ii for n identicaljobs with q = 1, n = α ( G ). By Lemma 2.1 all schedules are basic. Clearly, the optimummakespan opt is 1 as q = 1. Suppose there exists a γ -approximation algorithm for someconstant γ ≥ β denote the maximum numberof machines processing jobs in parallel in Π at any point in time. As we have at most n distinct starting times in Π we can compute β in polynomial time. Because Π is basic, wecan transform Π into a new schedule Π ′ that processes all jobs non-idling on β machineswithout increasing the makespan. Hence, we obtain ∥ Π ∥ ≥ ⌈ n / β ⌉ ≥ n / β . This fact togetherwith the assumption ∥ Π ∥ ≤ γ · opt = γ yields β ≥ n / ∥ Π ∥ ≥ n / γ = / γ · α ( G ). In otherwords, the γ -approximation algorithm implies an / γ -approximation algorithm for computinga maximum 1-IS for every graph G ; a contradiction to the fact that, unless P = N P , aconstant factor approximation does not exist [17]. ◀ As we can see, the hardness of scr-prop-i , scr-prop-ii and scr-prop-ii lies in thehardness of computing a maximum 1-IS for general graphs. However, if we are given amaximum 1-IS, the computational complexity dramatically changes, as already existing exactand approximation algorithms for P || C max can be translated to these problems. ▶ Theorem 2.3.
Given an oracle for maximum -ISs, (i) scr-prop-i can be solved in polynomial time, (ii) scr-prop-ii can be solved in polynomial time, and (iii) scr-prop-iii has a PTAS. In case we are given an approximate 1-IS, we obtain the following approximation resultsbased on a list scheduling approach.
Scheduling with Contact Restrictions ▶ Theorem 2.4.
Given an oracle for / γ -approximate 1-ISs, there exists (i) a ⌈ γ ⌉ -approximation for scr-prop-i , (ii) a ⌈ γ ⌉ -approximation for scr-prop-ii , and (iii) a ( γ + 1 − / m ) -approximation for scr-prop-iii , The last approximation generalizes the (2 − / m )-approximation by Graham [14]. In this section, we consider scr restricted to the case when all jobs are identical, which wedenote by scr-Id . Moreover, scr-Id ( G, n, ( ↼ b , p , ⇀ b )) denotes an instance with conflict graph G and n identical jobs with parameters ( ↼ b , p, ⇀ b ). Complementary to the last section, we focus onidentical jobs with short blocking times in which every instance scr-id-short ( G, n, ( ↼ b , p, ⇀ b ))has n identical jobs with parameters ( ↼ b , p, ⇀ b ) where ↼ b ≤ p and ⇀ b ≤ p . We start by introducingschedules which play an important role in this section. All omitted proofs of this section arepresented in Appendix C. c -Pattern Consider a graph G = ( M , E ) and a set of identical jobs with parameters ( ↼ b , p, ⇀ b ),where ↼ b ≥ ⇀ b and q = ↼ b + p + ⇀ b . Let c ∈ N ≥ with c ≤ ⌊ p / ↼ b ⌋ + 1 and let I = ( I , I , . . . , I c )be a c -tuple of disjoint independent sets of G . A partial schedule of length q + ( c − · ↼ b starting at time t has a c -pattern on I if on each machine i in I k , there is one job starting attime t + ( k − ↼ b . For an illustration consider Figure 3. We say that a schedule contains a pure c -pattern on I if in this interval no other job is present. I I I Figure 3
A 3-pattern
In order to show connections between conflict-free schedules and c -ISs, we extract a c -ISfrom a conflict-free schedule. We say a job j blocks a time t in schedule Π if one of itsblocking times contains t , i.e., t ∈ (cid:0) ( S Π j , S Π j + ↼ b ) ∪ ( C Π j − ⇀ b , C Π j ) (cid:1) . For a schedule Π, wedefine the quantity β Π c := max t <... For a schedule Π with n jobs and a constant c , β Π c can be computed in timepolynomial in n . The definition of β Π c helps us to lower bound the makespan of a policy Π. ▶ Lemma 3.2. A schedule Π of scr-id-short ( G, n, ( ↼ b , p, ⇀ b )) with ↼ b ≥ ⇀ b > has makespan ∥ Π ∥ ≥ q · ⌈ n / β Π k +1 ⌉ , where k := ⌈ p / ↼ b ⌉ . . Buchem, L. Kleist and D. Schmidt genannt Waldschmidt 7 Proof-Sketch. The proof is twofold. First, we show that the number of jobs starting withinan interval of length q is at most β Π k +1 (Lemma C.1). The idea is to divide the intervalinto specific subintervals of length at most ↼ b . Then, the jobs starting within a subintervalblock its right endpoint. Secondly, an upper bound on the number of jobs starting within aninterval of specific length implies a lower bound on the makespan (Lemma C.2). ◀ Using the fact that computing a max c -IS is NP-hard to approximate [25], we present aninapproximability result for the case of identical jobs for all fixed constants ↼ b , p, ⇀ b . ▶ Theorem 3.3. Unless P = NP, there exists no constant factor approximation for scr-Id even for any choice of fixed positive parameters ( ↼ b , p, ⇀ b ) . Proof. By Theorem 2.2 it suffices to restrict to the case in which the blocking timesare short. Suppose for a contradiction that there exists a γ -approximation A for someconstant γ ≥ 1. We assume ↼ b ≥ ⇀ b and define the constant k := ⌈ p / ↼ b ⌉ . For an instance scr-id-short ( G, n, ( ↼ b , p, ⇀ b )) with n ≥ α k +1 , we consider the schedule Π computed by A .Because A is a γ -approximation and by Lemma 3.2, the schedule Π computed by A hasmakespan ∥ Π ∥ ≤ γ · OP T and ∥ Π ∥ ≥ q · ⌈ n / β Π k +1 ⌉ ≥ qn · / β Π k +1 . Recall that, by Lemma 3.1,we can compute a ( k + 1)-IS from Π of size β Π k +1 in polynomial time.Moreover, we obtain a feasible schedule by repeatedly using pure ( k + 1)-patterns on amaximum ( k + 1)-IS. Recall that a ( k + 1)-pattern has length ( q + ⌊ p / ↼ b ⌋ ↼ b ) ≤ ( q + k ↼ b ) andschedules α k +1 jobs. This yields the upper bound opt ≤ ( q + k ↼ b ) ·⌈ n / α k +1 ⌉ ≤ q + k ↼ b ) · n / α k +1 , where the last inequality uses the fact n ≥ α k +1 . All together we obtain β Π k +1 ≥ qn ∥ Π ∥ ≥ qnγ · opt ≥ γ · q ( q + k ↼ b ) · α k +1 , where q / q + k ↼ b is a constant ≤ 1. By Lemma 3.1, a constant factor approximation algorithm forthe scheduling problem would imply a constant factor approximation algorithm for maximum( k + 1)-IS contradicting [25]. ◀ In contrast, we obtain a constant factor approximation for (general) identical jobs if weare given maximum c -ISs. ▶ Theorem 3.4. Given an oracle for maximum c -ISs for every c , scr-Id with ↼ b ≥ ⇀ b > allows for a (cid:0) ⌊ p / ↼ b ⌋ · ↼ b / q (cid:1) -approximation, where (cid:0) ⌊ p / ↼ b ⌋ · ↼ b / q (cid:1) < . Proof. By Theorem 2.3, we may focus on instances with short blocking times.Consider an instance scr-id-short ( G, n, ( ↼ b , p, ⇀ b )) and let k = ⌈ p / ↼ b ⌉ . In order to constructa schedule Π, we use ⌈ n / α k +1 ⌉ many pure ( k + 1)-patterns on a maximum ( k + 1)-IS I of G , yielding a makespan of ∥ Π ∥ ≤ ( q + ⌊ p / ↼ b ⌋ ↼ b ) · ⌈ n / α k +1 ⌉ . By Lemma 3.2, it holds that opt ≥ q · ⌈ n / α k +1 ⌉ . Moreover, note that ⌊ p / ↼ b ⌋ · ↼ b / q < (cid:0) ⌊ p / ↼ b ⌋ · ↼ b / q (cid:1) < ◀ Similarly, we provide an approximation when an approximate c -IS is given. ▶ Theorem 3.5. Given an oracle for / γ -approximate c -ISs for every c , scr-Id allows for a γ -approximation. In this section, we consider the special case scr-Unit in which we are given n identical unitjobs with ↼ b = p = ⇀ b = 1 for all jobs. By Theorem 3.3, scr-Unit does not admit a constantfactor approximation algorithm, unless P = N P . In the following, we present improved Scheduling with Contact Restrictions approximation algorithms for special graph classes or solve the problem in polynomial time.We begin with the simple case of complete graphs. All omitted proofs of this section arepresented in Appendix D. ▶ Observation 4.1. scr-Unit on a complete graph K m with n jobs can be solved in lineartime. In particular, for m ≥ , it coincides with an optimal schedule for K . In the remainder we focus on bipartite graphs. At several places, we use the fact thatwhen all parameters are integers, there exists an optimal schedule with integral startingtimes (see Lemma D.1). For bipartite graphs, Theorem 3.4 yields a / -approximation for scr-Unit . In the following,we present an improvement by deriving a more precise lower bound on the minimum makespan.To this end, we introduce a measure of productivity. The load density of schedule Π for scr ( G, J ) is defined as the average load ρ (Π) := P j ∈J q j m ∥ Π ∥ . In the case of unit jobs this simplifies to n / m ∥ Π ∥ . Clearly, the maximum load density yieldsa simple lower bound on the makespan. ▶ Observation 4.2. Let G be a graph and ρ ∗ be an upper bound on the load density of eachschedule on G for any number of jobs. Then, the makespan of every schedule Π for n unitjobs is ∥ Π ∥ ≥ n / m · / ρ ∗ . With the help of Observation 4.2, we derive a lower bound of n / m + α on the makespanfor scr-Unit on bipartite graphs. To obtain an upper bound, we construct a schedule Πas follows. If α / m ≤ / , Π consists of pure 2-patterns using all machines. If α / m > / , Πconsists of pure 1-patterns on a maximum independent set. ▶ Theorem 4.3. For every instance scr-Unit ( G, n ) where G is bipartite, we can compute aschedule Π with makespan ∥ Π ∥ ≤ ( / · opt + 4) in polynomial time. We now identify more interesting cases of bipartite graphs that can be solved efficiently. Agraph is called traceable if it contains a Hamiltonian path. ▶ Theorem 4.4. scr-Unit for complete bipartite and traceable bipartite graphs can be solvedin time polynomial in log n and m . To prove Theorem 4.4, we use two nice facts. Firstly, if optimal schedules are guaranteedto be composed of a constant number of subschedules, we can compute an optimal schedulein polynomial time with help of Lenstra’s algorithm [22, 24]. By a prefix of a schedule, wedenote a subschedule for which a set of last jobs is deleted. ▶ Proposition 4.5. Consider an instance scr-Id (G,n,( ↼ b , p, ⇀ b )). Let Π , . . . , Π c be scheduleson G of length ℓ , . . . , ℓ c for n , . . . , n c jobs, respectively. If an optimal schedule for n jobsconsists of the subschedules Π , . . . , Π c (where one subschedule may be a prefix), then anoptimal schedule can be computed in time O (max { n , . . . , n c } · log(max { ℓ , . . . , ℓ c , n } )) . . Buchem, L. Kleist and D. Schmidt genannt Waldschmidt 9 Secondly, to complete the proof of Theorem 4.4, we show that optimal schedules on theconsidered graphs consist of pure 1- or 2-patterns. ▶ Lemma 4.6. For every instance scr-Unit ( G, n ) , where G is either (i) complete bipartiteor (ii) bipartite and traceable, there exists an optimal schedule consisting only of the pure1-pattern on a maximum -IS of G and/or the pure 2-pattern on the two classes of thebipartition of G (where one subschedule may be a prefix). Proof-Sketch. We iteratively consider the machines with jobs starting in an interval oflength 4 starting with the very first; M i denotes the set of machines with a job starting attime i . Because the schedule is conflict-free, some jobs can start earlier, see Figures 4(a)and 4(b). This enables us to find pure 1- and 2-patterns in many cases. In a last step, weexploit graph properties of complete bipartite or traceable bipartite graphs explicitly to makefurther modifications to obtain the desired structure without increasing the makespan. ◀ M ∩ M M M M \ M M \ M (cid:102) M (a) Jobs on M and M \ M can beshifted by 2 time units. (cid:102) M (b) If M \ M = ∅ , jobson M can be shifted. M M ∩ M (cid:102) M M \ M (c) Conflict relationsbetween machine sets Figure 4 Illustration for the proof of Lemma 4.6. Using the fact that a maximum independent set of a bipartite graph can be computed inpolynomial time [23], Proposition 4.5 and Lemma 4.6 imply Theorem 4.4.For later reference, we note the following implications from Lemma 4.6. ▶ Corollary 4.7. For every instance of scr-Unit ( G, n ) , an optimal schedule Π for G where G is a path P m , m ≥ , has load density ρ (Π) ≤ / ,where G is a complete bipartite graph K m ,m , has load density ρ (Π) ≤ max { m / m , m / m , / } . In contrast to complete bipartite graphs and tracable bipartite graphs, optimal schedules donot always consist of the pure 1- or 2-pattern of a graph. ▶ Observation 4.8. For scr-Unit ( G, n ) , there exist a tree T such that a schedule withhighest load density does not consist of the 1- or 2-patterns of T . In particular, for all n large enough, no optimal schedule consists of the 1- and 2-pattern. Proof. Consider the tree and schedule illustrated in Figure 5. The load density is · / · = / and thus exceeds the density / of the 1-pattern and the density / of the 2-pattern. ◀ The trick of the above example is that a machine incident to many leaves remains idle.Thus, we define the leaf degree of a non-leaf as the number of adjacent leaves. The leafdegree of a graph is the maximum leaf degree over all its non-leaves.When seeking schedules of high load density, the following lemma allows us to concentrateon schedules where vertices with high leave degree are idle. Figure 5 ▶ Lemma 4.9. Let G be a graph with a vertex v of leaf degree at least 3. For every schedule Π ,there exists a schedule Π ′ (not necessarily for the same number of jobs) in which v is idleand ρ (Π ′ ) ≥ ρ (Π) . Proof. Let d denote the leaf degree of v and V ′ the vertex set consisting of v and its leaves.We consider the two disjoint subgraphs G := G [ V \ V ′ ] and G := G [ V ′ ].For G , let S be the schedule on G induced by Π with load density ρ . Let S bea schedule consisting of the pure 1-pattern on the leaves of v with density ρ ( S ) = d / d +1 ;Corollary 4.7 implies that S is a schedule of highest density for G = K ,d . In particular, v is idle. Consequently, we may concatenate three copies of S for the machines in G and ∥ Π ∥ copies of schedule S for the machines in G to obtain a schedule Π ′ in which v is idle.Because the load density on G remains unchanged and it does not decrease on G , it followsthat ρ (Π ′ ) ≥ ρ (Π). Figure 6 illustrates the resulting schedule. ◀ S S S S vG Figure 6 Illustration of Lemma 4.9 and its proof. ▶ Proposition 4.10. Let T be a tree on at least two vertices with leaf degree at most 2. Then,in each schedule processing n jobs, the load density on any interval of length ≥ is upperbounded by / . The 2-pattern has a highest load density. Proof. We partition T into a set of paths and stars on 4 vertices by deleting edges from T with the following steps:Firstly, choose an arbitrary non-leaf as the root r of T and orient all edges away from r .We will use the fact that each vertex has at most one incoming edge, see Figure 7(a).Secondly, we delete a set of edges. To this end, call an edge of T strong if its twovertices are non-leaves. If a vertex has degree at least 3, then delete all incident strong edges.This results in graph where each vertex has degree at most 2, see Figure 7(b). Thus, eachcomponent consists of a single vertex or a path on an least two vertices. Moreover, if a vertexhas out-degree 2, then its successors are leaves. . Buchem, L. Kleist and D. Schmidt genannt Waldschmidt 11 Thirdly, we re-insert some edges in case of isolated vertices. For an illustration considerFigure 7(c). For an isolated vertex v with smallest distance to r , we re-insert one designatedoutgoing edge. If each vertex has outdegree ≤ 1, the resulting component is P m for some m ≥ 2; otherwise the resulting component is the star K , . Here we use the fact that if avertex had outdegree 2, then its successors are leaves. We repeat this step if there exists anisolated vertices.Consequently, this procedure results in a set of paths and stars K , . By Corollary 4.7,the density of every schedule on the subgraphs is upper bounded by / . ◀ (a) A rooted tree. (b) Deletion of strong edges. (c) Re-insertion of edges. Figure 7 Illustration of the proof of Proposition 4.10. ▶ Theorem 4.11. For scr-Unit where G is a tree, there exists a schedule Π of highest loaddensity with length 12, which can be computed in polynomial time. Proof. Let U be the set of vertices of G with leaf degree at least 3 and G := G [ M \ U ]the graph induced by the vertices not in U . We iteratively define U k +1 as the set of verticesin G k with leaf degree at least 3 and G k +1 := G [ M \ S j ≤ k U j ] until all vertices have leafdegree of at most 2. We then define U = S U k , S as the set of isolated vertices in the graphof the last iteration, R := M \ ( U ∪ S ) and G ′ := G [ R ].By Lemma 4.9, we may restrict our attention to schedules in which all vertices in U areidle. We obtain a feasible schedule Π (of length 12) by assigning the 2-pattern on the twoparts of a bipartition of G ′ (3 times) and the 1-pattern on S (4 times), see Figure 8 for aschematic illustration. By Proposition 4.10, the 2-pattern has highest load density for everycomponent of G ′ . This implies that Π is a schedule with highest load density. ◀ RUS Figure 8 A schematice schedule of highest load density for a tree. With the help of these schedules of highest load density, we provide near optimal schedulesfor all numbers of jobs. ▶ Corollary 4.12. For every instance scr-Unit ( G, n ) where G is a tree, we can compute aschedule Π with ∥ Π ∥ ≤ opt + 2 in polynomial time. In this section, we consider scr-Id with symmetric blocking times, i.e., b := ↼ b = ⇀ b , denotedby scr-id-sym . While we considered the case p < b in Section 2 and the unit case p = b inSection 4, we now focus on p ≥ b and show that scr-id-sym can be solved in polynomialtime for bipartite graphs. ▶ Theorem 5.1. For bipartite graphs, scr-id-sym with p ≥ b is polynomial time solvable. In order to prove Theorem 5.1, we use schedules of special structure. By scaling, we mayassume without loss of generality that b = 1. Consider an instance scr-id-sym ( G, n, (1 , p, p ≥ G , and let I , I be two disjoint independent sets with | I | ≥ | I | and m ′ := | I | + | I | . Moreover, let µ := ⌈ n / m ′ ⌉ . A brick schedule on I and I hasthe following structure:for each machine i ∈ I and all z ∈ { , . . . , µ } , a job starts at time z · q ,for each machine i ∈ I and all z ∈ { , . . . , µ } , a job starts at time z · q + 2.To ensure that the correct number of jobs, we delete the last µ · m ′ − n jobs. Figure 9 displaysan example of a brick schedule. I I Figure 9 Example of a brick schedules on two independent sets of machines. Proof-Sketch for Theorem 5.1. Given a maximum independent of the conflict graph G andthe maximum independent set in the remaining subgraph, we show an optimal schedule iseither a brick schedule or a combination of pure 1- and 2-patterns using the partition classesor the maximum independent set(s). A full proof is presented in Appendix E. ◀ As the efficient allocation of scarce resources with contact restrictions is an optimization taskthat has become even more essential in pandemic times, we introduce and study the problem scr . Surprisingly and to the best of our knowledge, the problem has not been studied before.We show that the problem does not allow for constant factor approximations even in thecase of unit jobs and present approximation and exact algorithms for special graph classes.Various interesting questions remain open for future research. In particular, consideringgraph classes capturing geometric information might be of special interest for applications.We are positive that some of our insights and tools prove to be useful. References Noga Alon, Yossi Azar, Gerhard J. Woeginger, and Tal Yadid. Approximation schemes forscheduling on parallel machines. Journal of Scheduling , 1(1):55–66, 1998. Brenda S. Baker and Edward G. Coffman Jr. Mutual exclusion scheduling. TheoreticalComputer Science , 162(2):225–243, 1996. Mohamed Bendraouche and Mourad Boudhar. Scheduling jobs on identical machines withagreement graph. Computers & Operations research , 39(2):382–390, 2012. 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Clique is hard to approximate within 1- ε . Acta Mathematica , 182(1):105–142,1999. Dorit S. Hochbaum. Various notions of approximations: Good, better, best and more. Approximation algorithms for NP-hard problems , 1997. Dorit S. Hochbaum and David B. Shmoys. Using dual approximation algorithms for schedulingproblems theoretical and practical results. Journal of the ACM , 34(1):144–162, 1987. Klaus Jansen. An EPTAS for scheduling jobs on uniform processors: using an MILP relaxationwith a constant number of integral variables. SIAM Journal on Discrete Mathematics ,24(2):457–485, 2010. Klaus Jansen, Kim-Manuel Klein, and José Verschae. Closing the gap for makespan schedulingvia sparsification techniques. Mathematics of Operations Research , 45(4):1371–1392, 2020. Ravi Kannan. Minkowski’s convex body theorem and integer programming. Mathematics ofOperations Research , 12(3):415–440, 1987. Denis Kőnig. Gráfok és mátrixok. Matematikai és Fizikai Lapok , 38:116–119, 1931. Hendrik W. Lenstra Jr. Integer programming with a fixed number of variables. Mathematicsof Operations Research , 8(4):538–548, 1983. Carsten Lund and Mihalis Yannakakis. The approximation of maximum subgraph problems. In International Colloquium on Automata, Languages, and Programming , pages 40–51. Springer,1993. Amine Mohabeddine and Mourad Boudhar. New results in two identical machines schedulingwith agreement graphs. Theoretical Computer Science , 779:37–46, 2019. Sartaj K. Sahni. Algorithms for scheduling independent tasks. Journal of the ACM (JACM) ,23(1):116–127, 1976. B Details of Section 2 ▶ Theorem 2.3. Given an oracle for maximum -ISs, (i) scr-prop-i can be solved in polynomial time, (ii) scr-prop-ii can be solved in polynomial time, and (iii) scr-prop-iii has a PTAS. Proof. Let Π be an optimal schedule for one of the three problems. By Lemma 2.1 Π is basic,hence, at any point in time at most α ( G ) jobs are running. Therefore, we can modify thejob-to-machine assignment of Π such that all jobs are processed on a maximum 1-IS whilemaintaining the starting times of the jobs. As we did not increase the makespan, we havean optimal schedule where all jobs are assigned to a maximum 1-IS and hence, the problemreduces to P || C max on α ( G ) machines. Thus, evenly distributing all identical jobs on themachines (for scr-prop-i and scr-prop-ii ) and an already existing PTAS for P || C max (for scr-prop-iii ) yields the desired result. ◀▶ Theorem 2.4. Given an oracle for / γ -approximate 1-ISs, there exists (i) a ⌈ γ ⌉ -approximation for scr-prop-i , (ii) a ⌈ γ ⌉ -approximation for scr-prop-ii , and (iii) a ( γ + 1 − / m ) -approximation for scr-prop-iii , Proof. We consider an instance of scr-prop-i or scr-prop-ii . Without loss of generalitywe assume q = 1. Let I be a / γ -approximate 1-IS given by the oracle. An optimal scheduledistributes all jobs evenly over a maximum 1-IS, i.e., we have opt = ⌈ n / α ⌉ as q = 1. Sinceeach system time is 1, we can find a schedule Π with makespan ∥ Π ∥ = ⌈ n / |I| ⌉ · q ≤ ⌈ γ · n / α ⌉ ≤⌈ γ ⌉ · ⌈ n / α ⌉ = ⌈ γ ⌉ · opt by distributing all jobs evenly on I .Next, we consider an instance of scr-prop-iii . Let I be a / γ -approximate 1-IS givenby the oracle. In order to construct Π, we use a list scheduling approach on I . To this end,consider the jobs in an arbitrary order and iteratively assign the next job to a machine in I with minimum total completion time. Π is feasible because I is an independent set. Let k bea job ending the latest in Π, i.e., ∥ Π ∥ = C Π k = S Π k + q k . By construction of Π, all machinesin I are processing some job at least up to time S Π k . Therefore, S Π k ≤ / |I| P j ∈J \{ k } q j =( / |I| P j ∈J q j ) − / |I| · q k .As in the proof of Lemma 2.1, there exists an optimal schedule using only α machines.Therefore, the optimal makespan is lower bounded by the average load opt ≥ α P j ∈ J q j .Additionally, using the fact that q k ≤ opt and |I| ≤ m , we obtain the following chain ofinequalities ∥ Π ∥ = S Π k + q k ≤ ( / |I| X j ∈J q j ) + (1 − / |I| ) · q k ≤ α / |I| · opt + (1 − / |I| ) · opt ≤ ( γ + 1 − / m ) · opt ◀ C Details of Section 3 ▶ Lemma 3.1. For a schedule Π with n jobs and a constant c , β Π c can be computed in timepolynomial in n . Proof. The schedule has 4 n event times, namely the starting time of the blocking times andthe processing time and its completion time. For every time t between every two consecutiveevent points, we count the number of machines processing a blocking interval. By definitionof β Π c , it suffices to check O ( n c ) tuples. ◀ . Buchem, L. Kleist and D. Schmidt genannt Waldschmidt 15 ▶ Lemma C.1. For every schedule Π of an instance scr-id-short ( G, n, ( ↼ b , p, ⇀ b )) with p ≥ ↼ b ≥ ⇀ b > and every time t ≥ , the number of jobs starting within the interval I := [ t, t + q ) is at most β Π k +1 , where k := ⌈ p / ↼ b ⌉ . Proof. Because I has length q = ( ↼ b + p + ⇀ b ) and is half-open, at most one job starts oneach machine within I . We partition the interval I into k + 2 disjoint (left-closed, right-open)intervals I , . . . , I k +2 , where the first interval has length ⇀ b , the next ( k − 1) intervals havelength ↼ b , interval I k +1 has length ( p + ↼ b ) − ( ⇀ b + ( k − ↼ b ) < ↼ b and the last interval I k +2 has length ↼ b , see Figure 10. I I . . . I k +2 I I . . . I k +2 Figure 10 Illustration of proof of Lemma C.1 By M ℓ we denote the set of machines that have a job starting in I ℓ . For all ℓ the jobsprocessed on machine M ℓ block a point in time arbitrarily close to the right end of I ℓ .Additionally, every job processed on a machine in M as well as every job processed on M k +2 block a point in time arbitrarily close to the right end of the whole interval I . Thus, thenumber of jobs can be bounded from above by β Π k +1 . ◀▶ Lemma C.2. Let Π be a schedule for an instance scr-id-short ( G, n, ( ↼ b , p, ⇀ b )) with ↼ b ≥ ⇀ b such that in every interval of length L at most β jobs start. Then, for system time q := ↼ b + p + ⇀ b , it holds that ∥ Π ∥ ≥ L ⌊ n / β ⌋ + ( if β divides nq otherwise. Proof. We divide Π into intervals of length L starting with 0. By assumption, at most β jobs start in each interval. Hence, the number of these intervals (where some job is processed)is at least ⌊ n / β ⌋ . Moreover, if β does not divide n , then at least some job (of length q ) startsafter time L ⌊ n / β ⌋ + q . ◀▶ Theorem 3.5. Given an oracle for / γ -approximate c -ISs for every c , scr-Id allows for a γ -approximation. Proof. By Theorem 2.4, it suffices to consider instances with short blocking times.Consider an instance scr-id-short ( G, n, ( ↼ b , p, ⇀ b )) and let k = ⌈ p / ↼ b ⌉ . Moreover, let I bea ( k + 1)-IS of size β ≥ α k +1 / γ given by the oracle. We construct a schedule Π by using ⌈ n / β ⌉ many pure ( k + 1)-patterns on I , yielding ∥ Π ∥ ≤ ( q + ⌊ p / ↼ b ⌋ ↼ b ) · ⌈ n / β ⌉ . Moreover, Lemma 3.2implies opt ≥ q · ⌈ n / α k +1 ⌉ ≥ q · n / α k +1 . If n ≤ β , then also n ≤ α k +1 and we obtain ∥ Π ∥ opt ≤ ( q + ⌊ p / ↼ b ⌋ ↼ b ) · ⌈ n / β ⌉ q · ⌈ n / α k +1 ⌉ = ( q + ⌊ p / ↼ b ⌋ ↼ b ) q < . If n > β , we obtain ⌈ n / β ⌉ ≤ · n / β ≤ γ · n / α k +1 . Consequently, it holds ∥ Π ∥ opt ≤ ( q + ⌊ p / ↼ b ⌋ ↼ b ) · γ · n / α k +1 q · n / α k +1 ≤ γ · (cid:0) ⌊ p / ↼ b ⌋ · ↼ b / q (cid:1) < γ. ◀ D Details of Section 4 We first show a more general result needed to prove the following results. If the job parametersare integral, there exist optimal integral schedules. ▶ Lemma D.1. If ↼ b j , p j , ⇀ b j are integral for all j ∈ J , every feasible schedule Π can betransformed into a feasible schedule Π ∗ such that all starting times S Π ∗ j , j ∈ J , are integraland the makespan does not increase, i.e., ∥ Π ∗ ∥ ≤ ∥ Π ∥ . Proof. We modify Π by defining S Π ∗ j := ⌊ S Π j ⌋ for each job j . It remains to show that Π ∗ isfeasible. Let j and j ′ be two jobs such that two of their blocking times b and b ′ intersect inΠ ∗ . By integrality of Π ∗ and the blocking times, they intersect in at least one time unit.Without loss of generality, we assume that b ′ does not start before b in Π. With slightabuse of notation, S Π b denotes the starting time of the blocking time b in Π. Then, S Π b ′ − S Π b and S Π ∗ b ′ − S Π ∗ b differ by strictly less than 1. Hence, if they intersect in at least 1 time unitin Π ∗ , then they also intersect in Π. Therefore, by feasibility of Π, j and j ′ are scheduled onconflict-free machines. ◀▶ Observation 4.1. scr-Unit on a complete graph K m with n jobs can be solved in lineartime. In particular, for m ≥ , it coincides with an optimal schedule for K . Proof. An optimal schedule for K processes all jobs consecutively.It remains to consider m ≥ 2. Observe that every scheduled job specifies two unit timeintervals in which no other job can start. Moreover, at most one job may start (or end)during the processing time; in particular, the blocking time of one job and the processingtime of the other job must match. Hence, the jobs with overlapping system times fall intopairs and isolated jobs. Consequently, we may schedule all jobs on the same two machines.It is easy to observe, that two jobs with overlapping system time save time. Consequently,an optimal schedule consists of pure 2-patterns on two designated vertices and potentiallyone pure 1-pattern in the end. ◀ D.1 Proof of Theorem 4.3 In order to prove Theorem 4.3, we derive an upper bound on the load density for schedulesfor scr-Unit on bipartite graphs. ▶ Lemma D.2. Consider a schedule Π for an instance scr-Unit (G,n) where G is bipartite.For every interval I of length 4, the number of empty slots in I is at least m − α ) . Proof. Let Π be a feasible schedule. By Lemma D.1 we know that we can restrict ourselvesto schedules in which all starting times are integers. Now, consider some integral time point t and the interval [ t, t + 4]. Let M , M , M denote the set of machines with 0 , , ≥ m i := | M i | . Note that M is an independent set and M hasintersecting blocking times with at least one job on every machine in M . Because the inducedgraph G [ M ] is bipartite, its maximum independent set M ′ has size ≥ / m . Moreover, M ∪ M ′ is an independent set. Consequently, | M | = | M | − | M ∪ M ′ | − | M \ M ′ | ≥ m − α − / m and the number of empty slots is2 m + m ≥ m − α − / m ) + m = 2( m − α ) . ◀ Now, we can show how to find an improved approximation guarantee for scr-Unit onbipartite graphs. . Buchem, L. Kleist and D. Schmidt genannt Waldschmidt 17 ▶ Theorem 4.3. For every instance scr-Unit ( G, n ) where G is bipartite, we can compute aschedule Π with makespan ∥ Π ∥ ≤ ( / · opt + 4) in polynomial time. Proof. Consider an instance scr-Unit (G,n) on a bipartite graph G with m machines and amaximum independent set I of size α . Note that I can be computed in polynomial time [23].By Lemma D.2, every schedule has density at most m − m − α ) / m = m + α / m . Togetherwith Observation 4.2, this yields a lower bound on the makespan of n / m · m / m + α = n / m + α .To construct our schedule Π, we distinguish two cases. If α / m ≤ / , the schedule Πconsists of the pure 2-pattern on the two classes of the bipartition of G . This yields amakespan of ∥ Π ∥ = 4 ⌈ n / m ⌉ ≤ n / m + 4 = / · ( m + α / m ) · ( n / m + α ) + 4 ≤ / · (1 + α / m ) · opt + 4 ≤ / · (1 + / ) · opt + 4 = / · opt + 4 . If α / m > / , the schedule Π consists the pure 1-pattern on I with makespan ∥ Π ∥ = 3 ⌈ n / m ⌉ ≤ n / α + 3 < / · ( m + α / α ) · ( n / m + α ) + 4 ≤ / · (1 + m / α ) · opt + 4 ≤ / · (1 + / ) · opt + 4 = / · opt + 4 . ◀ D.2 Complete Bipartite and Traceable Bipartite Graphs – Proof ofTheorem 4.4 To prove Theorem 4.4, we use two nice facts. ▶ Proposition 4.5. Consider an instance scr-Id (G,n,( ↼ b , p, ⇀ b )). Let Π , . . . , Π c be scheduleson G of length ℓ , . . . , ℓ c for n , . . . , n c jobs, respectively. If an optimal schedule for n jobsconsists of the subschedules Π , . . . , Π c (where one subschedule may be a prefix), then anoptimal schedule can be computed in time O (max { n , . . . , n c } · log(max { ℓ , . . . , ℓ c , n } )) . Proof. We can find the optimum solution by first guessing the number r of jobs in the last(incomplete) subschedule. Clearly, r ≤ max { n , . . . , n c } . Then, for fixed r we formulate thefollowing Integer Linear Program to find the optimal schedule for exactly n − r jobs anddenote this solution by ILP ( n − r ).min c X k =1 ℓ k x k s.t. c X k =1 n k x k = n − rx , . . . , x c ∈ N Using Lenstra’s result for integer programs of fixed dimension [22, 24] we can solve this ILPin time O (log(max { n , . . . , n c , ℓ , . . . , ℓ c , n − r ) } . Next, we determine the subschedule Π k for k ∈ { , . . . , c } in which r jobs can be scheduled in the shortest amount of time, denoted by OP T ( r ). Finally, the optimal schedule for n jobs has makespan min r { ILP ( n − r ) + OP T ( r ) } .This gives a total running time of the algorithm of O (max { n , . . . , n c }· log(max { ℓ , . . . , ℓ c , n } ))because n , . . . , n c ≤ n . ◀▶ Lemma 4.6. For every instance scr-Unit ( G, n ) , where G is either (i) complete bipartiteor (ii) bipartite and traceable, there exists an optimal schedule consisting only of the pure1-pattern on a maximum -IS of G and/or the pure 2-pattern on the two classes of thebipartition of G (where one subschedule may be a prefix). Proof. Let Π be an optimal schedule of a bipartite graph G . By Lemma D.1, we may assumethat Π is integral. We modify Π iteratively to increase the number of 1- and 2-patterns whilemaintaining optimality. This proof is constructive and can be understood as an algorithm.We start with t = 0. For k = 0 , , , 3, let M k denote the set of machines that start toprocess some job at time t + k in Π for k ∈ { , , , } and let f M := M \ S k =0 M k . Notethat M and M might have a non-empty intersection. The optimality of Π implies that M ̸ = ∅ ; otherwise we may shift Π while decreasing the makespan. Furthermore, if M ̸ = ∅ ,the corresponding jobs could start at time t without violating feasibility. Similarly, for everymachine in M \ M , we can shift its job to time t + 1, see also Figure 11(a). Hence, we mayassume that M = M \ M = ∅ .Case I: M = ∅ . Because M = M = ∅ , | M | jobs start at time t and the next jobs startthe earliest at t + 3. Clearly, | M | ≤ α because all blocking times coincide. If | M | = α ,then Π on [ t, t + 3] has a 1-pattern. If | M | < α , we substitute Π on the interval [ t, t + 3] bythe 1-pattern and increase the number of scheduled jobs in this interval. We then continueas above for t + 3. M ∩ M M M M \ M M \ M (cid:102) M (a) Jobs on M and M \ M can beshifted by 2 time units. (cid:102) M (b) If M \ M = ∅ , jobson M can be shifted. M M ∩ M (cid:102) M M \ M (c) Conflict relationsbetween machine sets Figure 11 Illustration for the proof of Lemma 4.6. Case II: M ̸ = ∅ . If M = ∅ , then no job is running/processing at t + 4. Therefore, wemay replace Π on [ t, t + 4] with the 2-pattern. Note that Π processed | M | + | M | ≤ m jobs.Therefore, the 2-pattern processes at least as many jobs as before in this interval.If M ̸ = ∅ and M \ M = ∅ , we can shift the jobs on M to t because the blockingintervals of the jobs on M ∩ M and M intersect, see Figure 11(b). Thus all job start attime t and we obtain a 1-pattern as in Case I. Hence, the remaining case is that M ̸ = ∅ and M \ M ̸ = ∅ . For the remainder of the proof, we distinguish whether we are given (i) acomplete bipartite graph or (ii) a traceable bipartite graph. (i) Notice that M and M form two independent sets. If they belong to the same part ofthe bipartition of G , then we reschedule their jobs to start at time t . It follows that M = M = ∅ , and thus, we are in Case I. If M and M belong to different parts ofthe bipartition, then M ∩ M = ∅ . However, this yields a contradiction to our aboveassumption. (ii) We show that | M ∩ M | ≤ | f M | for traceable graphs. This implies that 2 | M ∩ M | + | M \ M | + | M | ≤ m and all jobs starting in [ t, t + 4] may be rescheduled by the2-pattern which processes m jobs in 4 time units.In order to prove | M ∩ M | ≤ | f M | , we present an injection f from M ∩ M to f M .Clearly, M ∩ M forms an independent set and all its neighbors are contained in f M ,see Figure 11(c). Consider some Hamiltonian path of G , say P . If P contains a leaf ℓ / ∈ M ∩ M , we orient P towards ℓ and denote the successor of every vertex ( ̸ = ℓ ) by . Buchem, L. Kleist and D. Schmidt genannt Waldschmidt 19 s ( v ). Then f ( i ) := s ( i ) for all i ∈ M ∩ M defines an injection. If both leaves ℓ , ℓ of P are in M ∩ M , we orient P away from ℓ and denote the successor of every vertex( ̸ = ℓ ) by s ( v ). Then f ( i ) := s ( i ) for all i ∈ ( M ∩ M ) \ { ℓ } defines an injection. Itremains to define f ( ℓ ). Because M is non-empty, there exists a shortest directedsubpath of P on M \ ( M ∩ M ) from some vertex k ∈ f M to a vertex k ′ ∈ f M . As aconsequence, we can set f ( ℓ ) to the predecessor of k ′ in P .Now, we update t to t + 4 and follow the procedure describe above. ◀ D.3 Trees ▶ Corollary 4.12. For every instance scr-Unit ( G, n ) where G is a tree, we can compute aschedule Π with ∥ Π ∥ ≤ opt + 2 in polynomial time. Proof. The schedule Π ∗ of highest load density with length 12 computed in the Theorem 4.11schedules N := 4 | S | + 3 | R | < m jobs. Let k, r be such that n = kN + r for 0 ≤ r < N . ByObservation 4.2 and integrality, every schedule has makespan ≥ ⌈ n / m · m / N ⌉ = ⌈ kN + r ) / N ⌉ = 12 k + ⌈ r / N ⌉ ≥ k + ⌈ r / m ⌉ . We construct a schedule as follows: For r = 0, k concatenated copies of Π ∗ yield anoptimal schedule Π ′ .For 0 < r ≤ α , Π ′ together with one (partial) pure 1-pattern on a maximum independentset of G has makespan ≤ k + 3. The lower bound is at least 12 k + 1.For α < r ≤ m , Π ′ together with one (partial) pure 2-pattern on the classes of thebipartition of G has makespan ≤ k + 4. The lower bound is at least 12 k + ⌈ r / m ⌉ ≥ k + ⌈ / ⌉ = 12 k + 2.For m < r ≤ | R | + 2 | S | , we use the fact that α ( G ) ≥ | S | + / | R | , because S and amaximum independent set of the bipartite graph G [ R ] yield an independent set of G . Hence, | R | + 2 | S | ≤ α and Π ′ together with two pure 1-patterns on a maximum independent sethas makespan 12 k + 6. The lower bound is at least ⌈ k + r / m ⌉ ≥ k + 4.For | R | + / | S | < r ≤ / | R | + 2 | S | , Π ′ together with a prefix of Π ∗ has makespan 12 k + 7.The lower bound is at least 12 k + ⌈ r / N ⌉ ≥ k + 5.For / | R | + 2 | S | < r ≤ | R | + 3 | S | , Π ′ together with a prefix of Π ∗ has makespan 12 k + 9.The lower bound is at least 12 k + ⌈ r / N ⌉ ≥ k + 7.For 2 | R | + 3 | S | < r ≤ . | R | + 3 | S | , Π ′ together with a prefix of Π ∗ has makespan 12 k + 11.The lower bound is at least 12 k + ⌈ r / N ⌉ ≥ k + 9.For 2 . | R | +3 | S | < r < N , k +1 concatenated copies of Π ∗ yield a schedule with makespan12 k + 12. The lower bound is at least 12 k + ⌈ r / N ⌉ ≥ k + 10. ◀ E Details of Section 5 Here, we show the details of the proof of Theorem 5.1. We show how to find an optimalschedule in polynomial time by distinguishing the cases n ≤ m + α and n > m + α . Westart with the first case. ▶ Lemma E.1. For bipartite graphs, an instance scr-id-sym ( G, n, (1 , p, with p ≥ and n ≤ m + α ( G ) can be solved in polynomial time. Proof. Consider an instance scr-id-sym ( G, n, (1 , p, G = ( V ∪ V , E ) is bipartiteand p ≥ 2. Furthermore, let m i := | V i | for i = 1 , 2, and I ∗ be the maximum 1-IS of G and I ∗ the maximum 1-IS of G [ V \ I ]. Note that I ∗ and I ∗ can be found in polynomial time [23].In order to prove the statement we make a case distinction on the number of jobs: (i) If n ≤ α , all jobs can be scheduled in a 1-pattern on I ∗ starting at time 0. As thisgives a makespan of q the schedule is optimal. (ii) If α < n ≤ m , not all jobs can start at time 0 and the makespan is at least q + 1.Scheduling all jobs in a 2-pattern on M gives a makespan of q + 1. (iii) If m < n , we know that at least one machine processes two jobs, i.e., the makespan isat least 2 q , and we must make a further case distinction: (i) If n ≤ max { m + m , α + | I ∗ |} , we can construct a brick schedule with respectto V and V or I ∗ and I ∗ , respectively. This yields a makespan of 2 q . (ii) If max { m + m , α + | I ∗ |} < n ≤ m + α , at least one pair of conflicting machinesmust schedule 2 jobs and, hence, the makespan must be strictly larger than 2 q .Using a combination of a 2-pattern on M starting at 0 and a 1-pattern on I ∗ starting at q + 1 gives a makespan of 2 q + 1. ◀ Next, we consider the case of many jobs. First, note that one can easily show that brickschedules defined in Section 5 are feasible. ▶ Observation E.2. For an instance scr-id-sym ( G, n, (1 , p, with p ≥ on a bipartitegraph G , the brick schedule on two independent sets I and I is feasible. Proof. The blocking times of the jobs scheduled on a machine i ∈ I d for d ∈ , d − , d − 1] (mod q ). Clearly, the blocking times of jobs scheduled on machinesin different independent sets never intersect. ◀ In order to show the optimality of brick schedules, we first present two properties of optimalschedules. ▶ Lemma E.3. Let Π ∗ be an optimal schedule for an instance scr-id-sym ( G, n, (1 , p, with p ≥ and a bipartite graph G . Then, every machine processes at most ⌈ n / m ⌉ jobs in Π ∗ . Proof. Consider an instance scr-id-sym ( G, n, (1 , p, G = ( V ∪ V , E ) being abipartite graph and p ≥ 2. The brick schedule on V and V is feasible and has a makespanof at most q ⌈ n / m ⌉ + 2. Because q = p + 2 ≥ 4, we obtain ∥ Π ∗ ∥ ≤ q ⌈ n / m ⌉ + 2 < q ( ⌈ n / m ⌉ + 1) . Hence, the number of jobs assigned to any machine by Π ∗ is at most ⌈ n / m ⌉ . ◀▶ Lemma E.4. Let Π be a feasible schedule for an instance scr-id-sym ( G, n, (1 , p, with p ≥ on a bipartite graph G . If there exists a pair of conflicting machines i and i ′ such thatboth i and i ′ are assigned at least jobs, then i or i ′ (or both) has idle time at least . Proof. Let Π be a feasible schedule such that there exists a pair of conflicting machines i and i ′ such that both i and i ′ are assigned at least 2 jobs. Clearly, one of the machines mustbe idle for some time. Suppose towards contradiction that the maximum idle time amongthe two machines is 1 and assume without loss of generality that this idle time appears atthe beginning. The first two jobs on i and i ′ are then scheduled in a 2-pattern on these twomachines which implies that at time q + 1 either a job can start on i or on i ′ . In both casesone of the machines would be idle for at least 2 time units which gives a contradiction. ◀ Using Lemmas E.3 and E.4 we can show the following. ▶ Lemma E.5. scr-id-sym on bipartite graphs with p ≥ and n > m + α can be solved inpolynomial time using brick schedules. . Buchem, L. Kleist and D. Schmidt genannt Waldschmidt 21 Proof. Consider an instance scr-id-sym ( G, n, (1 , p, G with p ≥ m i := | V i | for i = 1 , 2, and I ∗ be the maximum 1-IS of G and I ∗ themaximum 1-IS of G [ V \ I ]. Note that I ∗ and I ∗ can be found in polynomial time [23]. Weassume that G has at least one edge, otherwise pure 1-patterns on all machines yield anoptimal schedule. Let r := n (mod m ). Then, we construct schedules as follows:If r = 0 or ( r > m and n > ⌈ n / m ⌉ α + ⌊ n / m ⌋| I ∗ | ), a brick schedule with respect to V and V yields a makespan of q ⌈ n / m ⌉ + 2. By Lemma E.3 we know that the maximumnumber of jobs scheduled by a machine is at most ⌈ n / m ⌉ . If r = 0, this implies that allmachines must schedule exactly this amount of jobs and by Lemma E.4 we know thatat least one machine is idle for 2 time units or more. Hence, the yielded makespan isoptimal. If r > m and n > ⌈ n / m ⌉ α + ⌊ n / m ⌋| I ∗ | we know by Lemma E.3 that for anyindependent set of machines at least one machine not in this set needs to be assignedthe maximum amount of jobs and the same idea as earlier implies that the obtainedmakespan is optimal.If 0 < r ≤ m , a brick schedule with respect to V and V yields a makespan of q ⌈ n / m ⌉ which is the lower bound implied by Lemma 3.2.If r > m and n ≤ ⌈ n / m ⌉ α + ⌊ n / m ⌋| I ∗ | , a brick schedule on I ∗ and I ∗ gives a makespanof q ⌈ n / m ⌉ which is the lower bound implied by Lemma 3.2.which is the lower bound implied by Lemma 3.2.