A FIXED-PARAMETER ALGORITHM FOR A UNIT-EXECUTION-TIME UNIT-COMMUNICATION-TIME TASKS SCHEDULING PROBLEM WITH A LIMITED NUMBER OF IDENTICAL PROCESSORS

. This paper considers the minimization of the maximum lateness for a set of dependent tasks with unit duration, unit communication delays release times and due dates. The number of processors is limited, and each task requires one processor for its execution. A time window built from an upper bound of the minimum maximum lateness is associated to each task. The parameter considered is the pathwidth of the associated interval graph. A fixed-parameter algorithm based on a dynamic programming approach is developed to solve this optimization problem. This is, as far as we know, the first fixed-parameter algorithm for a scheduling problem with communication delays and a limited number of processors.


Introduction
Scheduling problems with communication delays have been intensively studied since 1990s because of the importance of practical applications.Several surveys are dedicated to this class of problems known to be mostly NP-hard [4,9,11,24].
This paper considers a scheduling problem with communication delays defined as follows: a set  = {1, 2, . . ., } of  tasks is to be executed on  identical machines (processors).Each machine can process at most one task at a time and each task is to be processed once.Tasks have a unit execution time and are partially ordered by a precedence graph  = ( , ).Let  be a feasible schedule; for any task  ∈  ,    denotes the starting time of the task  following .For any arc (, ) ∈ , the task  must finish its execution before the task  starts executing, i.e.    + 1 ≤    .If the tasks  and  are assigned to different processors, a unit communication delay must be added after the execution of the task  to send data to the task , thus    +2 ≤    .Moreover, we assume that release dates   ∈ N and due dates   ∈ N are given.Then, the inequality   ≤    holds for each task  ∈  .
The maximum lateness of a feasible schedule  is defined as () = max ∈ (   + 1 −   ); the minimization of the maximum lateness is denoted by  max .The problem considered in this paper is designated by  |  , prec,   = 1,   = 1| max using standard notations [13].The minimization of the maximum lateness includes the minimization of the makespan  max ; Indeed, the minimization of the maximum lateness for an instance with   = 0 for each task  ∈  corresponds to minimize the makespan.
Several authors developed algorithms for solving scheduling problems with similar constraints.Veltman provided in [23] an exact dynamic programming algorithm of time complexity (2 () ×  2() ) for  |prec,   = 1,   = 1| max where () is the width of the precedence graph , i.e. the size of the largest antichain.Zinder et al. [26] have developed and tested an exact branch-and-bound algorithm for the problem  |prec,   = 1,   = 1| max .For more general problems, several authors considered integer linear programming formulations (ILP in short).Davidović et al. [6] tackled scheduling problems for a fixed network of processors; communications are proportional to both the amount of exchanged data between pairs of dependent tasks and the distance between processors in the multiprocessor architecture.These authors developed two formulations and compared them experimentally.Later, Ait El Cadi et al. [1] improved this approach by reducing the size of the linear program (number of variables and constraints) and adding cuts; they compared positively to the previous works.Venugopalan and Sinnen [25] provided a new ILP formulation for  |prec,   | max and comparisons with Davidović et al. [6] for several classes of graphs and fixed number of processors.
Fixed-parameter algorithms for NP-complete problems allow to obtain polynomial-time algorithms when some parameters are fixed [5,8].More precisely, a fixed-parameter algorithm solves any instance of a problem of size  in time  () × poly(), where  is allowed to be a computable superpolynomial function and  the associated parameter.
Mnich and van Bevern [16] surveyed recently main results on parameterized complexity for scheduling problems and identified 15 challenging open problems.For the scheduling problem with usual precedence constraints, many researchers consider the width () of the precedence graph as a parameter, leading usually to negative results.Du et al. [10] proved that  2|chains| max is strongly NP-hard for unbounded width.Günther et al. [14] proved that  2|chains, () ≤ 3| max is weakly NP-hard.Bodlaender and Fellows [3] proved that  |prec,   = 1| max is W [2]-hard parameterized by the width and the number of machines.More recently, van Bevern et al. [22] proved that  2|prec,   ∈ {1, 2}| max is W [2]-hard parameterized by the width ().
Let us suppose that an upper bound L of the maximum lateness is fixed.We develop in this paper a fixedparameter algorithm for  |  , prec,   = 1,   = 1| max in time complexity ( 2 +  × ( L) × 2 3( L) ).Let us consider the interval graph ℐ( L) = ( , ( L)) associated to the time windows (  ,   + L),  ∈  .An edge  = (, ) ∈ ( L) if the intersection (  ,   + L) ∩ (  ,   + L) ̸ = ∅.The parameter ( L) is the pathwidth of the interval graph ℐ( L) [2]; It corresponds to the maximum number of intersecting time windows (  ,   + L),  ∈  minus 1.Our algorithm is as far as we know the first fixed-parameter algorithm for solving a scheduling problem with communication delays and a bounded number of processors.
The pathwidth was identified recently by several authors as an important parameter for several classes of scheduling problems.De Weerdt et al. [7] provided an exact fixed-parameter algorithm in the slack and the pathwidth for a sequencing problem with rejection, set-up times and penalties.Munier Kordon [17] developed a fixed-parameter algorithm in the pathwidth for the basic scheduling problem  |  , prec,   = 1| max to handle both precedence constraints and resource limitations.A similar approach was developed by Tang and Munier Kordon [21] who presented a fixed-parameter algorithm in the pathwidth in time complexity ( 3 +  × ( C) × 2 4( C) ) for the scheduling problem with communication delays and unlimited number of processors P |  , prec,   = 1,   = 1| max .The value C is here an upper bound of the minimum makespan.This approach was extended in this paper by considering a more complex criteria, namely the maximum lateness, and a limited number of parallel identical processors.Moreover, the structure of the algorithm was improved to limit the enumeration to active schedules with a slightly better worst-case time complexity.A schedule  is active if there is not another feasible schedule  ′ such that, for each task  ∈  ,   ′  ≤    with at least one of these inequalities is strict [19].This paper is organized as follows: Section 2 presents additional notations.In order to limit the combinatorial explosion of the method, we identify a structural property of the set of tasks schedulable at each time instant and a characterization of active schedules [19].Our algorithm is presented in Section 3, and its correctness established in Section 4. Section 5 is devoted to its complexity.Section 6 is our conclusion.

Problem definition and dominance properties
This section is devoted to some theoretical lemmas that will be considered for the correctness of our algorithm.A small example is also provided.The scheduling problem considered is described in Section 2.1, while a small example is presented in Section 2.2.Section 2.3 presents some important notations and a structural property of feasible schedules.Lastly, Section 2.3 presents a dominance property of active schedules that will be considered below.
We observe that a feasible schedule  is completely defined by the starting times vector   ∈ N  .Indeed, for any arc  = (, ) ∈ , we note    the communication delay between the tasks  and ; we set    = 0 if the execution of the task  starts right after the task .These two tasks are necessarily executed by a same processor and the communication delay is removed.Otherwise, a communication delay is required between the completion time of the task  and the starting time of the task  and thus    = 1.We then set The problem considered is expressed below.A time-indexed formulation should be considered to transform it into an integer linear program [20] for modelling the resource constraints.We set  max = max ∈   (resp. max = max ∈   ) the maximum due date (resp.release time); we also suppose that an upper bound of the maximum lateness L is fixed.We then observe that C = min( max + 2,  max + L) is an upper bound of the makespan of any active feasible schedule which maximum lateness is bounded by L.
Since communications delays and length of the tasks are unitary, starting times can be reduced to integer values; Inequalities (2) come from the definition of the maximum lateness and the release dates.Communication delays are defined from the starting time of the tasks (3).Inequalities (4)-( 6) express the communication delay constraints: any task  has at most one successor (resp.predecessor) performed at its completion time (resp.just before its starting time) on the same processor.Inequalities (7) express the limitation on the number of processors.

Example
Let us consider an instance of our scheduling problem defined by 7 tasks of unit length.The precedence graph, release dates and due dates are reported by Figure 1.The number of machines is fixed to 2. A feasible schedule  of maximum lateness () = 2 is given by Figure 2.

Time windows, pathwidth, and a structural property
Let us consider that an upper bound of the maximum lateness L is fixed.This value can be easily computed by extending the classical Graham priority list algorithm [12].Then for any optimal feasible schedule , each task  ∈  has to be completed in the time window [  ,   + L], which is equivalent to    ≥   and    ≤   + L − 1.Let us suppose that the tasks are numbered following increasing release times, that is  1 ≤  2 . . .≤   .We also suppose that the minimum value for a release date  min =  1 = 0. We can consider that release dates are compatible with respect to precedence, that is, if (, ) ∈ ,   + 1 ≤   .
Next lemma bounds the maximum value of the release dates and C: Lemma 2.1.We can suppose that  max = max ∈   ≤ 2( − 1) and C ≤ 4 − 2 without loss of generality.
Since L is an upper bound of the minimum maximum lateness, our optimization problem is solvable even if we restrict our algorithm to the determination of feasible schedules  with () ≤ L. Now, let suppose that  is a feasible schedule with () ≤ L. For every integer  ∈ {0, . . ., C − 1}, we set    = { ∈  ,    = } as the set of tasks performed at time  by .The following lemma will be considered further to reduce the size of the tasks sets built at each step of our algorithm.For the example given by Figure 1  Now, let us consider the feasible schedule given by Figure 2.For  = 3,

A general dominance property of active schedules
Let us consider that  is a feasible schedule of maximum lateness bounded by L. For every integer  ∈ {−1, . . ., C − 1}, we set   = ⋃︀  =0    and   =    .The set   contains all the tasks that are executed in time [0,  + 1), and   contains all the tasks that are executed at time .Notice that  −1 =  −1 = ∅.

Complexity analysis
We study in this section the complexity of Algorithm 1 to conclude that our scheduling problem is fixedparameter tractable in the pathwidth.Proof.The time complexity of the computation of the sets   and   for  ∈ {0, . . ., C} (lines 3 and 4) is ( 2 ) since C is bounded by 4 − 2 following Lemma 2.1.
The overall complexity of the algorithm is thus ( 2 +  × ( L) × 2 3( L) ), and the first part of the theorem holds.

Conclusion and perspectives
We proved in this paper that the scheduling problem  |  , prec,   = 1,   = 1| max is fixed-parameter tractable in the pathwidth ( L) of the interval graph ℐ( L) associated with the intervals (  ,   + L),  ∈  .We extended previous approaches [17,21] to tackle both communications delay, a limited number of machines, and to optimize the maximum lateness.We also limit our enumeration to active schedules, which allows to decrease the worst-case complexity of the method.
We believe that this work opens up many perspectives.From a theoretical point of view, many fundamental questions remain open as the existence of a fixed-parameter algorithm in the width, or the possible extension of this work to scheduling problems with large communication delays.From a practical point of view, our algorithm defines an original exploration scheme probably well suited to general scheduling problems.Similarly to branch-and-bound methods, dominance properties allow to reduce the size of the generated multistage graph.It would then be interesting to test this new class of algorithms to compare their performance with those from the literature.

Figure 2 .
Figure 2. A feasible schedule  of maximum lateness () = 2 associated to the example given in Figure 1.

Figure 3 .
Figure 3.The multistage auxiliary graph () = (, ) associated with the example given in Figure 1.Each node  ∈   is designated by the triple ( (), (), ()).The nodes  filled in gray are associated to a set of feasible schedules which minimum maximum lateness is ().

Theorem 4 . 5 (
Correctness of Algorithm 1).Algorithm 1 returns the minimum maximum lateness () ≤ L of a feasible schedule  if it exists, +∞ if there is no feasible schedule of maximum lateness bounded by L.