[AggounBeldiceanu93]

$\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}(\mathrm{𝚃𝙰𝚂𝙺𝚂},\mathrm{𝙻𝙸𝙼𝙸𝚃})$

$\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}\_\mathrm{𝚖𝚊𝚡}$.

Figure 5.82.1 shows the cumulated profile associated with the example. To each task of the $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ constraint corresponds a set of rectangles coloured with the same colour: the sum of the lengths of the rectangles corresponds to the duration of the task, while the height of the rectangles (i.e., all the rectangles associated with a task have the same height) corresponds to the resource consumption of the task. The $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ constraint holds since at each point in time we do not have a cumulated resource consumption strictly greater than the upper limit 8 enforced by the last argument of the $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ constraint.

Figure 5.82.1. Resource consumption profile

• Items of $\mathrm{𝚃𝙰𝚂𝙺𝚂}$ are permutable.

• $\mathrm{𝚃𝙰𝚂𝙺𝚂}.\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}$ can be decreased to any value $\ge 0$.

• One and the same constant can be added to the $\mathrm{𝚘𝚛𝚒𝚐𝚒𝚗}$ and $\mathrm{𝚎𝚗𝚍}$ attributes of all items of $\mathrm{𝚃𝙰𝚂𝙺𝚂}$.

• $\mathrm{𝙻𝙸𝙼𝙸𝚃}$ can be increased.

In the original $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ constraint of CHIP the $\mathrm{𝙻𝙸𝙼𝙸𝚃}$ parameter was a domain variable corresponding to the maximum peak of the resource consumption profile. Given a fixed time frame, this variable could be used as a cost in order to directly minimise the maximum resource consumption peak.

Some systems like Ilog CP Optimizer also assume that a zero -duration task overlaps a point $i$ if and only if (1) its origin is less than or equal to $i$, and (2) its end is greater than or equal to $i$. Under this definition, the height of a zero -duration task is also taken into account in the resource consumption profile.

Note that the concept of cumulative is different from the concept of rectangles non -overlapping even if, most of the time, each task of a ground solution of a $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ constraint is simply drawn as a single rectangle. As illustrated by Figure 5.100.5, this is in fact not always possible (i.e., some rectangles may need to be broken apart). In fact the $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ constraint is only a necessary condition for rectangles non -overlapping (see Figure 5.100.4 and the corresponding explanation in the Algorithm slot of the $\mathrm{𝚍𝚒𝚏𝚏𝚗}$ constraint).

The first filtering algorithms were related to the notion of compulsory part of a task [Lahrichi82]. They compute a cumulated resource profile of all the compulsory parts of the tasks and prune the origins of the tasks with respect to this profile in order to not exceed the resource capacity. These methods are sometimes called time tabling. Even if these methods are quite local, i.e., a task has a non -empty compulsory part only when the difference between its latest start and its earliest start is strictly less than its duration, it scales well and is therefore widely used. Later on, more global algorithmsEven if these more global algorithms usually can prune more early in the search tree, these algorithms do not catch all deductions derived from the cumulated resource profile of compulsory parts. based on the resource consumption of the tasks on specific intervals were introduced [ErschlerLopez90], [CaseauLaburthe96b], [Lock96]. A popular variant, called edge finding, considers only specific intervals [MercierVanHentenryck08]. An efficient implementation of edge finding in $O(knlogn)$, where $k$ is the number of distinct task heights and $n$ is the number of tasks, based on a specific data structure, so called a cumulative $\Phi$ -tree [Vilim09a], is provided in [Vilim09b]. A $O(n^{2}logn)$ filtering algorithm based on tasks that can not be the earliest (or not be the latest) is described in [SchuttWolf10].

Within the context of linear programming, the reference [HookerYan02] provides a relaxation of the $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ constraint.

A necessary condition for the $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ constraint is obtained by stating a $\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}$ constraint on a subset of tasks $𝒯$ such that, for each pair of tasks of $𝒯$, the sum of the two corresponding minimum heights is strictly greater than $\mathrm{𝙻𝙸𝙼𝙸𝚃}$. This can be done by applying the following procedure:

• Let $h$ be the smallest minimum height strictly greater than $\lfloor \frac{\mathrm{𝙻𝙸𝙼𝙸𝚃}}{2}\rfloor$ of the tasks of the $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ constraint. If no such task exists then the procedure is stopped without stating any $\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}$ constraint.

• Let ${𝒯}_h$ denote the set of tasks of the $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ constraint for which the minimum height is greater than or equal to $h$. By construction, the tasks of ${𝒯}_h$ cannot overlap. But we can eventually add one more task as shown by the next step.

• When it exists, we can add one task that does not belong to ${𝒯}_h$ and such that its minimum height is strictly greater than $\mathrm{𝙻𝙸𝙼𝙸𝚃}-h$. Again, by construction, this task cannot overlap all the tasks of ${𝒯}_h$.

When the tasks are involved in several $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ constraints more sophisticated methods are available for extracting $\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}$ constraints [BaptisteLePape00], [BaptisteDemassey04].

In the context where, both the duration and height of all the tasks are fixed, [BeldiceanuCarlssonPoder08] provides two kinds of additional filtering algorithms that are specially useful when the slack $\sigma$ (i.e., the difference between the available space and the sum of the surfaces of the tasks) is very small:

• The first one introduces bounds for the so called cumulative longest hole problem. Given an integer $ϵ$ that does not exceed the resource limit, and a subset of tasks ${𝒯}^{\text{'}}$ for which the resource consumption is a most $ϵ$, the cumulative longest hole problem is to find the largest integer ${\mathrm{𝑙𝑚𝑎𝑥}}_{\sigma }^{ϵ}\left({𝒯}^{\text{'}}\right)$ such that there is a cumulative placement of maximum height $ϵ$ involving a subset of tasks of ${𝒯}^{\text{'}}$ where, on one interval $[i,i+{\mathrm{𝑙𝑚𝑎𝑥}}_{\sigma }^{ϵ}\left({𝒯}^{\text{'}}\right)-1]$ of the cumulative profile, the area of the empty space does not exceed $\sigma$.

• The second one used dynamic programming for filtering so called balancing knapsack constraints. When the slack is 0, such constraints express the fact that the total height of tasks ending at instant $i$ must equal the total height of tasks starting at instant $i$. Such constraints can be generalized to non -zero slack.

cumulativeMax in Choco, cumulative in Gecode, cumulative in JaCoP, cumulative in SICStus.

assignment dimension added: $\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}\_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎𝚜}$ (sum of $\mathrm{𝚝𝚊𝚜𝚔}$ $\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝𝚜}$ replaced by number of distinct $\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚜}$, assignment dimension added), $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎𝚜}$ (negative $\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝𝚜}$ allowed and assignment dimension added).

common keyword: $\mathrm{𝚌𝚊𝚕𝚎𝚗𝚍𝚊𝚛}$ (scheduling constraint), $\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}\_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ (resource constraint, sum of $\mathrm{𝚝𝚊𝚜𝚔}$ $\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝𝚜}$ replaced by number of distinct values), $\mathrm{𝚌𝚘𝚕𝚘𝚞𝚛𝚎𝚍}\_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎𝚜}$ (resource constraint), $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}\_\mathrm{𝚌𝚘𝚗𝚟𝚎𝚡}$ (resource constraint, $\mathrm{𝚝𝚊𝚜𝚔}$ defined by a set of $\mathrm{𝚙𝚘𝚒𝚗𝚝𝚜}$), $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}\_\mathrm{𝚙𝚛𝚘𝚍𝚞𝚌𝚝}$ (resource constraint, sum of $\mathrm{𝚝𝚊𝚜𝚔}$ $\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝𝚜}$ replaced by product of $\mathrm{𝚝𝚊𝚜𝚔}$ $\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝𝚜}$), $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}\_\mathrm{𝚠𝚒𝚝𝚑}\_\mathrm{𝚕𝚎𝚟𝚎𝚕}\_\mathrm{𝚘𝚏}\_\mathrm{𝚙𝚛𝚒𝚘𝚛𝚒𝚝𝚢}$ (resource constraint, a $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ constraint for each set of $\mathrm{𝚝𝚊𝚜𝚔𝚜}$ having a priority less than or equal to a given threshold).

generalisation: $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}\_\mathrm{𝚝𝚠𝚘}\_𝚍$ ($\mathrm{𝚝𝚊𝚜𝚔}$ replaced by $\mathrm{𝚛𝚎𝚌𝚝𝚊𝚗𝚐𝚕𝚎}$ with a $\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}$).

implied by: $\mathrm{𝚍𝚒𝚏𝚏𝚗}$ ($\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ is a neccessary condition for each dimension of the $\mathrm{𝚍𝚒𝚏𝚏𝚗}$ constraint).

related: $\mathrm{𝚕𝚎𝚡}\_\mathrm{𝚌𝚑𝚊𝚒𝚗}\_\mathrm{𝚕𝚎𝚜𝚜}$, $\mathrm{𝚕𝚎𝚡}\_\mathrm{𝚌𝚑𝚊𝚒𝚗}\_\mathrm{𝚕𝚎𝚜𝚜𝚎𝚚}$ (lexicographic ordering on the origins of $\mathrm{𝚝𝚊𝚜𝚔𝚜}$, $\mathrm{𝚛𝚎𝚌𝚝𝚊𝚗𝚐𝚕𝚎𝚜}$, $...$), $\mathrm{𝚘𝚛𝚍𝚎𝚛𝚎𝚍}\_\mathrm{𝚐𝚕𝚘𝚋𝚊𝚕}\_\mathrm{𝚌𝚊𝚛𝚍𝚒𝚗𝚊𝚕𝚒𝚝𝚢}$ (controlling the shape of the cumulative profile for breaking symmetry).

soft variant: $\mathrm{𝚜𝚘𝚏𝚝}\_\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$.

specialisation: $\mathrm{𝚊𝚝𝚖𝚘𝚜𝚝}$ ($\mathrm{𝚝𝚊𝚜𝚔}$ replaced by $\mathrm{𝚟𝚊𝚛𝚒𝚊𝚋𝚕𝚎}$), $\mathrm{𝚋𝚒𝚗}\_\mathrm{𝚙𝚊𝚌𝚔𝚒𝚗𝚐}$ (all $\mathrm{𝚝𝚊𝚜𝚔𝚜}$ have a $\mathrm{𝚍𝚞𝚛𝚊𝚝𝚒𝚘𝚗}$ of 1 and a fixed $\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}$), $\mathrm{𝚍𝚒𝚜𝚓𝚞𝚗𝚌𝚝𝚒𝚟𝚎}$ (all $\mathrm{𝚝𝚊𝚜𝚔𝚜}$ have a $\mathrm{𝚑𝚎𝚒𝚐𝚑𝚝}$ of 1).

used in graph description: $\mathrm{𝚜𝚞𝚖}\_\mathrm{𝚌𝚝𝚛}$.

characteristic of a constraint: core, automaton, automaton with array of counters.

complexity: sequencing with release times and deadlines.

constraint type: scheduling constraint, resource constraint, temporal constraint.

filtering: linear programming, dynamic programming, compulsory part, cumulative longest hole problems, Phi-tree.

modelling: zero-duration task.

problems: producer-consumer.

puzzles: squared squares.

$\mathrm{𝚃𝙰𝚂𝙺𝚂}$

$\mathrm{𝚃𝙰𝚂𝙺𝚂}$ $\mathrm{𝚃𝙰𝚂𝙺𝚂}$

The first graph constraint enforces for each task the link between its origin, its duration and its end. The second graph constraint makes sure, for each time point $t$ corresponding to the start of a task, that the cumulated heights of the tasks that overlap $t$ does not exceed the limit of the resource.

Parts (A) and (B) of Figure 5.82.2 respectively show the initial and final graph associated with the second graph constraint of the Example slot. On the one hand, each source vertex of the final graph can be interpreted as a time point. On the other hand the successors of a source vertex correspond to those tasks that overlap that time point. The $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ constraint holds since for each successor set $𝒮$ of the final graph the sum of the heights of the tasks in $𝒮$ does not exceed the limit $\mathrm{𝙻𝙸𝙼𝙸𝚃}=8$.

Figure 5.82.2. Initial and final graph of the $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ constraint

(a)

(b)

Since $\mathrm{𝚃𝙰𝚂𝙺𝚂}$ is the maximum number of vertices of the final graph of the first graph constraint we can rewrite $\mathrm{𝐍𝐀𝐑𝐂}$ $=$ $\left|\mathrm{𝚃𝙰𝚂𝙺𝚂}\right|$ to $\mathrm{𝐍𝐀𝐑𝐂}$ $\ge$ $\left|\mathrm{𝚃𝙰𝚂𝙺𝚂}\right|$. This leads to simplify $\underline{\overline{\mathrm{𝐍𝐀𝐑𝐂}}}$ to $\overline{\mathrm{𝐍𝐀𝐑𝐂}}$.

Figure 5.82.3 depicts the automaton associated with the $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ constraint. To each item of the collection $\mathrm{𝚃𝙰𝚂𝙺𝚂}$ corresponds a signature variable ${𝚂}_i$ that is equal to 1.

Figure 5.82.3. Automaton of the $\mathrm{𝚌𝚞𝚖𝚞𝚕𝚊𝚝𝚒𝚟𝚎}$ constraint