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been discovered in $m1$ in $m2$ and viceversa thanks to different +propagation algorithms in different models $m1$ and $m2$. This improves +propagation because benefits from a model are brought to another model. + +\paragraph{Meta-constraints} +A \textit{meta-constraint} is a constraint occurring in another constraint. For +example $ \sum_{i} (X_i > t_i) <= n $; in this case, the inner constraint is +$ (X_i > t_i) $. This type of constraints is useful when we want to make choices +according to the results of other constraints. + +\paragraph{Implied and redundant constraints} +An \textit{implied constraint} is a semantically redundant and they speed up the +solver. On the other hand, we refer to redundant constraint as a constraint +which is semantically redundant, but it doesn't affect the solver performance. +It happens to have a redudant constraint $r$ when we have another constraint $s$ +which makes the propagation shrink the domains of variables in the same way $r$ +would do. For example, look at this model: + +\begin{lstlisting}[style=none] +include "alldifferent.mzn"; +int: n; +array [0..n-1] of var 0..n-1: x; + +constraint forall(i in 0..n-1) + (x[i] = sum (j in 0..n-1)(x[j] == i)); + +constraint sum(i in 0..n-1)(x[i]) = n; + +constraint sum(i in 0..n-1)(x[i]*i) = n; + +solve satisfy; +\end{lstlisting} + +In this model, we are expressing the sequence puzzle problem, namely we want to +make a sequence $x$ with length $n$, where for each $i \in {1, ..., n}$, $i$ +appears exactly $x_i$ times in the sequence $x$. Note that we don't used global +constraints. The last two constraints are implied, so they affect solver +performance. Now consider the following model: + +\begin{lstlisting}[style=none] +include "globals.mzn"; + +int: n; +array [1..n] of var 0..n-1: x; + +constraint let { + array[1..n] of 0..n-1: cover = 0..n-1 +} in global_cardinality(x, cover, x); + +constraint sum(i in 1..n) (x[i]) = n; + +constraint sum(i in 1..n) (x[i] * (i-1)) = n; + +solve satisfy; +\end{lstlisting} +This model is semantically equivalent to the previous one, but the first implied +constraint (so the second constraint) became redundant. This happened because +the propagation algorithm for \texttt{global\_cardinality\_constraint} is more +effective than decomposition and so it ``discovers" all the first (ex-)implied +constraint would do. This concept will be developed better nextly. + +\subsection{Symmetry} +Often, when we try to solve a problem, tehre are many solutions which are +``symmetric". Two solutions are symmetric if one of them is a permutation of the +other one. The solver cannot know if two solutions are symmetric and it will +look for all of them, but this is a waste of time because two symmetric +solutions are actually the same solution, they didn't bring something new. Thus, +to avoid symmetry, we can add some constraints which somehow impose an order in +order to accept just one solution and exclude all the symmetric ones. Usually, +useful constraints are the ones of \texttt{lex*} family, namely constraints +which pretend lexicographic order on array of variables. +TODO + +\section{Constraints propagation} +Propagation is the action of restricting the domains of variables. + +\subsection{Local consistency} +This is a form of inference which detects inconsistent partial assignments. What +is a partial assignment? An assignment is literally an assignment of a value to +a decision variable: + +\[ X_i = j \] + +If that assignment is inconsistent, then $j$ can be removed from $D(X_i)$ and, +as a consequence, it helps propagation. + +\subsubsection{Generalized Arc Consistency} +A \textit{support} $(d_1, ..., d_k) \in (D(X_1), ..., D(X_k))$ for a constraint +$C$ is an assignment of decision variables which satisfies $C$. +A constraint $C$ is \textit{GAC} iff $\forall X_i \in {X_i, ..., X_n}$, +$\forall v \in D(X_i)$, $v \in d_i$, where $d_i$ is a support of $C$. +This is called \textit{Arc consistency} ($AC$) when $k = 2$. + +In other words, a constraint $C$ restricts the domains $D(X_i)$. A question can +arise: how does a constraint $C$ grant the property of $GAC$ for domains? The +answer is that global constraints have specialized propagation algorithms. These +algorithms tries to keep the domains of variables as restricted as possible in +order to have always the lowest number assignments. + +\subsubsection{Bounds consistency} +It relaxes the domain of a decision variable $X_i$ to be in a range such that + +\[ D(X_i) = [min(X_i)...max(X_i)] \] + +A \textit{bound support} is a tuple $(d_1, ..., d_k) \in C$ where +$d_i \in [min(X_i)...max(X_i)]$. + +$C(X_i, ..., X_k)$ is BC iff for all $X_i \in {X_1, ..., X_k}$, $min(X_i)$ and +$max(X_i)$ belong to a bound support. GAC is stronger than BC, however, it's +more expensive to achieve sometimes. + +Look at the following minizinc programs to solve sudoku: + +\begin{lstlisting}[style=none] +include "globals.mzn"; + +int: n; + +array[1..(n*n), 1..(n*n)] of var 1..(n*n): x; + +% Rows must be all different +constraint forall(i in 1..(n*n)) (alldifferent(x[i, ..]) + ::domain_propagation + ); + +% Columns must be all different +constraint forall(j in 1..(n*n)) (alldifferent(x[.., j]) + ::domain_propagation + ); + +% Squares must be all different +constraint forall (i in 1..n) (forall (j in 1..n) + (alldifferent(x[(1+((i-1)*n))..(n+((i-1)*n)), + (1+((j-1)*n))..(n+((j-1)*n))])) + ::domain_propagation + ); + +solve satisfy; +\end{lstlisting} + +and: + +\begin{lstlisting}[style=none] +include "globals.mzn"; + +int: n; + +array[1..(n*n), 1..(n*n)] of var 1..(n*n): x; + +% Rows must be all different +constraint forall(i in 1..(n*n)) (alldifferent(x[i, ..]) + ::bounds_propagation + ); + +% Columns must be all different +constraint forall(j in 1..(n*n)) (alldifferent(x[.., j]) + ::bounds_propagation + ); + +% Squares must be all different +constraint forall (i in 1..n) (forall (j in 1..n) + (alldifferent(x[(1+((i-1)*n))..(n+((i-1)*n)), + (1+((j-1)*n))..(n+((j-1)*n))])) + ::bounds_propagation + ); + +solve satisfy; +\end{lstlisting} + +and: + +\begin{lstlisting}[style=none] +include "globals.mzn"; + +int: n; + +array[1..(n*n), 1..(n*n)] of var 1..(n*n): x; + +% Rows must be all different +constraint forall(i in 1..(n*n)) (alldifferent(x[i, ..])); + +% Columns must be all different +constraint forall(j in 1..(n*n)) (alldifferent(x[.., j])); + +% Squares must be all different +constraint forall (i in 1..n) (forall (j in 1..n) + (alldifferent(x[(1+((i-1)*n))..(n+((i-1)*n)), + (1+((j-1)*n))..(n+((j-1)*n))])) + ); + +solve satisfy; +\end{lstlisting} + +We try to solve it with $n = 7$, so we are solving a sudoku 49X49. In the +first program, we specify (or better, we suggest) to use domain propagation, +namely we keep $GAC$. Results are these: + +\includegraphics[scale=0.2]{sudoku_res.png} + +Just 3 seconds, not bad! But when we try to use bounds propagation (namely $BC$) +or even not to suggest the propagation way (the third case), the results is that +the solver goes over five minutes with millions of failures. Clearly, this is +just a case where domain propagation is better than bounds propagation. + +\subsection{Propagation} +It is the action of achieving a certain level of consistency which is a property +(not an action). Indeed, we talk about propagation algorithms. During search, +multiple propagation algorithms can interact, usually in way that at a given +momentum, only an algorithm is running and it does it until it reaches a level +of consistency. Then, there are no other actions to do, so another algorithm +runs. An algorithm can run multiple times, since, because of propagation, the +decision variables domains change. What is important is that algorithms run +until $GAC$ property is granted, if they can, however, $GAC$ is not always +possible to be kept, sometimes only $BC$ (or other weaker consistency +properties) can be kept. + +\paragraph{Complexity} +What about complexity of algorithms? Assume $ | D(X_i) | = d $, from definitions +we have that one run of $C(X_1, X_2)$ takes $ O(d^2) $. However, we can improve, +for example we can think cases where the running time is better, sometimes even +$O(1)$. + +\subsection{Global constraints} +They are specific constraints which helps propagation and can state expressions +impossible to state with primitive constraints. Often, the propagation +algorithms for these constraints keep $GAC$ in polynomial time, namely they +simply run in polynomial time and at the end of the process, the domains are +exactly the supports for constraints. + +\paragraph{Counting constraints} +They restrict the number of variables satisfying a condition. + +\subsubsection{Sequencing constraints} +They ensures a sequence of variables have certain patterns. + +\subsubsection{Scheduling constraints} +They are all about assigning resources to activities. + +\subsubsection{Ordering constraints} +They obviously care of order of variables values. They include also +lexicographic ordering constraint. + +The propagation for global constraints is developed by decomposition into +smaller constraints for which the propagation algorithm is known. Generally, +global constraints integrates a specialized propagation algorithm. + +The \texttt{alldifferent} constraint propagation algorithm is based on bipartite +graphs. One part of the graph is the variables, the other part is the possible +values of the variables. A \textit{matching M} in a graph \textit{G} is a set of +non-adjacent edges such that no two edges share common vertices. A +\textit{maximal matching} for a graph \textit{G} is a matching which is not +subset of any other matching in \textit{G}. The propagation algorithm for +\texttt{alldifferent} constraint considers edges as the assignments of variables +to the values; finding a maximal matching means finding a possible assignment +for variables. Before analyzing it, let's give some definition: + +\begin{itemize} + \item an edge is \textit{matching} if it belongs in a matching; + \item an edge is \textit{free} if it is not matching; + \item a node is \textit{matched} if it is incident to a matching edge; + \item a node is \textit{free} if it is not mathed; + \item an edge is \textit{vital} if it belongs to every maximal matching; +\end{itemize} + +Let's see the algorithm: + +\begin{itemize} + \item compute all maximal matchings; + \item if no maximal matching exists, then fail; + \item if an edge is free in all maximal matchings, then: + \begin{itemize} + \item remove the edge; + \item remove the corresponding value to the domain of the associated + variable; + \end{itemize} + \item if an edge is vital, then: + \begin{itemize} + \item keep the edge; + \item assign the corresponding value to the associated variable; + \end{itemize} + \item if an edge is matching (but not vital), then keep the edge. +\end{itemize} + +The problem with this algorithm is that calculating all maximal matching in a +naive way is too expensive. Let's see these further definitions: + +\begin{itemize} + \item An \textit{alternating path} is a simple path with edges alternating + free and matching; + \item An \textit{alternating cycle} is a path with edges alternating free + and matching; + \item An \textit{even path} or \textit{cycle} is such if the number of edges + is even. +\end{itemize} + +An important can help us to in finding maximal matchings. An edge \textit{e} +belongs to a maximal matching iff for some arbitrary maximal matching +\textit{M}: + +\begin{itemize} + \item either \textit{e} belongs to \textit{M}; + \item or \textit{e} belongs to even alternating path starting at a free + node; + \item or \textit{e} belongs to an even alternating cycle. +\end{itemize} + +Next, we have to perform the following actions: + +\begin{itemize} + \item we choose an arbitrary maximal matching \textit{M}; + \item all free edges in \textit{M} changes their direction in order to they + go from values to variables; + \item all matching edges in \textit{M} changes their direction in order to + they go from variables to values; + \item starting from a free node, search for all nodes on directed simple + path and mark all the edges; + \item find the strongly connected components and mark edges which are part + of the SCCs; + \item all the marked edges and the edges which are parts of \textit{M} are + part of some maximal matching. Other edges are discarded and domains can be + updated. +\end{itemize} + +\subsection{Table constraint} +Sometimes, we know the exact combinations of values that variables can take. +In those cases, table constraint can be really useful. An example can be +crossword puzzle: + +\[ table([X_1, X_2, X_3], dictionary) \] +\[ table([X_1, X_{13}, X_{16}], dictionary) \] +\[ table([X_4, X_5, X_6, X_7], dictionary) \] +\[ ... \] + +\subsection{Regular constraint} +Sometimes, we need that variables follow a certain pattern. In those cases, +deterministic finite-state automaton can be very useful, indeed, +\textit{regular} constraint is based on dfsa: $regular([X_1, ..., X_k], A)$ +holds iff $\langle X_1, ... X_k \rangle$ forms a string accepted by $A$. This +constraint has an efficient propagation algorithm which keeps $GAC$ property, +however, decomposition is very efficient for this constraint as well. The +advantage over table constraint is that the latter needs all solutions have to +be computed firstly. Anyway, even if many constraints are just instances of +regular, it is advisable to use specific constraints when possible, firstly for +readability of the code; regular constraint are useful for complicated patterns. + +\section{Search} +This phase is carried out by the constraint solver which performs a backtracking +tree search, where nodes are variables and branches are decisions on variables. +The first point to focus on is that the tree is not built immediately as whole, +but subtrees are built gradually (at the need). The search is helped by +propagation: without it, the search should build all the possible subtrees to +guess a solution. + +\noindent\fbox{\parbox{\textwidth}{ +Remember that propagation and search are two different phases: propagation is +concerned with domains shrinking and it's done when "evaluating" a constraint; +search phase is done after propagation and it is concerned with the "growing" +of the tree. Before doing search, it's necessary to perform propagation. +}} + +\subsection{Depth-first visit} +There are two types of branching: + +\begin{description} + \item[d-way branching] for a variable $X$ (which corresponds to a node), $k$ + branches are created, where $k$ is the cardinality of the actual domain of + $X$; + \item[2-way branching] a subset $S$ of the domain of a variable $X$ is set, + then two branches are created, one if $X \in S$, one if $X \notin S$. +\end{description} + +There are different types of heuristics, which are related to the choice of +variables that have to come in the next branches and their values. + +\paragraph{Static variable ordering heuristics} +A variable is associated a priori to each level of the search tree, regardless +how the search will be carried out. + +\paragraph{Dynamic variable ordering heuristics} +At any node, any variable can be considered. This can be more expensive than +static heuristics, but the current state of the tree can be kept, while state +heuristics cannot do this. There's no a best heuristics from this point of view, +it depends from the problem. + +\paragraph{Generic dynamic variable ordering heuristics} +There are many types of heuristics, two of them are fail-first, namely +discarding inconsistent subtrees as soon as possible. The advantage is that +making a choice like this allows to propagation to find a larger number of +inconsistent values with a greater probability. + +\begin{description} + \item[Minimum domain (dom)] the variable to be chosen is the one with the + minimum domain size. + \item[Most constrained (deg)] the variable to be chosen is the one with the + greatest number of constraints. +\end{description} + +\textit{dom} and \textit{deg} are often combined to take advantages from both. +A variation of \textit{deg} is the weighted degree heuristic, where weights are +assigned to constraints, initially set to 1, then during propagation of a +constraint, its weight is increased by 1 if the constraint fails. Minizinc has +an annotation (\texttt{dom\_w\_deg}) which consists in choosing the variable with +the following smallest value: domain size divided by the weighted degree, namely +the number of times a variable caused a failure earlier in the search. + +\subsubsection{Heavy tail behaviour} +Sometimes, particular instances of a problem can be difficult to solve and they +take much more time than other instances. But this is not due to instances +directly! Sometimes, the combination of instances and heuristics make solution +of those instances difficult to find, but this can be solved by changing +heuristics. Sometimes, randomization can make problems easier to solve! +Randomization can regard to parameters of heuristics in search but also the +choice of variables and even values. Restarting the search can be sometimes a +good idea as well, performing it under certain conditions, for instance when a +certain amount of resources have been consumed. Any kind of depth first search +for solving optimization problems suffers from the problem that wrong decisions +made at the top of the search tree can take an exponential amount of search to +undo. One common way to ameliorate this problem is to restart the search from +the top thus having a chance to make different decisions. The usefulness comes +when, at the restart, the search is performed differently. But how do we +restart? + +\begin{description} + \item[constant restart], restart after using $L$ resources; + \item[geometric restart], restart after using $L$ resources, but setting a + new limit at the next restart, so we restart after $a^n*L$, where $n$ is the + number of restarts. + \item[Luby restart], restart after using $s[i]*L$, where $s[i]$ is the i-th + number of the Luby sequence which is a sequence of powers of 2, but before + adding a new power in the sequence, the previous subsequence is repeated in + the sequence, so: + \[ [1] \] + \[ [1, 1, 2] \] + \[ [1, 1, 2, 1, 1, 2, 4] \] + \[ [1, 1, 2, 1, 1, 2, 4, 1, 1, 2, 1, 1, 2, 4, 8] \] + \[ ... \] + This is the most used. +\end{description} + +\subsubsection{Limited discrepancy search} +A \textit{discrepancy} is any decision in a search tree that does not follow the +heuristic. The point of LDS is that it starts with left depth-first search at +the first loop, then, if the solution is not found, the search is repeated by +going at the first right branch of each variable encountered at the previous +loop. + +\includegraphics[scale=0.25]{lds.png} + +\subsubsection{Depth-bounded discrepancy search} +Since it is more convenient to perform discrepancy at the highest levels of the +tree instead of the deepest ones, the discrepancy performing is done a bit +different from LDS: at the i-th iteration, discrepancy branches are visited only +if they are at the i-th level, otherwise the original heuristic is followed. + +\includegraphics[scale=0.25]{dds.png} + +The problem with LDS is that it acts on the deepest branches, but often it is +useful to make different decisions at the highest levels because the search is +less informative and the bad decisions are likely to be taken tehre. DDS is +better than LDS from this point of view. + +\subsection{Optimization problems} +An optimization problem is a 4-tuple $\langle X, D, C, f \rangle$ where $X$, +$D$ and $C$ are the decision variables, domains of variables and constraints +respectively while $f$ is an optimization criterion in the form of an objective +function. For example we can set the goal of a problem as the $min f$, so +minimizing the value of $f$ (remember that $f$ is a variable, not a function). +Another way to set the goal can be: + +\begin{itemize} + \item defining $f = max(X)$; + \item defining the goal as the $min f$. +\end{itemize} +In this way, we are searching the minimum number of $f$ value such that it is +useful to solve a certain problem. + +\paragraph{Destructive lower bound} +A possible technique to solve minimization (or maximization or stuff like those) +can be searching ``through the domain of f", namely try assigning values to $f$ +at each iteration starting from the minimum of $D(f)$ until we reach a solution. +At each iteration, the value is increased. Is is named ``destructive" because +intermediate results are discarded. The solution is proved to be optimal. +Disadvantages for this algorithm are: if resources of search are limited, +then it's not granted that it finds a solution; it performs small steps (just +one). An advantage is that it makes constraints tighter at each iteration, so +propagation is helped. Another advantage is that it provides lower bounds. + +\paragraph{Destructive upper bound} +It is the same mechanism of DLB, but it starts with $max D(f)$. However, when we +find a solution, it isn't proved that it is the optimal one, so we continue to +decrease values and search solutions. When we arrive at the case where it fails +to find the solution, then the previous iteration of the current case is proven +to be the optimal solution. The pros and cons of this algorithm are exactly +complementary to DLB. + +\paragraph{Binary search} +The main idea is to consider upper bound and lower bound in an array (the +solutions domain). At each iteration, these bounds are restricted and come +closer in this way: +\begin{itemize} + \item we set $lb < f < (lb + ub / 2)$, where $lb$ is lower bound and $ub$ is + upper bound; + \item we try to solve the problem for that value of $f$; + \item if the problem is feasible, then we update $ub$, else we update $lb$; + \item we repeat these procedures until we reach $f = lb + 1$, which is the + optimal solution. +\end{itemize} +It takes advantages from both DLB and DUB. But it's not the best way, because +all information is discarded at each iteration, so we are performing duplicated +work. + +\paragraph{Branch and bound} +It uses a single search tree, incorporating bounds in the search. At each +iteration, when a solution is found, a new bound constraint is set to ensure the +future solutions are better, then it backtracks and looks for a new solution. +This is repeated until the problem is infeasible, at that point, the last +solution is demonstrated to be optimal. + +\section{Scheduling} +It concerns with ordering resources and/or tasks over time and it has a lot of +applications. We have: +\begin{itemize} + \item a set of resources with fixed capacities; + \item a set of tasks with durations and resource requirements; + \item a set of temporal constraints, for example we may want task 1 to run + before or after task 2; + \item a performance metric, namely something to measure the goodness + (usually the objective variable). +\end{itemize} + +In this kind of problem, usually, we have to decide when tasks have to start. +The decision variables correspond to the operations/activities to be performed. +An activity $a_i$ has: a starting time $s_i$, a duration $d_i$ and an ending +time $e_i$. We will refer to the earliest start time as $EST_i$, to the latest +start time as $LST_i$, to the earliest end time as $EET_i$ and to latest end +time $LET_i$. Note that $EST_i$, $EET_i$, $LST_i$ and $LET_i$ (the latter called +deadline) correspond to the possible earliest or latest times (not necessarily +the real ones). There are two types of activity: +\begin{itemize} + \item preemptive activities, which can be interrupted any time, and it holds + $s_i + d_i \leq e_i $ + \item non-preemtpive activities, which cannot be interrupted by external + agents and it holds $s_i + d_i = e_i $. +\end{itemize} + +A resource is something available and limited for tasks to run. We have: +\begin{description} + \item[cumulative/parallel] resources which allow multiple activities to run + at the same time, for example, they can be a multi-core CPU. Formally, a + resource $r_k$ is associated to a capacity $c_k$. Each activity $a_i$ has + requirements $rq_{ik} \geq 0$ on resource $r_k$. Clearly, it must hold + $rq_{ik} \leq c_k$. We have cumulative constraint which handles the resource + requirements for activities; + \item[unary/disjunctive/sequential] resources which allow activities to + execute only one at a time regardless the resource capacity. The disjunctive + constraint handles this type of resources. +\end{description} + +\subsubsection{Temporal constraints} +Temporal constraints can model the fact some actvities must come before or after +other activities. But they can be finer: they can define \textit{time-legs}, +namely they bound the difference between the end time and the start time of two +activities; they can define also \textit{time-windows}, namely pretending one or +more activities runs in a certain time window. + +\subsubsection{Cost function} +We define \textit{makespan} as the completion time of the last activity. +The RCPSP (cumulative resources) and job shop (disjunctive resources) scheduling +cost functions are makespan and the objective is usually to minimize the +makespan. A makespan can be modeled in different ways, for example introducing +a dummy activity with duration 0 and that must run as the last activity or +taking the maximum of the starting times of activities plus their durations +(obtaining the ending times). + +\subsection{RCPSP} +Take a look at a sample for RCPSP: +\begin{itemize} + \item a project graph $\langle A, E \rangle$, where $A$ is the set of + activities and $E$ are their ending time; + \item a set $R$ of resource $r_k$ with a capacity $c_k$; + \item each activity $a_i$ has a duration $d_i$ and a resource requirement + $r_{ik}$ +\end{itemize} +We can see activities as: + +\includegraphics[scale=0.35]{activity.png} + +\subsubsection{Search heuristics} +What are best heuristics for RCPSP? We must pose the following questions: +which variable to pick next? Which value to assign? Let's focus on the value +selection. Since the makespan has to be minimized, a good choice can be using +the minimum value (the $EST_i$). As far as variables choice is concerned, a good +choice can be selecting the one with the minimum $LET_i$. These are just +heuristics, so the optimal solution is not granted to be found immediately, +sometimes backtracking is necessary. Generally, since the objetive is to +minimize the makespan, increasing an $S_i$ cannot improve the makespan (we say +these problems have \textit{regular cost metrics}), so a good choice is the +$EST_i$ and this is true for many scheduling problems. But this is just a greedy +solution for a fast first solution, indeed, in order to reach optimality we have +to do backtracking. How to backtrack? Given a non-optimal solution, we select an +activity $a_i$ and instead of choosing the value for that variable, we postpone +it and we visit a subtree. Doing this procedure will grant optimality, but it's +often very expensive in terms of time. This technique regards both value and +variable choices. + +And what about variable selection? Precedence constraints help us because they +make variables domains shrink. A good greedy choice can be selecting the task +with the minimum $LET_i$, so the one with the first deadline in order to discard +it immediately and to have the lowest probability for it to get a blind spot. To +reach optimality, refer to the technique explained before. + +\section{Heuristic search} +We have two types of methods: complete methods, which guarantee the optimal +solution to each finite-size instance of a problem in a finite time; approximate +methods, which don't guarantee neither optimality, nor termination in case of +infeasibility, but they can find better solutions than complete methods in a +lower amount of time. We have three types approximate methods: constructive +heuristics, local search, metaheuristics. + +\subsection{Constructive heuristics} +This is the fastest approximation and it consists in constructing the solution +from scratch by repeatedly extending the current partial assignment until a +solution is found or the stopping conditions are satisfied. An example can be +the greedy heuristics, but they don't guarantee optimality, but even though +this, they are quick and they are used as initialization step for other methods. + +\subsection{Local search} +It gives often better solutions than constructive heuristics in terms of +optimality. It starts with an initial solution and it iteratively tries to +replace the current solution a with a better ``neighbourhood" (a near solution), +by applying small changes. Now we have to define what neighbourhood is, let's +see combinatorial optimization from a different perspective: given the problem +$\langle X, D, C, f \rangle$, $S$ the set of all solutions, we have to find +$s^* \in S$ such that $\forall s \in S. \; f(s^*) \leq f(s)$. We define a +function $N: S \mapsto \wp(S)$ that assigns to every $s \in S$ a set of +neighbours $N(s) \subseteq S$. $N(s)$ is called the \textit{neighbourhood} of +$s$. It is often implicitly defined by defining the modifications to $s$ to +reach $N(s)$. We now define a \textit{locally minimal solution} with respect to +a neighbourhood $N$ as a solution $s'$ such that +$\forall s \in N(s'). \; f(s') \leq f(s)$. A structure of local search algorithm +is done like this: +\begin{itemize} + \item Generate a solution $s$; + \item while $\exists s' \in N(s). \; f(s') \leq f(s)$, assign to $s$ an + improving neighbor. +\end{itemize} +This algorithm stops when it finds a local minimum. There are different choices +of the neighbor: the first one or the best one. The first solution can be +generated randomly or heuristically. But how to define neighbourhood structure? +It comes useful the notion \textit{K-exchange neighbourhood}, where $K$ is the +number of modifications to do on the graph. In the problem of travelling +salesman problem, we can have that a starting solution is a hamiltonian tour +of the graph, then if we choose 2-exchange, we have to switch two archs, if +we choose 3-exchange, we have to switch three archs, etc.. Clearly, the more $K$ +grows, the more neighbourhoods grows exponentially. Generally, a key issue is +how to define the neighbourhood. Too small neighbourhood is fast to find, but +the quality of local minima is low, while too large neighbourhood improves the +quality of local minima, but they are expensive to calculate. + +\subsection{Metaheuristics} +They consist in higher level strategies which ``guide" heuristics to find a +solution. They can be seen, in simply terms, as heuristics on heuristics. We +have two types of policies: \textit{intensification} which exploits previous +experience; \textit{diversification} which explores the search space in the +large. A good balance between them is at the base of effectiveness of +metaheuristics. + +\subsubsection{Local search methods} +It is similar to local search, but differently to it, this method tries to +escape to local minimum and it does it by allowing worsening solutions, by +changing neighbourhood structure during search or by changing objective function +during search. Clearly, we have to interrupt the algorithm at a certain point +since it cannot stop; some criteria are setting a maximum CPU time, number of +iterations, etc.. Here some methods. + +\paragraph{Simulated annealing} +Like local search, but it accepts worsening moves with a certain probability. +The probability decreases at each iteration. It favours intensification over +diversification. + +\paragraph{Variable neighbourhood search} +It changes neighbourhood structure during search, more precisely this happens +whenever a new local optima is reached. + +\paragraph{Tabu search} +It keeps track of a ``tabu list" of solutions or moves and it forbids them. So +it doesn't change the neighbourhood, but it restricts the neighbors which it can +move to. Anyway, storing solutions is very inefficient, moves are cheaper even +if that could eliminate good solutions which haven't been visited yet. The tabu +list size is another important issue: the more it's big, the more +diversification grows against intensification. In general, the size can be +increased in case of repetitions (so when diversification is needed) or +decreased when there are no improvements (so intensification is needed). + +\paragraph{Guided local search} +It changes the objective function. But one could argue it changes the semantics +of the claimed solution! The idea is to penalize some solution characteristics +which occur frequently, so the result is that some solutions get worse than how +they are really. + +\subsubsection{Population-based methods} +We work on a set of solutions and the basic principle is to find common +characteristics among solutions. We have a probabilistic model from which we +obtain sample solutions to analyse. After this analysis, we update our model and +we restart the process. This process is similar to what ants do when they come +across an obstacle: the ants will try to find a path to escape from the +obstacle, then they will start to take the shortest path because it will be the +one which is walked by the majority of ants and the ants drop pheromones when +they walk and follow paths where the concentration of pheromones is higher. From +this phenomenon, we have the so-called \textit{pheromone model}: we have a set +of \textit{pheromone values} which act as the memory with the goal to focus the +search on certain paths. A way to make this concept formal can be: a +\textit{pheromone value} is a value $t(X_i, v_i)$, $\forall X_i \in X$ and +$v_i \in D(X_i)$ which represents the desirability of assigning $v_i$ to $X_i$. +A common usage is to have bounding for pheromone values $t_{min}$ and $t_{max}$ +and to set initially values to $t_{max}$. As a consequence, we have that +decreasing values augment diversification because, we will have different values +from what we have had before (so it's like ants forget older solutions to get +close to newer solutions), while increasing will help intensification. + +In order to begin, we use ``artificial ants". They simply consist in using +constructive heuristics for an initial solution. So the algorithm has the +following schema: +\begin{itemize} + \item start from an initial solution, using constructive heuristics; + \item iteratively choose a variable $X_i$ according to some heuristic (this + is parametric) and then choose the value $v_i \in D(X_i)$ according to the + probability: + + \includegraphics[scale=0.2]{prob_ants.png} + + where $\alpha$ and $\beta$ are parameters used to balance the pheromone and + heuristic factors. +\end{itemize} +Pheromone values in the $i$-th round are the ones which comes from $(i-1)$-th +round solutions. Then, at each round, pheromone values are updated. If they are +decreased, diversification is boosted, else if they are increased, then +intensification is boosted. The update is done in order to improve the quality +of future solutions. + +At the end, which one to use? Local search metaheuristics or population-based +metaheuristics? It depends from the problem. If in the problem, moving from +neighbor to neighbor is easy and cheap in terms of computational cost, then +local search-based algorithms is good. On the other hand, if neighbourhood is +difficult to build, the cost of moves is not so cheap and solutions can be built +as composition of blocks, then population-based methods is a good idea. +Generally, local-based methods boost intensification because we act on subsets +of all solutions, while population-based methdos have the ability to boost +diversification (always according to how the pheromones are updated). Hybrid +methods tries to get the best from both methods types. + +Metaheuristics is effective in finding first good-quality solutions, but it +struggles with complex constraints. For this reason, mixing metaheuristics and +complete methods in different ways can improve the quality of solutions in a +time unit, since complete methods, on the other hand, can be bad and inefficient +if we have loose bounds of objective function. For example, a complete method +can apply a metaheuristic to improve a solution or, on the contrary, +metaheuristics use a complete methods to efficiently to explore neighbourhood. + +\paragraph{Large neighbourhood search} +And local search? How can it be combined with complete methods? Remember the +issue about small and large neighbourhoods; from this point, we can use large +neighbourhood and explore it with a complete method. We can see exploration of +neighbourhood as the solution of a sub-problem. So, given a solution $s$, we +have to possible moves to apply to each variable: +\begin{itemize} + \item fix part of the variables value; + \item relax the remaining variables. +\end{itemize} +\includegraphics[scale=0.25]{fix_relax.png} + +The good fact about this neighbourhood structure is that it works with all +problems. Now, two issues are: which percentage of variables to fix? Which +variables to fix? There are many ways to follow and it can depend from the +problem. + +Generally, after wee saw all these different methods, the important thing is to +know how to mix these methods; to fix an optimization problem, the general good +way is to use a hybrid method. + +\end{document} diff --git a/appunti/appunti-luca-borghi/cp.png b/appunti/appunti-luca-borghi/cp.png new file mode 100644 index 0000000000000000000000000000000000000000..fb9491d2fa099ffc1b7223faa6468d829838e8f1 GIT binary patch literal 19477 zcmeIaWl)?^&@Ko;g1Zwu!QCN9kO0A5f;++8Ef6etNCF8C!Gc3@*C9B;-Q9I?_RRfu zYj5q{s{OuOdw=bwNM*>(yz`!Ox}Sdf>F$YARhGp>BSV9OgTs6$C#?<#2k#61-k~6a z|LIiJJOuwBx=Fs%L;-*NQ7j_C@1*W;b=@_btlYg!T`l3P9i1F3+1$)sEiE10Y@FPW z5!%GSi`ZZhg!zcjGCL1o0poCUznRyn2U>ARf$^a zt!6@VRWckLHQYOC2~F>`gGDcIO|3`q6X?ae)QbenIv^ zox!u6ao&Iz-D3uiiyQU*PdWJ;X=%}D2^Hy=t{A?ww29`U__$FoLNg-eHb^d{rD-8q z477Mc$fCH3eOx;`2u|2)`(eM^4KIzS~D8z)nzf}7-Pc&JwPqpc~ zm!C_~hH&x-!LLVPAKf5BKi=jO4SKI&b_Cps1(C_Wg+5+3p6z}4uW(D>2;v;d|GHhsZ)fiqh 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At each -iteration, when a solution is found, a new bound constraint is set to ensure the -future solutions are better, then it backtracks and looks for a new solution. -This is repeated until the problem is infeasible, at that point, the last -solution is demonstrated to be optimal. +iteration, when a solution is found, a new bounding constraint is set to ensure +the future solutions are better, then it backtracks and looks for a new +solution. This is repeated until the problem is infeasible, at that point, the +last solution is demonstrated to be optimal. We can see an example of this in +map colouring problem: + +\frame{\includegraphics[scale=0.15]{mapcol1.png}} +\frame{\includegraphics[scale=0.15]{mapcol2.png}} + +\frame{\includegraphics[scale=0.15]{mapcol3.png}} +\frame{\includegraphics[scale=0.15]{mapcol4.png}} + +We found a solution, namely 4, which is not prooved to optimal, so we set a +bounding constraint which pretends the actual objective variable is lower than +the our last solution (lower because the actual objective variable has to be +minimized): + +\pagebreak + +\frame{\includegraphics[scale=0.11]{mapcol5.png}} +\frame{\includegraphics[scale=0.11]{mapcol6.png}} +\frame{\includegraphics[scale=0.11]{mapcol7.png}} + +\frame{\includegraphics[scale=0.11]{mapcol8.png}} +\frame{\includegraphics[scale=0.11]{mapcol9.png}} +\frame{\includegraphics[scale=0.11]{mapcol10.png}} + +\frame{\includegraphics[scale=0.11]{mapcol11.png}} +\frame{\includegraphics[scale=0.11]{mapcol12.png}} +\frame{\includegraphics[scale=0.11]{mapcol13.png}} + +We found another solution, namely 3; at this point we would set another bounding +constraint and we repeat the previous steps. TODO: pros and cons \section{Scheduling} It concerns with ordering resources and/or tasks over time and it has a lot of diff --git a/appunti/appunti-luca-borghi/mapcol1.png b/appunti/appunti-luca-borghi/mapcol1.png new file mode 100644 index 0000000000000000000000000000000000000000..07019ee0b030b567926fb52d431ada36b8f8df97 GIT binary patch literal 113250 zcmdqJi9c3r`##)mP)U>|Ni;~dge1vW%8<+<$xP;X$}CBeBuSDGk_>l}WD1#*%n5O4 z&OB$H-(zX_{Jwv~`+Pq8*}HIC>sr@!p2u`9e?@tz?Uam^>(;H?eo^{@^15{!gVwEE z&$ERBe?yaf=_&rT(O&$b$`<_3eT(5;{CTf~gu26JYa@pndUn^>8CzLfUO#ScV0Znx zmA#3z!`OxbQGC;J;+xLfUDtCkwYEB-VrqGP-IeQx2Y5~&ki4#cfR~4t@4yKj!BeLN zPYN7Rlsj-*fDII!hrNq$X!jl-<) zW|_>bk2(&~uDhre%|T~q^Qvz&JjQ>e%26lmn^}g&mV>uKBF#k^OoPOC#U1#UH~svcY#q$iG}y>2`6_xExAw|MPNw9I**sEW_l^VwxH zWotgTaOfN#V{6l?-2oEc2KcD`HUyRV)btg1PYe$|bBdT3H00ge*wplISjBg7Om~dv2P_hwV5@1U0B`0d9hEy3{Raft* zjucW!Rvnxc^gesjginTtft86`>>Y1PgP9%!#T~s`vy8Xoksl*7A|f=C6Wfc=HaQM? z7`-VqN*`gk87gz+Nm$s`Yu64(%I~7~yT}&u!-AiQdb3sLVyMTm@1Nb%MV2=ev0!=d z<4W(Ql@W4gw);Hq$jAs4H8rjOg)M%?i#Z?cvSniwJ7z}6va+(Zt*vqj3N-uo?=R_| z@X#V#g+G7(tC&XYr8;MqU3UN2n6Zh;7FE>=XWA5(t3X_mp*Z1H4%-sM$7!d72#e5Z

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diff --git a/appunti/appunti-luca-borghi/appunti-bogo8liuk.tex b/appunti/appunti-luca-borghi/appunti-bogo8liuk.tex index ef9ce2e..341174f 100644 --- a/appunti/appunti-luca-borghi/appunti-bogo8liuk.tex +++ b/appunti/appunti-luca-borghi/appunti-bogo8liuk.tex @@ -130,12 +130,13 @@ \subsection{Constraints} according to the results of other constraints. \paragraph{Implied and redundant constraints} -An \textit{implied constraint} is a semantically redundant and they speed up the -solver. On the other hand, we refer to redundant constraint as a constraint -which is semantically redundant, but it doesn't affect the solver performance. -It happens to have a redudant constraint $r$ when we have another constraint $s$ -which makes the propagation shrink the domains of variables in the same way $r$ -would do. For example, look at this model: +An \textit{implied constraint} is a semantically redundant constraint with the +additional of speeding up the solver. On the other hand, we refer to +\textit{redundant constraint} as a constraint which is semantically redundant, +but it doesn't affect the solver performance. It happens to have a redudant +constraint $r$ when we have another constraint $s$ which makes the propagation +shrink the domains of variables in the same way $r$ would do. For example, look +at this model: \begin{lstlisting}[style=none] include "alldifferent.mzn"; @@ -153,8 +154,8 @@ \subsection{Constraints} \end{lstlisting} In this model, we are expressing the sequence puzzle problem, namely we want to -make a sequence $x$ with length $n$, where for each $i \in {1, ..., n}$, $i$ -appears exactly $x_i$ times in the sequence $x$. Note that we don't used global +make a sequence $x$ with length $n$, where for each $i \in {0, ..., n-1}$, $i$ +appears exactly $x_i$ times in the sequence $x$. Note that we don't use global constraints. The last two constraints are implied, so they affect solver performance. Now consider the following model: @@ -181,7 +182,7 @@ \subsection{Constraints} constraint would do. This concept will be developed better nextly. \subsection{Symmetry} -Often, when we try to solve a problem, tehre are many solutions which are +Often, when we try to solve a problem, there are many solutions which are ``symmetric". Two solutions are symmetric if one of them is a permutation of the other one. The solver cannot know if two solutions are symmetric and it will look for all of them, but this is a waste of time because two symmetric @@ -189,48 +190,55 @@ \subsection{Symmetry} to avoid symmetry, we can add some constraints which somehow impose an order in order to accept just one solution and exclude all the symmetric ones. Usually, useful constraints are the ones of \texttt{lex*} family, namely constraints -which pretend lexicographic order on array of variables. -TODO +which pretend lexicographic order on array of variables. Retake the example on +n-queens problem, in particular the second model: + +\[ n \times n \; B_{ij} \in {0, 1} \] +\[ \sum_{i \in {1, ..., n}} B_{ij} = 1, \; \forall j \in {1, ..., n} \] +\[ \sum_{j \in {1, ..., n}} B_{ij} = 1, \; \forall i \in {1, ..., n} \] +\[ \sum B_{ij} \leq 1 \; \textrm{on all diagonals} \] +\[ \textrm{lex-lesser-eq}(B, \pi(B)), \; \forall \pi \] +Here, we are using a \texttt{lex} constraint to impose an order on variables in +order not to get other equivalent (mirrored) solutions: + +\includegraphics[scale=0.2]{nqueens-perm.png} \section{Constraints propagation} Propagation is the action of restricting the domains of variables. \subsection{Local consistency} This is a form of inference which detects inconsistent partial assignments. What -is a partial assignment? An assignment is literally an assignment of a value to -a decision variable: +is a partial assignment? An assignment is literally a choice of a value to a +decision variable: \[ X_i = j \] - If that assignment is inconsistent, then $j$ can be removed from $D(X_i)$ and, -as a consequence, it helps propagation. +as a consequence, it helps propagation. Partial is referred to the fact that +only some variables get a value in a certain moment. \subsubsection{Generalized Arc Consistency} A \textit{support} $(d_1, ..., d_k) \in (D(X_1), ..., D(X_k))$ for a constraint $C$ is an assignment of decision variables which satisfies $C$. A constraint $C$ is \textit{GAC} iff $\forall X_i \in {X_i, ..., X_n}$, -$\forall v \in D(X_i)$, $v \in d_i$, where $d_i$ is a support of $C$. +$\forall v \in D(X_i)$, $v \in d$, where $d$ is a support of $C$. This is called \textit{Arc consistency} ($AC$) when $k = 2$. In other words, a constraint $C$ restricts the domains $D(X_i)$. A question can arise: how does a constraint $C$ grant the property of $GAC$ for domains? The answer is that global constraints have specialized propagation algorithms. These -algorithms tries to keep the domains of variables as restricted as possible in -order to have always the lowest number assignments. +algorithms tries to keep the domains of variables as restricted as possible. \subsubsection{Bounds consistency} It relaxes the domain of a decision variable $X_i$ to be in a range such that \[ D(X_i) = [min(X_i)...max(X_i)] \] - A \textit{bound support} is a tuple $(d_1, ..., d_k) \in C$ where -$d_i \in [min(X_i)...max(X_i)]$. - -$C(X_i, ..., X_k)$ is BC iff for all $X_i \in {X_1, ..., X_k}$, $min(X_i)$ and -$max(X_i)$ belong to a bound support. GAC is stronger than BC, however, it's -more expensive to achieve sometimes. +$d_i \in [min(X_i)...max(X_i)]$. $C(X_i, ..., X_k)$ is BC iff for all $X_i \in +{X_1, ..., X_k}$, $min(X_i)$ and $max(X_i)$ belong to a bound support. GAC is +stronger than BC, however, it's more expensive to achieve sometimes. Look at the +following minizinc programs to solve sudoku: -Look at the following minizinc programs to solve sudoku: +\pagebreak \begin{lstlisting}[style=none] include "globals.mzn"; @@ -289,6 +297,7 @@ \subsubsection{Bounds consistency} \end{lstlisting} and: +\pagebreak \begin{lstlisting}[style=none] include "globals.mzn"; @@ -323,23 +332,23 @@ \subsubsection{Bounds consistency} the solver goes over five minutes with millions of failures. Clearly, this is just a case where domain propagation is better than bounds propagation. -\subsection{Propagation} -It is the action of achieving a certain level of consistency which is a property -(not an action). Indeed, we talk about propagation algorithms. During search, -multiple propagation algorithms can interact, usually in way that at a given -momentum, only an algorithm is running and it does it until it reaches a level -of consistency. Then, there are no other actions to do, so another algorithm -runs. An algorithm can run multiple times, since, because of propagation, the -decision variables domains change. What is important is that algorithms run -until $GAC$ property is granted, if they can, however, $GAC$ is not always -possible to be kept, sometimes only $BC$ (or other weaker consistency +\subsection{Propagation in action} +It is the action of achieving a certain level of consistency (which is a +property, not an action). Indeed, we talk about propagation algorithms. During +search, multiple propagation algorithms can interact, usually in way that at a +given momentum, only an algorithm is running and it does it until it reaches a +level of consistency. Then, there are no other actions to do, so another +algorithm runs. An algorithm can run multiple times, since, because of +propagation, the decision variables domains change. What is important is that +algorithms run until $GAC$ property is granted, if they can, however, $GAC$ is +not always possible to be kept, sometimes only $BC$ (or other weaker consistency properties) can be kept. \paragraph{Complexity} What about complexity of algorithms? Assume $ | D(X_i) | = d $, from definitions we have that one run of $C(X_1, X_2)$ takes $ O(d^2) $. However, we can improve, -for example we can think cases where the running time is better, sometimes even -$O(1)$. +for example there are specialized propagation algorithms which are more +efficient. \subsection{Global constraints} They are specific constraints which helps propagation and can state expressions @@ -348,24 +357,49 @@ \subsection{Global constraints} simply run in polynomial time and at the end of the process, the domains are exactly the supports for constraints. -\paragraph{Counting constraints} -They restrict the number of variables satisfying a condition. +\subsubsection{Counting constraints} +These constraints count and restrict how many times certain values occur in an +array of variables. \subsubsection{Sequencing constraints} They ensures a sequence of variables have certain patterns. \subsubsection{Scheduling constraints} -They are all about assigning resources to activities. +They are all about activities organization. For example, \texttt{cumulative} is +about assigning resources to activities and \texttt{disjunctive} makes +activities not overlap in time. \subsubsection{Ordering constraints} They obviously care of order of variables values. They include also lexicographic ordering constraint. +\subsubsection{Table constraint} +Sometimes, we know the exact combinations of values that variables can take. +In those cases, table constraint can be really useful. An example can be +crossword puzzle: + +\[ table([X_1, X_2, X_3], dictionary) \] +\[ table([X_1, X_{13}, X_{16}], dictionary) \] +\[ table([X_4, X_5, X_6, X_7], dictionary) \] +\[ ... \] + +\subsubsection{Regular constraint} +Sometimes, we need that variables follow a certain pattern. In those cases, +deterministic finite-state automaton can be very useful, indeed, +\textit{regular} constraint is based on dfsa: $regular([X_1, ..., X_k], A)$ +holds iff $\langle X_1, ... X_k \rangle$ forms a string accepted by $A$. This +constraint has an efficient propagation algorithm which keeps $GAC$ property, +however, decomposition is very efficient for this constraint as well. The +advantage over table constraint is that the latter needs all solutions have to +be computed firstly. Anyway, even if many constraints are just instances of +regular, it is advisable to use specific constraints when possible, firstly for +readability of the code; regular constraint are useful for complicated patterns. + +\subsection{Propagation of global constraints} The propagation for global constraints is developed by decomposition into smaller constraints for which the propagation algorithm is known. Generally, -global constraints integrates a specialized propagation algorithm. - -The \texttt{alldifferent} constraint propagation algorithm is based on bipartite +global constraints integrates a specialized propagation algorithm. For example, +the \texttt{alldifferent} constraint propagation algorithm is based on bipartite graphs. One part of the graph is the variables, the other part is the possible values of the variables. A \textit{matching M} in a graph \textit{G} is a set of non-adjacent edges such that no two edges share common vertices. A @@ -379,10 +413,9 @@ \subsubsection{Ordering constraints} \item an edge is \textit{matching} if it belongs in a matching; \item an edge is \textit{free} if it is not matching; \item a node is \textit{matched} if it is incident to a matching edge; - \item a node is \textit{free} if it is not mathed; + \item a node is \textit{free} if it is not matched; \item an edge is \textit{vital} if it belongs to every maximal matching; \end{itemize} - Let's see the algorithm: \begin{itemize} @@ -401,7 +434,6 @@ \subsubsection{Ordering constraints} \end{itemize} \item if an edge is matching (but not vital), then keep the edge. \end{itemize} - The problem with this algorithm is that calculating all maximal matching in a naive way is too expensive. Let's see these further definitions: @@ -413,7 +445,6 @@ \subsubsection{Ordering constraints} \item An \textit{even path} or \textit{cycle} is such if the number of edges is even. \end{itemize} - An important can help us to in finding maximal matchings. An edge \textit{e} belongs to a maximal matching iff for some arbitrary maximal matching \textit{M}: @@ -424,7 +455,6 @@ \subsubsection{Ordering constraints} node; \item or \textit{e} belongs to an even alternating cycle. \end{itemize} - Next, we have to perform the following actions: \begin{itemize} @@ -442,28 +472,6 @@ \subsubsection{Ordering constraints} updated. \end{itemize} -\subsection{Table constraint} -Sometimes, we know the exact combinations of values that variables can take. -In those cases, table constraint can be really useful. An example can be -crossword puzzle: - -\[ table([X_1, X_2, X_3], dictionary) \] -\[ table([X_1, X_{13}, X_{16}], dictionary) \] -\[ table([X_4, X_5, X_6, X_7], dictionary) \] -\[ ... \] - -\subsection{Regular constraint} -Sometimes, we need that variables follow a certain pattern. In those cases, -deterministic finite-state automaton can be very useful, indeed, -\textit{regular} constraint is based on dfsa: $regular([X_1, ..., X_k], A)$ -holds iff $\langle X_1, ... X_k \rangle$ forms a string accepted by $A$. This -constraint has an efficient propagation algorithm which keeps $GAC$ property, -however, decomposition is very efficient for this constraint as well. The -advantage over table constraint is that the latter needs all solutions have to -be computed firstly. Anyway, even if many constraints are just instances of -regular, it is advisable to use specific constraints when possible, firstly for -readability of the code; regular constraint are useful for complicated patterns. - \section{Search} This phase is carried out by the constraint solver which performs a backtracking tree search, where nodes are variables and branches are decisions on variables. @@ -474,8 +482,8 @@ \section{Search} \noindent\fbox{\parbox{\textwidth}{ Remember that propagation and search are two different phases: propagation is -concerned with domains shrinking and it's done when "evaluating" a constraint; -search phase is done after propagation and it is concerned with the "growing" +concerned with domains shrinking and it's done when ``evaluating" a constraint; +search phase is done after propagation and it is concerned with the ``growing" of the tree. Before doing search, it's necessary to perform propagation. }} @@ -501,7 +509,7 @@ \subsection{Depth-first visit} At any node, any variable can be considered. This can be more expensive than static heuristics, but the current state of the tree can be kept, while state heuristics cannot do this. There's no a best heuristics from this point of view, -it depends from the problem. +it depends on the problem. \paragraph{Generic dynamic variable ordering heuristics} There are many types of heuristics, two of them are fail-first, namely @@ -515,7 +523,6 @@ \subsection{Depth-first visit} \item[Most constrained (deg)] the variable to be chosen is the one with the greatest number of constraints. \end{description} - \textit{dom} and \textit{deg} are often combined to take advantages from both. A variation of \textit{deg} is the weighted degree heuristic, where weights are assigned to constraints, initially set to 1, then during propagation of a @@ -530,7 +537,7 @@ \subsubsection{Heavy tail behaviour} directly! Sometimes, the combination of instances and heuristics make solution of those instances difficult to find, but this can be solved by changing heuristics. Sometimes, randomization can make problems easier to solve! -Randomization can regard to parameters of heuristics in search but also the +Randomization can regard parameters of heuristics in search but also the choice of variables and even values. Restarting the search can be sometimes a good idea as well, performing it under certain conditions, for instance when a certain amount of resources have been consumed. Any kind of depth first search @@ -542,11 +549,11 @@ \subsubsection{Heavy tail behaviour} restart? \begin{description} - \item[constant restart], restart after using $L$ resources; - \item[geometric restart], restart after using $L$ resources, but setting a + \item[constant restart] restart after using $L$ resources; + \item[geometric restart] restart after using $L$ resources, but setting a new limit at the next restart, so we restart after $a^n*L$, where $n$ is the number of restarts. - \item[Luby restart], restart after using $s[i]*L$, where $s[i]$ is the i-th + \item[Luby restart] restart after using $s[i]*L$, where $s[i]$ is the i-th number of the Luby sequence which is a sequence of powers of 2, but before adding a new power in the sequence, the previous subsequence is repeated in the sequence, so: @@ -577,14 +584,14 @@ \subsubsection{Depth-bounded discrepancy search} The problem with LDS is that it acts on the deepest branches, but often it is useful to make different decisions at the highest levels because the search is -less informative and the bad decisions are likely to be taken tehre. DDS is +less informative and the bad decisions are likely to be taken there. DDS is better than LDS from this point of view. \subsection{Optimization problems} An optimization problem is a 4-tuple $\langle X, D, C, f \rangle$ where $X$, $D$ and $C$ are the decision variables, domains of variables and constraints respectively while $f$ is an optimization criterion in the form of an objective -function. For example we can set the goal of a problem as the $min f$, so +function. For example we can set the goal of a problem as the $min \; f$, so minimizing the value of $f$ (remember that $f$ is a variable, not a function). Another way to set the goal can be: @@ -597,22 +604,22 @@ \subsection{Optimization problems} \paragraph{Destructive lower bound} A possible technique to solve minimization (or maximization or stuff like those) -can be searching ``through the domain of f", namely try assigning values to $f$ -at each iteration starting from the minimum of $D(f)$ until we reach a solution. -At each iteration, the value is increased. Is is named ``destructive" because -intermediate results are discarded. The solution is proved to be optimal. -Disadvantages for this algorithm are: if resources of search are limited, -then it's not granted that it finds a solution; it performs small steps (just -one). An advantage is that it makes constraints tighter at each iteration, so -propagation is helped. Another advantage is that it provides lower bounds. +can be searching ``through the domain of $f$", namely try assigning values to $f +$ at each iteration starting from the minimum of $D(f)$ until we reach a +solution. At each iteration, the value is increased. It is named ``destructive" +because intermediate results are discarded. The solution is proved to be +optimal. Disadvantages for this algorithm are: if resources of search are +limited, then it's not granted that it finds a solution; it performs small steps +(just one). An advantage is that it makes constraints tighter at each iteration, +so propagation is helped. Another advantage is that it provides lower bounds. \paragraph{Destructive upper bound} -It is the same mechanism of DLB, but it starts with $max D(f)$. However, when we -find a solution, it isn't proved that it is the optimal one, so we continue to -decrease values and search solutions. When we arrive at the case where it fails -to find the solution, then the previous iteration of the current case is proven -to be the optimal solution. The pros and cons of this algorithm are exactly -complementary to DLB. +It is the same mechanism of DLB, but it starts with $max \; D(f)$. However, when +we find a solution, it isn't proved that it is the optimal one, so we continue +to decrease values and search solutions. When we arrive at the case where it +fails to find the solution, then the previous iteration of the current case is +proven to be the optimal solution. The pros and cons of this algorithm are +exactly complementary to DLB. \paragraph{Binary search} The main idea is to consider upper bound and lower bound in an array (the @@ -627,8 +634,8 @@ \subsection{Optimization problems} optimal solution. \end{itemize} It takes advantages from both DLB and DUB. But it's not the best way, because -all information is discarded at each iteration, so we are performing duplicated -work. +whole information is discarded at each iteration, so we are performing +duplicated work. \paragraph{Branch and bound} It uses a single search tree, incorporating bounds in the search. At each @@ -648,7 +655,6 @@ \section{Scheduling} \item a performance metric, namely something to measure the goodness (usually the objective variable). \end{itemize} - In this kind of problem, usually, we have to decide when tasks have to start. The decision variables correspond to the operations/activities to be performed. An activity $a_i$ has: a starting time $s_i$, a duration $d_i$ and an ending @@ -860,7 +866,7 @@ \subsubsection{Population-based methods} of future solutions. At the end, which one to use? Local search metaheuristics or population-based -metaheuristics? It depends from the problem. If in the problem, moving from +metaheuristics? It depends on the problem. If in the problem, moving from neighbor to neighbor is easy and cheap in terms of computational cost, then local search-based algorithms is good. 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