Combinatorial optimization. In computer science and artificial intelligence, combinatorial search studies search algorithms for solving instances of problems that are believed to be hard in general, by efficiently exploring the usually large solution space of these instances. Combinatorial search algorithms achieve this efficiency by reducing the effective size of the search space or employing heuristics. Some algorithms are guaranteed to find the optimal solution, while others may only return the best solution found in the part of the state space that was explored. Classic combinatorial search problems include solving the eight queens puzzle or evaluating moves in games with a large game tree, such as reversi or chess.
A study of computational complexity theory helps to motivate combinatorial search. Combinatorial search algorithms are typically concerned with problems that are NP-hard. Such problems are not believed to be efficiently solvable in general. Examples[edit] Lookahead[edit] See also[edit] References[edit] Constraint satisfaction problem. Examples of simple problems that can be modeled as a constraint satisfaction problem Examples demonstrating the above are often provided with tutorials of ASP, boolean SAT and SMT solvers. In the general case, constraint problems can be much harder, and may not be expressible in some of these simpler systems. "Real life" examples include planning and resource allocation. Formal definition[edit] Formally, a constraint satisfaction problem is defined as a triple , where [1] is a set of variables, is a set of the respect domains of values, and is a set of constraints.
Each variable can take on the values in the nonempty domain . Is in turn a pair , where is a subset of variables and is an . . Satisfies a constraint if the values assigned to the variables satisfies the relation An evaluation is consistent if it does not violate any of the constraints. Resolution of CSPs[edit] Constraint satisfaction problems on finite domains are typically solved using a form of search.
Theoretical aspects of CSPs[edit] Operations research. Operations research, or operational research in British usage, is a discipline that deals with the application of advanced analytical methods to help make better decisions.[1] It is often considered to be a sub-field of mathematics.[2] The terms management science and decision science are sometimes used as synonyms.[3] Employing techniques from other mathematical sciences, such as mathematical modeling, statistical analysis, and mathematical optimization, operations research arrives at optimal or near-optimal solutions to complex decision-making problems.
Because of its emphasis on human-technology interaction and because of its focus on practical applications, operations research has overlap with other disciplines, notably industrial engineering and operations management, and draws on psychology and organization science. Overview[edit] The major subdisciplines in modern operational research, as identified by the journal Operations Research,[6] are: History[edit] Historical origins[edit] Lecture06.pdf (application/pdf objekt) Backmarking. In constraint satisfaction, backmarking is a variant of the backtracking algorithm. Backmarking works like backtracking by iteratively evaluating variables in a given order, for example, . It improves over backtracking by maintaining information about the last time a variable was instantiated to a value and information about what changed since then.
In particular: An example, in which search has reached xi=d the first time. for each variable and value , the algorithm records information about the last time has been set to ; in particular, it stores the minimal index such that the assignment to was then inconsistent;for each variable , the algorithm stores some information relative to what changed since the last time it has evaluated ; in particular, it stores the minimal index of a variable that was changed since then.
The first information is collected and stored every time the algorithm evaluates a variable to , and is done by simply checking consistency of the current assignments for . Backjumping. In backtracking algorithms, backjumping is a technique that reduces search space, therefore increasing efficiency. While backtracking always goes up one level in the search tree when all values for a variable have been tested, backjumping may go up more levels.
In this article, a fixed order of evaluation of variables is used, but the same considerations apply to a dynamic order of evaluation. A search tree visited by regular backtrackingA backjump: the grey node is not visited Definition[edit] Whenever backtracking has tried all values for a variable without finding any solution, it reconsiders the last of the previously assigned variables, changing its value or further backtracking if no other values are to be tried. Is the current partial assignment and all values for have been tried without finding a solution, backtracking concludes that no solution extending exists. . , changing its value if possible, backtracking again otherwise. lead to a solution.
Such that . Instead of reconsidering . . Look-ahead (backtracking) In a general constraint satisfaction problem, every variable can take a value in a domain. A backtracking algorithm therefore iteratively chooses a variable and tests each of its possible values; for each value the algorithm is recursively run. Look ahead is used to check the effects of choosing a given variable to evaluate or to decide the order of values to give to it. In this example, x1=2 and the tentative assignment x2=1 is considered. Forward checking only checks whether each of the unassigned variables x3 and x4 is consistent with the partial assignment, removing the value 2 from their domains. The simpler technique for evaluating the effect of a specific assignment to a variable is called forward checking. That is still unassigned, and checks whether there exists an evaluation of that is consistent with the extended partial solution.
That are consistent with the extended assignment. Two other methods involving arc consistency are full and partial look ahead. With. Local consistency. In constraint satisfaction, local consistency conditions are properties of constraint satisfaction problems related to the consistency of subsets of variables or constraints. Several such conditions exist, the most known being node consistency, arc consistency, and path consistency.
Local consistency can be enforced via transformations of the problem called constraint propagation. Local consistency conditions can be grouped into various classes. The original local consistency conditions require that every consistent assignment can be consistently extended to another variable. Directional consistency only requires this condition to be satisfied when the other variable is higher than the ones in the assignment, according to a given order. Relational consistency includes extensions to more than one variable, but this extension is only required to satisfy a given constraint or set of constraints. Assumptions[edit] Local consistency[edit] Node consistency[edit] For example, given a variable and .
AC-3 algorithm. The AC-3 algorithm (short for Arc Consistency Algorithm #3) is one of a series of algorithms used for the solution of constraint satisfaction problems (or CSP's). It was developed by Alan Mackworth in 1977. The earlier AC algorithms are often considered too inefficient, and many of the later ones are difficult to implement, and so AC-3 is the one most often taught and used in very simple constraint solvers. The algorithm[edit] The current status of the CSP during the algorithm can be viewed as a directed graph, where the nodes are the variables of the problem, with edges or arcs between variables that are related by symmetric constraints, where each arc in the worklist represents a constraint that needs to be checked for consistency. AC-3 proceeds by examining the arcs between pairs of variables (x, y). It removes those values from the domain of x which aren't consistent with the constraints between x and y.
AC-3 is expressed in pseudocode as follows: References[edit] Constraint learning. In constraint satisfaction backtracking algorithms, constraint learning is a technique for improving efficiency. It works by recording new constraints whenever an inconsistency is found. This new constraint may reduce the search space, as future partial evaluations may be found inconsistent without further search. Clause learning is the name of this technique when applied to propositional satisfiability. Definition[edit] Backtracking algorithms work by choosing an unassigned variable and recursively solve the problems obtained by assigning a value to this variable.
Whenever the current partial solution is found inconsistent, the algorithm goes back to the previously assigned variable, as expected by recursion. A constraint learning algorithm differs because it tries to record some information, before backtracking, in form of a new constraint. If the partial solution is inconsistent, the problem instance implies the constraint stating that cannot be true for all at the same time.
All contain. GECODE home. Distributed constraint optimization. Distributed constraint optimization (DCOP or DisCOP) is the distributed analogue to constraint optimization. A DCOP is a problem in which a group of agents must distributedly choose values for a set of variables such that the cost of a set of constraints over the variables is either minimized or maximized. Distributed Constraint Satisfaction is a framework for describing a problem in terms of constraints that are known and enforced by distinct participants (agents). The constraints are described on some variables with predefined domains, and have to be assigned to the same values by the different agents.
Problems defined with this framework can be solved by any of the algorithms that are proposed for it. The framework was used under different names in the 1980s. Definitions[edit] DCOP[edit] A DCOP can be defined as a tuple , where: The objective of a DCOP is to have each agent assign values to its associated variables in order to either minimize or maximize Context[edit] implies that the agent. Constraint optimization. General form[edit] A general constrained minimization problem may be written as follows: where and Solution methods[edit] Many unconstrained optimization algorithms can be adapted to the constrained case, often via the use of a penalty method.
However, search steps taken by the unconstrained method may be unacceptable for the constrained problem, leading to a lack of convergence. This is referred to as the Maratos effect.[1] Equality constraints[edit] If the constrained problem has only equality constraints, the method of Lagrange multipliers can be used to convert it into an unconstrained problem whose number of variables is the original number of variables plus the original number of equality constraints. Inequality constraints[edit] With inequality constraints, the problem can be characterized in terms of the Karush–Kuhn–Tucker conditions, in which simple problems may be solvable. Linear programming[edit] Quadratic programming[edit] Constraint optimization problems[edit] Branch and bound[edit] Bartak - prednaska Programovani s omezujicimi podminkami. Bartak - Guide to Constraint Programming. Some of my constraint related tutorials (with slides to download): Filtering Techniques in Planning and Scheduling, ICAPS 2006, June 6-10, 2006, Cumbria, England [Slides] Constraint propagation and backtracking-based search, First international summer school on CP, September 11-15, 2005, Maratea, Italy [Lecture Notes], [Slides] Programming with Logic and Constraints, ESSLLI 2005, August 8-12, 2005, Edinburgh, UK Constraint Processing, SAC 2005, March 13, 2005, Santa Fe, U.S.A.
Constraint Satisfaction for Planning and Scheduling, ICAPS 2004, June 3, 2004, Whistler, Canada Foundations of Constraint Satisfaction, IJCAI 2003, August 11, 2003, Acapulco, Mexico Foundations of Constraint Satisfaction, NASSLLI 2003, June 17-21, 2003, Bloomington, Indiana, U.S.A. Foundations of Constraint Programming, ETAPS 2003, April 12, 2003. Warshaw, Poland Foundations of Constraint Satisfaction, ESSLLI 2002, August 5-9, 2002, Trento, Italy Now you can download a survey in the form of PDF file (146 KB) MindMap Combinatorial Optimization. MindMap DCOP.
Aplikace DCOP.