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Combinatorial optimization

Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, where the set of feasible solutions is discrete or can be

Linear programming relaxation

In mathematics, the relaxation of a (mixed) integer linear program is the problem that arises by removing the integrality constraint of each variable. For example, in a 0–1 integer program, all constr

Branch and bound

Branch and bound (BB, B&B, or BnB) is an algorithm design paradigm for discrete and combinatorial optimization problems, as well as mathematical optimization. A branch-and-bound algorithm consists of

Vertex k-center problem

The vertex k-center problem is a classical NP-hard problem in computer science. It has application in facility location and clustering. Basically, the vertex k-center problem models the following real

Minimum k-cut

In mathematics, the minimum k-cut, is a combinatorial optimization problem that requires finding a set of edges whose removal would partition the graph to at least k connected components. These edges

Greedy randomized adaptive search procedure

The greedy randomized adaptive search procedure (also known as GRASP) is a metaheuristic algorithm commonly applied to combinatorial optimization problems. GRASP typically consists of iterations made

Gilbert–Pollack conjecture

No description available.

Parametric search

In the design and analysis of algorithms for combinatorial optimization, parametric search is a technique invented by Nimrod Megiddo for transforming a decision algorithm (does this optimization probl

Travelling salesman problem

The travelling salesman problem (also called the travelling salesperson problem or TSP) asks the following question: "Given a list of cities and the distances between each pair of cities, what is the

Extremal combinatorics

Extremal combinatorics is a field of combinatorics, which is itself a part of mathematics. Extremal combinatorics studies how large or how small a collection of finite objects (numbers, graphs, vector

Submodular set function

In mathematics, a submodular set function (also known as a submodular function) is a set function whose value, informally, has the property that the difference in the incremental value of the function

Maximum weight matching

In computer science and graph theory, the maximum weight matching problem is the problem of finding, in a weighted graph, a matching in which the sum of weights is maximized. A special case of it is t

Steiner travelling salesman problem

The Steiner traveling salesman problem (Steiner TSP, or STSP) is an extension of the traveling salesman problem. Given a list of cities, some of which are required, and the lengths of the roads betwee

Graph bandwidth

In graph theory, the graph bandwidth problem is to label the n vertices vi of a graph G with distinct integers so that the quantity is minimized (E is the edge set of G).The problem may be visualized

Linear bottleneck assignment problem

In combinatorial optimization, a field within mathematics, the linear bottleneck assignment problem (LBAP) is similar to the linear assignment problem. In plain words the problem is stated as follows:

Bridge and torch problem

The bridge and torch problem (also known as The Midnight Train and Dangerous crossing) is a logic puzzle that deals with four people, a bridge and a torch. It is in the category of river crossing puzz

Symmetry-breaking constraints

In the field of mathematics called combinatorial optimization, the method of symmetry-breaking constraints can be used to take advantage of symmetries in many constraint satisfaction and optimization

B*

In computer science, B* (pronounced "B star") is a best-first graph search algorithm that finds the least-cost path from a given initial node to any goal node (out of one or more possible goals). Firs

Configuration linear program

The configuration linear program (configuration-LP) is a particular linear programming used for solving combinatorial optimization problems. It was introduced in the context of the cutting stock probl

Bottleneck traveling salesman problem

The Bottleneck traveling salesman problem (bottleneck TSP) is a problem in discrete or combinatorial optimization. The problem is to find the Hamiltonian cycle (visiting each node exactly once) in a w

Quadratic knapsack problem

The quadratic knapsack problem (QKP), first introduced in 19th century, is an extension of knapsack problem that allows for quadratic terms in the objective function: Given a set of items, each with a

Weight function

A weight function is a mathematical device used when performing a sum, integral, or average to give some elements more "weight" or influence on the result than other elements in the same set. The resu

Smallest-circle problem

The smallest-circle problem (also known as minimum covering circle problem, bounding circle problem, smallest enclosing circle problem) is a mathematical problem of computing the smallest circle that

Metric k-center

In graph theory, the metric k-center or metric facility location problem is a combinatorial optimization problem studied in theoretical computer science. Given n cities with specified distances, one w

Combinatorial search

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

Gomory–Hu tree

In combinatorial optimization, the Gomory–Hu tree of an undirected graph with capacities is a weighted tree that represents the minimum s-t cuts for all s-t pairs in the graph. The Gomory–Hu tree can

Edge cycle cover

In mathematics, an edge cycle cover (sometimes called simply cycle cover) of a graph is a family of cycles which are subgraphs of G and contain all edges of G. If the cycles of the cover have no verti

Maximum satisfiability problem

In computational complexity theory, the maximum satisfiability problem (MAX-SAT) is the problem of determining the maximum number of clauses, of a given Boolean formula in conjunctive normal form, tha

Matroid parity problem

In combinatorial optimization, the matroid parity problem is a problem of finding the largest independent set of paired elements in a matroid. The problem was formulated by as a common generalization

Integer programming

An integer programming problem is a mathematical optimization or feasibility program in which some or all of the variables are restricted to be integers. In many settings the term refers to integer li

Max-flow min-cut theorem

In computer science and optimization theory, the max-flow min-cut theorem states that in a flow network, the maximum amount of flow passing from the source to the sink is equal to the total weight of

Quadratic assignment problem

The quadratic assignment problem (QAP) is one of the fundamental combinatorial optimization problems in the branch of optimization or operations research in mathematics, from the category of the facil

1-center problem

The 1-center problem, also known as minimax problem or minmax location problem, is a classical combinatorial optimization problem in operations research of facilities location type. In its most genera

Combinatorial data analysis

In statistics, combinatorial data analysis (CDA) is the study of data sets where the order in which objects are arranged is important. CDA can be used either to determine how well a given combinatoria

Kernighan–Lin algorithm

The Kernighan–Lin algorithm is a heuristic algorithm for finding partitions of graphs.The algorithm has important practical application in the layout of digital circuits and components in electronic d

Network flow problem

In combinatorial optimization, network flow problems are a class of computational problems in which the input is a flow network (a graph with numerical capacities on its edges), and the goal is to con

Closure problem

In graph theory and combinatorial optimization, a closure of a directed graph is a set of vertices C, such that no edges leave C. The closure problem is the task of finding the maximum-weight or minim

Knapsack problem

The knapsack problem is a problem in combinatorial optimization: Given a set of items, each with a weight and a value, determine the number of each item to include in a collection so that the total we

Hungarian algorithm

The Hungarian method is a combinatorial optimization algorithm that solves the assignment problem in polynomial time and which anticipated later primal–dual methods. It was developed and published in

Lin–Kernighan heuristic

In combinatorial optimization, Lin–Kernighan is one of the best heuristics for solving the symmetric travelling salesman problem. It belongs to the class of local search algorithms, which take a tour

Greedoid

In combinatorics, a greedoid is a type of set system. It arises from the notion of the matroid, which was originally introduced by Whitney in 1935 to study planar graphs and was later used by Edmonds

Assignment problem

The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows: The problem instance has a number of agents and a number of tasks. Any

Branch and cut

Branch and cut is a method of combinatorial optimization for solving integer linear programs (ILPs), that is, linear programming (LP) problems where some or all the unknowns are restricted to integer

Utility functions on indivisible goods

Some branches of economics and game theory deal with indivisible goods, discrete items that can be traded only as a whole. For example, in combinatorial auctions there is a finite set of items, and ev

Cut (graph theory)

In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut-set, the set of edges that have one endpoint in each subset of the partition. These

Dijkstra's algorithm

Dijkstra's algorithm (/ˈdaɪkstrəz/ DYKE-strəz) is an algorithm for finding the shortest paths between nodes in a graph, which may represent, for example, road networks. It was conceived by computer sc

Vehicle routing problem

The vehicle routing problem (VRP) is a combinatorial optimization and integer programming problem which asks "What is the optimal set of routes for a fleet of vehicles to traverse in order to deliver

Criss-cross algorithm

In mathematical optimization, the criss-cross algorithm is any of a family of algorithms for linear programming. Variants of the criss-cross algorithm also solve more general problems with linear ineq

A* search algorithm

A* (pronounced "A-star") is a graph traversal and path search algorithm, which is used in many fields of computer science due to its completeness, optimality, and optimal efficiency. One major practic

Cutting stock problem

In operations research, the cutting-stock problem is the problem of cutting standard-sized pieces of stock material, such as paper rolls or sheet metal, into pieces of specified sizes while minimizing

Generalized assignment problem

In applied mathematics, the maximum generalized assignment problem is a problem in combinatorial optimization. This problem is a generalization of the assignment problem in which both tasks and agents

Subadditive set function

In mathematics, a subadditive set function is a set function whose value, informally, has the property that the value of function on the union of two sets is at most the sum of values of the function

List of knapsack problems

The knapsack problem is one of the most studied problems in combinatorial optimization, with many real-life applications. For this reason, many special cases and generalizations have been examined. Co

Branch and price

In applied mathematics, branch and price is a method of combinatorial optimization for solving integer linear programming (ILP) and mixed integer linear programming (MILP) problems with many variables

Quadratic bottleneck assignment problem

In mathematics, the quadratic bottleneck assignment problem (QBAP) is one of fundamental combinatorial optimization problems in the branch of optimization or operations research, from the category of

Ellipsoid method

In mathematical optimization, the ellipsoid method is an iterative method for minimizing convex functions. When specialized to solving feasible linear optimization problems with rational data, the ell

Continuous knapsack problem

In theoretical computer science, the continuous knapsack problem (also known as the fractional knapsack problem) is an algorithmic problem in combinatorial optimization in which the goal is to fill a

Minimum relevant variables in linear system

MINimum Relevant Variables in Linear System (Min-RVLS) is a problem in mathematical optimization. Given a linear program, it is required to find a feasible solution in which the number of non-zero var

Weapon target assignment problem

The weapon target assignment problem (WTA) is a class of combinatorial optimization problems present in the fields of optimization and operations research. It consists of finding an optimal assignment

Change-making problem

The change-making problem addresses the question of finding the minimum number of coins (of certain denominations) that add up to a given amount of money. It is a special case of the integer knapsack

Floorplan (microelectronics)

In electronic design automation, a floorplan of an integrated circuit is a schematics representation of tentative placement of its major functional blocks. In modern electronic design process floorpla

Matroid intersection

In combinatorial optimization, the matroid intersection problem is to find a largest common independent set in two matroids over the same ground set. If the elements of the matroid are assigned real w

Matching (graph theory)

In the mathematical discipline of graph theory, a matching or independent edge set in an undirected graph is a set of edges without common vertices. Finding a matching in a bipartite graph can be trea

Maximum cut

For a graph, a maximum cut is a cut whose size is at least the size of any other cut. That is, it is a partition of the graph's vertices into two complementary sets S and T, such that the number of ed

Graph cut optimization

Graph cut optimization is a combinatorial optimization method applicable to a family of functions of discrete variables, named after the concept of cut in the theory of flow networks. Thanks to the ma

Multidimensional assignment problem

The multidimensional assignment problem (MAP) is a fundamental combinatorial optimization problem which was introduced by . This problem can be seen as a generalization of the linear assignment proble

Egalitarian item allocation

Egalitarian item allocation, also called max-min item allocation is a fair item allocation problem, in which the fairness criterion follows the egalitarian rule. The goal is to maximize the minimum va

Snow plow routing problem

The snow plow routing problem is an application of the structure of Arc Routing Problems (ARPs) and Vehicle Routing Problems (VRPs) to snow removal that considers roads as edges of a graph. The proble

Superadditive set function

In mathematics, a superadditive set function is a set function whose value when applied to the union of two disjoint sets is greater than or equal to the sum of values of the function applied to each

Quadratic pseudo-Boolean optimization

Quadratic pseudo-Boolean optimisation (QPBO) is a combinatorial optimization method for quadratic pseudo-Boolean functions in the form in the binary variables , with . If is submodular then QPBO produ

European Chapter on Combinatorial Optimization

The European Chapter on Combinatorial Optimization (also, EURO Working Group on Combinatorial Optimization, or EWG ECCO) is a working group whose objective is to promote original research in the field

Cederbaum's maximum flow theorem

Cederbaum's theorem defines hypothetical analog electrical networks which will automatically produce a solution to the minimum s–t cut problem. Alternatively, simulation of such a network will also pr

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