The travelling salesman problem asks the following question: "Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city?" It is an NP-hard problem in combinatorial optimization, important in theoretical computer science and operations research.
Linear programming (LP), also called linear optimization, is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming.
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 the edges in a minimum cut, i.e., the smallest total weight of the edges which if removed would disconnect the source from the sink.
The assignment problem is a fundamental combinatorial optimization problem. In its most general form, the problem is as follows:
In optimization theory, maximum flow problems involve finding a feasible flow through a flow network that obtains the maximum possible flow rate.
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 reduced to a discrete set. Typical combinatorial optimization problems are the travelling salesman problem ("TSP"), the minimum spanning tree problem ("MST"), and the knapsack problem. In many such problems, such as the ones previously mentioned, exhaustive search is not tractable, and so specialized algorithms that quickly rule out large parts of the search space or approximation algorithms must be resorted to instead.
A randomized algorithm is an algorithm that employs a degree of randomness as part of its logic or procedure. The algorithm typically uses uniformly random bits as an auxiliary input to guide its behavior, in the hope of achieving good performance in the "average case" over all possible choices of random determined by the random bits; thus either the running time, or the output are random variables.
In graph theory, a flow network is a directed graph where each edge has a capacity and each edge receives a flow. The amount of flow on an edge cannot exceed the capacity of the edge. Often in operations research, a directed graph is called a network, the vertices are called nodes and the edges are called arcs. A flow must satisfy the restriction that the amount of flow into a node equals the amount of flow out of it, unless it is a source, which has only outgoing flow, or sink, which has only incoming flow. A network can be used to model traffic in a computer network, circulation with demands, fluids in pipes, currents in an electrical circuit, or anything similar in which something travels through a network of nodes.
In combinatorial optimization, network flow problems are a class of computational problems in which the input is a flow network, and the goal is to construct a flow, numerical values on each edge that respect the capacity constraints and that have incoming flow equal to outgoing flow at all vertices except for certain designated terminals.
In graph theory, a minimum cut or min-cut of a graph is a cut that is minimal in some metric.
In computational complexity theory, Polynomial Local Search (PLS) is a complexity class that models the difficulty of finding a locally optimal solution to an optimization problem. The main characteristics of problems that lie in PLS are that the cost of a solution can be calculated in polynomial time and the neighborhood of a solution can be searched in polynomial time. Therefore it is possible to verify whether or not a solution is a local optimum in polynomial time. Furthermore, depending on the problem and the algorithm that is used for solving the problem, it might be faster to find a local optimum instead of a global optimum.
In constraint satisfaction, a decomposition method translates a constraint satisfaction problem into another constraint satisfaction problem that is binary and acyclic. Decomposition methods work by grouping variables into sets, and solving a subproblem for each set. These translations are done because solving binary acyclic problems is a tractable problem.
The minimum-cost flow problem (MCFP) is an optimization and decision problem to find the cheapest possible way of sending a certain amount of flow through a flow network. A typical application of this problem involves finding the best delivery route from a factory to a warehouse where the road network has some capacity and cost associated. The minimum cost flow problem is one of the most fundamental among all flow and circulation problems because most other such problems can be cast as a minimum cost flow problem and also that it can be solved efficiently using the network simplex algorithm.
The multi-commodity flow problem is a network flow problem with multiple commodities between different source and sink nodes.
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 to a given set of customers?" It generalises the travelling salesman problem (TSP). It first appeared in a paper by George Dantzig and John Ramser in 1959, in which the first algorithmic approach was written and was applied to petrol deliveries. Often, the context is that of delivering goods located at a central depot to customers who have placed orders for such goods. The objective of the VRP is to minimize the total route cost. In 1964, Clarke and Wright improved on Dantzig and Ramser's approach using an effective greedy algorithm called the savings algorithm.
In mathematics, a graph partition is the reduction of a graph to a smaller graph by partitioning its set of nodes into mutually exclusive groups. Edges of the original graph that cross between the groups will produce edges in the partitioned graph. If the number of resulting edges is small compared to the original graph, then the partitioned graph may be better suited for analysis and problem-solving than the original. Finding a partition that simplifies graph analysis is a hard problem, but one that has applications to scientific computing, VLSI circuit design, and task scheduling in multiprocessor computers, among others. Recently, the graph partition problem has gained importance due to its application for clustering and detection of cliques in social, pathological and biological networks. For a survey on recent trends in computational methods and applications see Buluc et al. (2013). Two common examples of graph partitioning are minimum cut and maximum cut problems.
David Bernard Shmoys is a Professor in the School of Operations Research and Information Engineering and the Department of Computer Science at Cornell University. He obtained his Ph.D. from the University of California, Berkeley in 1984. His major focus has been in the design and analysis of algorithms for discrete optimization problems.
In applied mathematics, Graver bases enable iterative solutions of linear and various nonlinear integer programming problems in polynomial time. They were introduced by Jack E. Graver. Their connection to the theory of Gröbner bases was discussed by Bernd Sturmfels. The algorithmic theory of Graver bases and its application to integer programming is described by Shmuel Onn.
Approximate max-flow min-cut theorems are mathematical propositions in network flow theory. They deal with the relationship between maximum flow rate ("max-flow") and minimum cut ("min-cut") in a multi-commodity flow problem. The theorems have enabled the development of approximation algorithms for use in graph partition and related problems.
In mathematical optimization, the network simplex algorithm is a graph theoretic specialization of the simplex algorithm. The algorithm is usually formulated in terms of a minimum-cost flow problem. The network simplex method works very well in practice, typically 200 to 300 times faster than the simplex method applied to general linear program of same dimensions.
