682 research outputs found
Approximately Fair Cost Allocation in Metric Traveling Salesman Games
A traveling salesman game is a cooperative game . Here N, the set of players, is the set of cities (or the vertices of the complete graph) andc D is the characteristic function where D is the underlying cost matrix. For all S⊆N, define c D (S) to be the cost of a minimum cost Hamiltonian tour through the vertices of S∪{0} where is called as the home city. Define Core ({\mathcal{G}})=\{x\in \Re^{|N|}:x(N)=c_{D}(N)\ \mbox{and}\ \forall S\subseteq N,x(S)\le c_{D}(S)\} as the core of a traveling salesman game . Okamoto (Discrete Appl. Math. 138:349-369, [2004]) conjectured that for the traveling salesman game with D satisfying triangle inequality, the problem of testing whether Core is empty or not is -hard. We prove that this conjecture is true. This result directly implies the -hardness for the general case when D is asymmetric. We also study approximately fair cost allocations for these games. For this, we introduce the cycle cover games and show that the core of a cycle cover game is non-empty by finding a fair cost allocation vector in polynomial time. For a traveling salesman game, let \epsilon\mbox{-Core}({\mathcal{G}})=\{x\in \Re^{|N|}:x(N)\ge c_{D}(N) and ∀ S⊆N, x(S)≤ε⋅c D (S)} be an ε-approximate core, for a given ε>1. By viewing an approximate fair cost allocation vector for this game as a sum of exact fair cost allocation vectors of several related cycle cover games, we provide a polynomial time algorithm demonstrating the non-emptiness of the log 2(|N|−1)-approximate core by exhibiting a vector in this approximate core for the asymmetric traveling salesman game. We improve it further by finding a -approximate core in polynomial time for some constantc. We also show that there exists an ε 0>1 such that it is -hard to decide whether ε 0-Core is empty or no
Operations Research Games: A Survey
This paper surveys the research area of cooperative games associated with several types of operations research problems in which various decision makers (players) are involved.Cooperating players not only face a joint optimisation problem in trying, e.g., to minimise total joint costs, but also face an additional allocation problem in how to distribute these joint costs back to the individual players.This interplay between optimisation and allocation is the main subject of the area of operations research games.It is surveyed on the basis of a distinction between the nature of the underlying optimisation problem: connection, routing, scheduling, production and inventory.cooperative games;operational research
Exploiting Bounds in Operations Research and Artificial Intelligence
Combinatorial optimization problems are ubiquitous in scientific research, engineering, and even our daily lives. A major research focus in developing combinatorial search algorithms has been on the attainment of efficient methods for deriving tight lower and upper bounds. These bounds restrict the search space of combinatorial optimization problems and facilitate the computa-tion of what might otherwise be intractable problems. In this paper, we survey the history of the use of bounds in both AI and OR. While research has been extensive in both domains, until very recently it has been too narrowly focused and has overlooked great opportunities to exploit bounds. In the past, the focus has been on the relaxations of constraints. We present methods for deriving bounds by tightening constraints, adding or deleting decision variables, and modifying the objective function. Then a formalization of the use of bounds as a two-step procedure is introduced. Finally, we discuss recent developments demonstrating how the use of this framework is conducive for eliciting methods that go beyond search-tree pruning
Reformulation and decomposition of integer programs
In this survey we examine ways to reformulate integer and mixed integer programs. Typically, but not exclusively, one reformulates so as to obtain stronger linear programming relaxations, and hence better bounds for use in a branch-and-bound based algorithm. First we cover in detail reformulations based on decomposition, such as Lagrangean relaxation, Dantzig-Wolfe column generation and the resulting branch-and-price algorithms. This is followed by an examination of Benders’ type algorithms based on projection. Finally we discuss in detail extended formulations involving additional variables that are based on problem structure. These can often be used to provide strengthened a priori formulations. Reformulations obtained by adding cutting planes in the original variables are not treated here.Integer program, Lagrangean relaxation, column generation, branch-and-price, extended formulation, Benders' algorithm
Combinatorial Optimization
Combinatorial Optimization is an active research area that developed from the rich interaction among many mathematical areas, including combinatorics, graph theory, geometry, optimization, probability, theoretical computer science, and many others. It combines algorithmic and complexity analysis with a mature mathematical foundation and it yields both basic research and applications in manifold areas such as, for example, communications, economics, traffic, network design, VLSI, scheduling, production, computational biology, to name just a few. Through strong inner ties to other mathematical fields it has been contributing to and benefiting from areas such as, for example, discrete and convex geometry, convex and nonlinear optimization, algebraic and topological methods, geometry of numbers, matroids and combinatorics, and mathematical programming. Moreover, with respect to applications and algorithmic complexity, Combinatorial Optimization is an essential link between mathematics, computer science and modern applications in data science, economics, and industry
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