Operations Research
Operations Research
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New in 6.0

Enhanced Coverage of Intractable (NP-Complete) Problems

  • Clique problem
  • Vertex cover problem
  • Independent set problem
  • 3-satisfiability (3-SAT) problem
  • 3-coloring of planar graphs

New in 5.5

Graph Algorithms

  • ShortestPathOneToAll calculates the shortest path from a starting node to all other nodes on a directed graph.
  • ShortestPathAllToAll calculates simultaneously the shortest path from all nodes to all nodes on a directed graph.

Improved in 5.5

Graph Algorithms

  • AVLTreeInsert inserts a new key into a balanced binary search tree. The tree is rebalanced if necessary. AVLTreeInsert may be used to generate a balanced tree step by step from an empty tree.
  • AVLTreeDelete deletes the node carrying the given key in a balanced tree. The tree is rebalanced if necessary.

New in 5.4

Game Theory

  • NashEquilibrium returns one equilibrium point for Bimatrix Games where A and B are the payoff matrices of players 1 and 2, respectively.
  • NashEquilibriumQ — NashEquilibriumQ tests whether strategies x, y correspond to an equilibrium point with respect to payoff matrices A, B.
  • DynamicEvolution computes the time evolution of strategies based on a payoff matrix with initial strategies up to a final time.

Binary Search Trees

  • GenerateAVLTree generates a balanced binary search tree from a given list of keys.
  • AVLTreeInsert inserts a new key into a balanced binary search tree. The tree is rebalanced if necessary.
  • AVLTreeDelete deletes the node carrying the given key in a tree. The tree is rebalanced if necessary.
  • OptimalBinarySearchTree generates a binary search tree such that the average access time is minimized. The method is based on dynamic programming.
  • ShowTree generates graph of binary tree.

General Features

  • The graph algorithms have been extended by routines for binary search trees.
  • The matrix game branch has given new routines.

New in 5.3

  • MatrixGames calculates optimal strategies for two-person zero-sum games. The optimal strategy is in general a mixed one, where players 1 and 2 independently pick decisions (alternatives) si and tj, with probability xi and yj, respectively. Such optimal mixed strategies always exist. Payment to a player will be less if he deviates from his optimal strategy whereas the other player sticks to his optimal strategy (Nash equilibrium).

General Features

  • Implemented a new routine DualSimplex that solves linear constrained problems with the dual simplex method
  • Representation of graph routines graphically
  • Linear optimization: revised simplex with sensitivity analysis and branch-and-bound methods
  • Shortest-path tasks: one-to-all and all-to-all shortest connections on a graph
  • Graph algorithms: calculations of flows on a graph
  • Combinatorial optimization and heuristics: several routines that can be used to solve assignment, traveling salesman and other problems and can be used for implementation of additional effective heuristics
  • Documentation fully compatible with Mathematica 7
  • Completely new branch of problems: graph algorithms
  • MaximumFlow: calculates the maximum flow on a network defined by a list of links from a source node to a sink node
  • MinCostsFlow: calculates the minimum costs flow on a network from a source node to a sink node, at given net flow
  • MinimalSpanningTree: calculates a minimal spanning tree of a network with not directed links; the algorithm of Prim is implemented
  • BiConnectComp: calculates the list of articulation points and the biconnected components of a graph given by its link list
  • StronglyConnectedComponents: calculates a list of subsets of nodes ("block nodes") each inducing a strongly connected component of the directed graph
  • TopologicalOrdering: calculates a reordered list of nodes if topological ordering is possible; an empty is returned otherwise
  • LCShortestPath: calculates the shortest path in a network defined through a list of links; negative weights are allowed as long as no negative cycles arise