For instance, investment problems in which only a subset of the projects can be undertaken at any one time scheduling problems in which precedence among the activities must be respected problems in which a fixed cost is incurred only if the activity is undertaken, otherwise no cost is incurred (the so-called fixed-charge problems) and many other “matching,” “covering,” and “route selection” problems, to name but a few, have been successfully modeled using integer variables. Interest in IP, and in particular in ILP, stems from the ease with which numerous real-life problems can be modeled with the use of IP. For instance, a LP in one constraint is trivially solvable in one pass by the so-called “greedy algorithm,” while the same problem in integer variables is the notorious knapsack problem (KnP), which is NP-complete! An example of IP would be any LP whose variables are restricted to be integers, in which case one speaks of integer LP (ILP), which is clearly an abuse of language since the integer requirement vitiates the host of properties commonly associated with linearity! An example of combinatorial optimization would be to determine the minimum set of arcs in a connected graph G = (N, A) which “covers” all the nodes.Īs was hinted above, the presence of the integer requirement often transforms a solvable problem into an unsolvable one, or at least one that is several orders of magnitude more difficult than its continuous-variable analog. Combinatorial optimization is closely linked to IP, and is concerned with seeking the best subset of items (decisions, activities, etc.) satisfying particular criteria from a structured finite set of alternatives. Loosely speaking, IP is the domain of mathematical optimization in which some or all of the problem variables are restricted to be integers. Elmaghraby, in Encyclopedia of Physical Science and Technology (Third Edition), 2003 II.B Integer Programming (IP), Graphs, and Combinatorial Optimization
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