115-853Page115-853:Algorithms in the Real WorldLinear and Integer Programming III– Integer Programming• Applications• Algorithms15-853Page2Integer (linear) ProgrammingRelated Problems–Mixed Integer Programming (MIP)–Zero-one programming–Integer quadratic programming–Integer nonlinear programmingx ∈Znx ≥0Ax ≤bsubject to:cTxminimize:15-853Page3History•Introduced in 1951 (Dantzig)•TSP as special case in 1954 (Dantzig)•First convergent algorithm in 1958 (Gomory)•General branch-and-bound technique 1960 (Land and Doig)•Frequently used to prove bounds on approximation algorithms (late 90s)15-853Page4Current Status•Has become “dominant” over linear programming in past decade•Saves industry Billions of Dollars/year•Can solve 10,000+ city TSP problems•1 million variable LP approximations•Branch-and-bound, Cutting Plane, and Separation all used in practice•General purpose packages do not tend to work as well as with linear programming --- knowledge of the domain is critical.
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215-853Page5Subproblems/Applications•Facility locationLocating warehouses or franchises (e.g. a Burger King)•Set covering and partitioningScheduling airline crews•Multicomodity distributionDistributing auto parts•Traveling salesman and extensionsRouting deliveries•Capital budgeting•Other ApplicationsVLSI layout, clustering15-853Page6Knapsack Problemwhere:b = maximum weightci= utility of item iai= weight of item ixi= 1 if item i is selected, or 0 otherwiseThe problem is NP-hard.Integer (zero-one) Program:x binaryax ≤bsubject to:cTxmaximize15-853Page7Traveling Salesman ProblemFind shortest tours that visit all of n cities.courtesy: Applegate, Bixby, Chvátal, and Cook15-853Page8Traveling Salesman Problemcij= cji= distance from city i to city j (assuming symmetric version)xijif tour goes from i to j or j to i, and 0 otherwiseAnything missing?∑∑==ninjijijxc11nixnjij≤≤=∑=120(path enters and leaves)minimize:subject to:binary,ijjixx=