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# linear3 - Integer(linear Programming 15-853:Algorithms in...

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1 15-853 Page1 15-853:Algorithms in the Real World Linear and Integer Programming III – Integer Programming • Applications • Algorithms 15-853 Page2 Integer (linear) Programming Related Problems Mixed Integer Programming (MIP) Zero-one programming Integer quadratic programming Integer nonlinear programming x Z n x 0 Ax b subject to: c T x minimize: 15-853 Page3 History 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) 15-853 Page4 Current 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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2 15-853 Page5 Subproblems/Applications Facility location Locating warehouses or franchises (e.g. a Burger King) Set covering and partitioning Scheduling airline crews Multicomodity distribution Distributing auto parts Traveling salesman and extensions Routing deliveries Capital budgeting Other Applications VLSI layout, clustering 15-853 Page6 Knapsack Problem where: b = maximum weight c i = utility of item i a i = weight of item i x i = 1 if item i is selected, or 0 otherwise The problem is NP-hard. Integer (zero-one) Program: x binary ax b subject to: c T x maximize 15-853 Page7 Traveling Salesman Problem Find shortest tours that visit all of n cities. courtesy: Applegate , Bixby , Chvátal , and Cook 15-853 Page8 Traveling Salesman Problem c ij = c ji = distance from city i to city j (assuming symmetric version ) x ij if tour goes from i to j or j to i, and 0 otherwise Anything missing? ∑∑ = = n i n j ij ij x c 1 1 n i x n j ij = = 1 2 0 (path enters and leaves) minimize: subject to: binary , ij ji x x =