Linear3 - Integer(linear Programming 15-853:Algorithms in the Real World Linear and Integer Programming III Integer Programming Applications

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1 15-853:Algorithms in the Real World g Linear and Integer Programming III – Integer Programming • Applications •A lgorithms 15-853 Page1 g Integer (linear) Programming minimize: c T x subject to: Ax b Related Problems – Mixed Integer Programming (MIP) ero- ne programming x 0 x Z n 15-853 Page2 Zero one programming – Integer quadratic programming – Integer nonlinear programming History • Introduced in 1951 (Dantzig) •T S P a s 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 Page3 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 eneral purpose packages do not tend to work as 15-853 Page4 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 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 15-853 Page5 g Capital budgeting Other Applications VLSI layout, clustering Knapsack Problem Integer (zero-one) Program: maximize c T x where: b = maximum weight = utility of item i subject to: ax b x binary 15-853 Page6 c i ut ty of t m a i = weight of item i x i = 1 if item i is selected, or 0 otherwise The problem is NP-hard. Traveling Salesman Problem Find shortest tours that visit all of n cities. 15-853 Page7 courtesy: Applegate , Bixby , Chvátal , and Cook Traveling Salesman Problem ∑∑ == n i n j ij ij x c 11 minimize: c ij = c ji = distance from city i to city j ssuming ymmetric version n i x n j ij = = 1 2 0 (path enters and leaves) subject to: binary , ij ji x x = 15-853 Page8 (assuming symmetric version ) x ij if tour goes from i to j or j to i, and 0 otherwise Anything missing?
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3 Traveling Salesman Problem ∑∑ == n i n j ij ij x c 11 minimize: n j i n nx t t n j x n i x ij j i n i ij n j ij + = = = = , 2 1 1 1 1 1 0 0 (out degrees = 1) (in degrees = 1) (??) subject to: 15-853 Page9 c ij = distance from city i to city j x ij = 1 if tour visits i then j, and 0 otherwise (binary) t i = arbitrary real numbers we need to solve for Traveling Salesman Problem n i n j ij ij x c minimize : n j i n nx t t n j x n i x ij j i n i ij n j ij + = = = = , 2 1 1 1 1 1 0 0 subject to : (out degrees = 1) (in degrees = 1) (??) 15-853 Page10 c ij = distance from city i to city j x ij = 1 if tour visits i then j, and 0 otherwise (binary) t i = arbitrary real numbers we need to solve for Traveling Salesman Problem The last set of constraints: prevents “subtours”: n j i n nx t t ij j i + , 2 1
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This note was uploaded on 01/26/2010 for the course COMPUTER S 15-853 taught by Professor Guyblelloch during the Fall '09 term at Carnegie Mellon.

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Linear3 - Integer(linear Programming 15-853:Algorithms in the Real World Linear and Integer Programming III Integer Programming Applications

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