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 •A lgor ithms 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 11 n i x n j ij = = 1 2 0 (path enters and leaves) minimize: subject to: binary , ij ji x x =
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3 15-853 Page9 Traveling Salesman Problem 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 ∑∑ == n i n j ij ij x c 11 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) (??) minimize: subject to: 15-853 Page10 Traveling Salesman Problem 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 n i n j ij ij x c 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 : minimize : (out degrees = 1) (in degrees = 1) (??) 15-853 Page11 Traveling Salesman Problem The last set of constraints: prevents “subtours”: n j i n nx t t ij j i + , 2 1 Consider a cycle that goes from some node 4 to 5, t 4 –t 5 + nx 4,5 · n-1 ) t 5 ¸ t 4 + 1 Similarly t has to increase by 1 along each edge of the cycle that does not include vertex 1.
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This note was uploaded on 11/09/2008 for the course COMPUTER S 15853 taught by Professor Guyblelloch during the Fall '07 term at Carnegie Mellon.

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

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