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15-853:Algorithms in the Real World

15-853:Algorithms in the Real World. Linear and Integer Programming III Integer Programming Applications Algorithms. Integer (linear) Programming. Related Problems Mixed Integer Programming (MIP) zero-one programming Integer quadratic programming Integer nonlinear programming.

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15-853:Algorithms in the Real World

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  1. 15-853:Algorithms in the Real World Linear and Integer Programming III Integer Programming Applications Algorithms 15-853

  2. Integer (linear) Programming • Related Problems • Mixed Integer Programming (MIP) • zero-one programming • Integer quadratic programming • Integer nonlinear programming 15-853

  3. 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) 15-853

  4. 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 wrok as well as with linear programming --- knowledge of the domain is critical. 15-853

  5. Subproblems/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, clustering 15-853

  6. Knapsack Problem Integer (zero-one) Program: • where: b = maximum weight ci = utility of item i ai = weight of item i xi = 1 if item i is selected, or 0 otherwise • The problem is NP-hard. 15-853

  7. Traveling Salesman Problem • cij = distance from city i to city j • xij = 1 if tour visits i then j, and 0 otherwise (binary) • ti = arbitrary real numbers we need to solve for (out degrees = 1) (in degrees = 1) (??) 15-853

  8. Traveling Salesman Problem • The last set of constraints: • prevents “subtours”: Consider a cycle that goes from some node 4 to 5, t4 – t5 + nx4,5· n-1 ) t5¸ t4 + 1 Similarly t has to increase by 1 along each edge of the cycle that does not include vertex 1. Therefore, for a tour of length m that does not go through vertex 1, t4¸ t4 + m, a contradiction. Every cycle must go through vertex 1. Together with other constraints, it forces one cycle. 15-853

  9. Traveling Salesman Problem • Many “Real World” applications based on the TSP. • They typically involve more involved constraints • Not just routing type problems.Consider a drug company with k drugs they can make at a lab. They make them one at a time.The cost of converting the equipment from making drug i to drug j is cij • Current best solutions are based on IP • Applegate, Bixby, et. al., have solutions for more than 15K cities in Germany > 150,000 CPU hours (more info) • Involves “branch-and-bound” and “cutting planes” 15-853

  10. Set Covering and Partitioning • Given m sets and n items: Columns = sets Rows = items Set covering: Set partitioning: 15-853

  11. Set Covering and Partitioning • Best cover: s2, s4, s5 = .5 • Best partition: s4, s6 = .7 15-853

  12. Set Covering and Partitioning • Applications: • Facility location. Each set is a facility (e.g. warehouse, fire station, emergency response center).Each item is an area that needs to be covered • Crew scheduling.Each set is a route for a particular crew member (e.g. NYC->Pit->Atlanta->NYC).Each item is a flight that needs to be covered. 15-853

  13. Constraints Expressible with IP • Logical constraints (x1, x2 binary): Combining constraints: y is a binary variable, M needs to be “large”,a1, a2, and x can be vectors 15-853

  14. Piecewise Linear Functions • Intentionally left blank. 15-853

  15. Algorithms • Use a linear program • round to integer solution (what if not feasible?) • Search • Branch and bound (integer ranges) • Implicit (0-1 variables) • Cutting planes • Many variants 15-853

  16. Important Properties • LP solution is an upper bound on IP solution (assuming maximization) • If LP is infeasible then IP is infeasible • If LP solution is integral (all variables have integer values), then it is the IP solution. 15-853

  17. Linear Programming Solution • Some LP problems will always have integer solutions • transportation problem • assignment problem • min-cost network flow These are problems with a unimodular matrix A. • Solve as linear program and round. Can violate constraints, and be non-optimal. Works OK if • integer variables take on large values • accuracy of constraints is questionable 15-853

  18. x2 = 1 x2 = 0 x1 = 1 x1 = 0 x3 = 1 x3 = 0 Branch and Bound • Lets first consider 0-1 programs. • Exponential solution: try all {0,1}n • Branch-and-bound solution: Traverse tree keeping current best solution. If it can be shown that a subtree never improves on the current solution, or is infeasable, prune it. 15-853

  19. Zero-One Branch and Bound • minimize: z = cTx, subject to: Ax · b, x ¸ 0, x 2 {0,1}n • functionIP(C, c, xf, z*) • // xf a subset of the variables which are fixed • // z* is the cost of current best solution x = xf + 0 // set free variables to zero if cx > z* or no feasible completion return z* if feasible(C,x) then cx pick a non-fixed variable xi from x z* = IP(xf[ {xi = 0}, c, z*) z* = IP(xf[ {xi = 1}, c, z*) return z* • end 15-853

  20. Zero-One Branch and Bound • Checking for feasible completions: check each constraint and find if minimum of left is greater than right. • Example: • xf = {x1 = 1, x3 = 0} • and one of the constraints is • 3x1 + 2x2 – x3 + x4· 2 • then • 3 + 2x2 - 0 + x4· 2 • 2x2 + x4 · -1 • which is impossible. 15-853

  21. Integer Branch and Bound • The zero-one version is sometimes called “implicit enumeration” since it might enumerate all possibilities. • An integer version cannot branch on all possible integer values for a variable. Even if the integer range is bounded, it is not practical. • Will “bound” by adding inequalities to split the two branches. Since solutions are integral, each split can remove a strip of width 1 15-853

  22. Integer Branch and Bound • maximize: z = cTx, subject to: Ax · b, x ¸ 0, x 2 Zn • functionIP(C, c, z*) • // C is a set of constraints Ax · b • // z* is the cost of current best solution z, x, f = LP(C,c) // f indicates whether feasible if not(f) or z < z* return z* if integer(x) return z pick a non-integer variable xi’ from x z* = IP(C with xi·bxi’c, c, z*) z* = IP(C with –xi· -dxi’e, c, z*) return z* • end 15-853

  23. Example l z u feasible Find optimal solution. Cut along y axis, and make two recursive calls 15-853

  24. Example • Find optimal solution. • Solution is integral, so return it as current best z* l z u 15-853

  25. Example • Find optimal solution. It is better than z*. • Cut along x axis, and make two recursive calls z u l = z* 15-853

  26. Example • Infeasible, Return. z u l 15-853

  27. Example z l u u l = z* Find optimal solution. It is better than z*. Cut along y axis, and make two recursive calls 15-853

  28. Example z l u = z* Find optimal solution. Solution is integral and better than z*. Return as new z*. 15-853

  29. Example z l u = z* Find optimal solution. Not as good as z*, return. 15-853

  30. Cutting Plane • General Algorithm: • function IP(C, c) x = LP(C,c) while not(integer(x) or infeasible(x)) add constraints to C x = LP(C,c) • The constraints must have the properties that • they do not cut off any feasible integer solutions • they do cut off the current optimal linear solution 15-853

  31. Cutting Plane • Note that we are removing a corner, and no integer solutions are being excluded. z feasible New plane 15-853

  32. Picking the Plane • Method 1: Gomory cuts (1958) • Cuts are generated from the LP TableauEach row defines a potential cut • Guaranteed to converge on solution • General purpose, but inefficient in practice • Method 2: problem specific cuts • Consider the problem at hand and generate cuts based on its structure • Not general purpose, but can work very well in practice 15-853

  33. Underconstraint Problems • It is possible to underconstrain the initial problem, and add constraints (cuts) later as needed. • For example, in TSP formulation, one could leave out the subtour elimination constraints. Constraints would only be added when a subtour is formed, and then only enough to break the subtour.Such constraints can take advantage of properties of graphs and tours. (more info) • This technique is used frequently in practice. 15-853

  34. Practical Developments • Good formulations, heuristics and theoryGoal: to get LP solution as close as possible to IP solutionDisaggregation, adding constraints (cuts) • PreprocessingAutomatic methods for reformulationSome interesting graph theory is involved • Cut generation (branch-and-cut)Add cuts during the branch-and-bound • Column generationImprove formulation by introducing an exponential number of variables. 15-853

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