CPS 296.1Brief introduction to linear and mixed integer programmingVincent Conitzer conitzer@cs.duke.edu

Linear programs: examplemaximize 3x + 2ysubject to4x + 2y 16x + 2y 8x + y 5x 0y 0

We make reproductions of two paintingsPainting 1 sells for $30, painting 2 sells for $20Painting 1 requires 4 units of blue, 1 green, 1 redPainting 2 requires 2 blue, 2 green, 1 redWe have 16 units blue, 8 green, 5 red

Solving the linear program graphicallymaximize 3x + 2ysubject to4x + 2y 16x + 2y 8x + y 5x 0y 0

204682468optimal solution: x=3, y=2

Modified LPmaximize 3x + 2ysubject to4x + 2y 15x + 2y 8x + y 5x 0y 0

Optimal solution: x = 2.5, y = 2.5Solution value = 7.5 + 5 = 12.5

Half paintings?

Integer (linear) programmaximize 3x + 2ysubject to4x + 2y 15x + 2y 8x + y 5x 0, integery 0, integer

204682468optimal LP solution: x=2.5, y=2.5 (objective 12.5)optimal IP solution: x=2, y=3 (objective 12)

Mixed integer (linear) programmaximize 3x + 2ysubject to4x + 2y 15x + 2y 8x + y 5x 0y 0, integer

204682468optimal LP solution: x=2.5, y=2.5 (objective 12.5)optimal IP solution: x=2, y=3 (objective 12)optimal MIP solution: x=2.75, y=2 (objective 12.25)

Solving linear/integer programsLinear programs can be solved efficientlySimplex, ellipsoid, interior point methods(Mixed) integer programs are NP-hard to solveQuite easy to model many standard NP-complete problems as integer programs (try it!)Search type algorithms such as branch and boundStandard packages for solving theseGNU Linear Programming Kit, CPLEX, LP relaxation of (M)IP: remove integrality constraintsGives upper bound on MIP (~admissible heuristic)

Exercise in modeling: knapsack-type problemWe arrive in a room full of precious objectsCan carry only 30kg out of the roomCan carry only 20 liters out of the roomWant to maximize our total valueUnit of object A: 16kg, 3 liters, sells for $11There are 3 units availableUnit of object B: 4kg, 4 liters, sells for $4There are 4 units availableUnit of object C: 6kg, 3 liters, sells for $9Only 1 unit availableWhat should we take?

Exercise in modeling: cell phones (set cover)We want to have a working phone in every continent (besides Antarctica) but we want to have as few phones as possiblePhone A works in NA, SA, AfPhone B works in E, Af, AsPhone C works in NA, Au, EPhone D works in SA, As, EPhone E works in Af, As, AuPhone F works in NA, E

Exercise in modeling: hot-dog standsWe have two hot-dog stands to be placed in somewhere along the beachWe know where the people that like hot dogs are, how far they are willing to walkWhere do we put our stands to maximize #hot dogs sold? (price is fixed)location: 4#customers: 1willing to walk: 2location: 7#customers: 3willing to walk: 3location: 9#customers: 4willing to walk: 3location: 1#customers: 2willing to walk: 4location: 15#customers: 3willing to walk: 2