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# formulations - ILP Formulations Spanning Trees...

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ILP Formulations Spanning Trees Cutting-plane separation k-MST C/G Cutting Planes ILP Formulation Properties and Strenghtening Techniques Version 2009.2 Andreas M. Chwatal Algorithms and Data Structures Group Institute of Computer Graphics and Algorithms Vienna University of Technology VU Fortgeschrittene Algorithmen und Datenstrukturen Mai 2009 1 ILP Formulations Spanning Trees Cutting-plane separation k-MST C/G Cutting Planes What is a strong/good (I)LP formulation? Linear program: low number of variables low number of constraints complexity grows polynomially in these entities Integer Linear Program: Let F = { x 1 , . . . , x k } be an feasible integer solution to some ILP Convex hull conv ( F ) = k i =1 λ i x i k i =1 λ i = 1 , λ i 0 Let P denote the polyhedron associated to the linear relaxation of the ILP: conv ( F ) P Optimal case: conv ( F ) = P , but this is often not achievable In a strong formulation P closely approximates conv ( F ) 2 ILP Formulations Spanning Trees Cutting-plane separation k-MST C/G Cutting Planes The pigeonhole Suppose, we want to place n + 1 items into n holes, such that each hole contains exactly one item. (This is clearly infeasible) n j =1 x ij = 1 , i = 1 , . . . , n + 1 (1a) x ij ∈ { 0 , 1 } i = 1 , . . . , n + 1 , j = 1 , . . . , n (1b) Two alternatives for the additional constraints: x ij + x kj 1 , j = 1 , . . . , n , i = k , i , k = 1 , . . . , n + 1 (2) n +1 i =1 x ij 1 , j = 1 , . . . , n (3) Linear relaxation of formulation with (2) is feasible (with x ij = 1 / n ), but infeasible with (3). 3 ILP Formulations Spanning Trees Cutting-plane separation k-MST C/G Cutting Planes Example: Facility Location Given: n potential facility locations with opening costs c j , m clients with service costs d ij (for client i from facility j ) Formulation 1: min n j =1 c j y j + m i =1 n j =1 d ij x ij (4a) s . t . n j =1 x ij = 1 for all i (4b) x ij y j for all i , j (4c) 0 x ij 1 , y i ∈ { 0 , 1 } for all i , j (4d) 4

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ILP Formulations Spanning Trees Cutting-plane separation k-MST C/G Cutting Planes Facility Location (cont.) Formulation 2: min n j =1 c j y j + m i =1 n j =1 d ij x ij (5a) s . t . n j =1 x ij = 1 for all i (5b) m i =1 x ij m · y j for all j (5c) 0 x ij 1 , y i ∈ { 0 , 1 } for all i , j (5d) Which formulation is better? F1 has n + nm constraints, whereas F2 has n + m constraints But, P F 1 P F 2 !! (where P X denotes the polyhedron corresponding to the LP relaxation of formulation X ) 5 ILP Formulations Spanning Trees Cutting-plane separation k-MST C/G Cutting Planes Spanning Trees In this section we study various ST formulations. Why? Spanning tree (ST) problems arise in various contexts, often as subproblems to more complex problems. While the MST is solvable in polynomial time, further constraints usually make the problem NP-hard. Variants: Minimum (Weight) Spanning Tree Steiner Tree, Price Collecting ST { Delay, Resource, Hop, Diameter, ... } - Constrained S.T. Minimum Label Spanning Tree . . .
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