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INTEGER LINEAR PROGRAMMING - INTRODUCTION

INTEGER LINEAR PROGRAMMING - INTRODUCTION INTEGER LINEAR PROGRAMMING a11x1+a12x2+ +a1nxn b1 Feasible Region: Z-Polyhedron (n dimensional) maxc1x1+c2x2+ + +a12x2+ +a1nxn +am2x2+ +amnxn bmx1,..,xn2 ZIntegrality Constraint INTEGER LINEAR PROGRAMMING Relaxation to a (real-valued) LINEAR Program How does the LP relaxation answer relate to the ILP answer? Integrality Gap Complexity of INTEGER LINEAR Programs NP-Completeness Some special cases of ILPs. Algorithms: Branch-And-Bound Gomory-Chvatal Cuts INTEGER LINEAR PROGRAMMING : LP RELAXATION 1. Relax an ILP to an LP 2. Examples with same answers and different answers. 3. Integrality gap. INTEGER LINEAR PROGRAMMING a11x1+a12x2+ +a1nxn b1 Feasible Region: Z-Polyhedron (n dimensional) maxc1x1+c2x2+ + +a12x2+ +a1nxn +am2x2+ +amnxn bmx1,..,xn2 ZInteger LINEAR Program Feasibility of ILP: INTEGER feasible solution. Unbounded ILP: INTEGER feasible solutions can achieve arbitrarily large values for the objective.

INTRODUCTION . Integer Linear Programming a 11 x 1 ... ILP AND COMBINATORIAL OPTIMIZATION Reducing 3-SAT to ILP . 3-SAT Problem x 1,x 2,x 3,x 4 Boolean Variables (x 1 OR x 2 OR ¬x 3) (¬x 2 OR ¬x 4 OR x 1) (x 1 OR x 2 OR ¬x 3) Find values for Boolean variables such that All the Clauses are True.

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