1. Course: Advanced Algorithms CSG713, Fall 2008 CCIS Department, Northeastern University Dimitrios Kanoulas

2. Maximizecx Subject to:Ax ≤ b Objective Function (c vector) Constraints where:x and c are two d-size vectors A is a nxd matrix b is a n-size right-hand vector Convex Polyhedron: Set of feasible points P:={x|Ax ≤ b} where: the vertices can be defined by the d constraints Polytope:A bounded Convex Polyhedron The form of the LP we will analyze:

3. Picture from: commons.wikimedia.org Starting from a vertex Try to find that vertex that optimizes the objective function TIME: (given random input constraints) Worst-Case: exponential Thus: An algorithm that uses the Simplex method in worst-case polynomial time is needed. (open problem for 50 years) Solution: Randomization Opt. Start

4. Given a Linear Program, we can randomly perturb it, in such a way that: We can use (Shadow) Simplex Method to solve this new nearby but different Linear Program and The solution of this new Linear Program to be the solution of the starting one. and It is polynomial in time w.h.p. A Randomized Polynomial-Time LP Simplex Method

5. 1.Reduce the starting LP to a problem of certifying boundedness Boundedness does not depend on RHS vector b, of the constraints Loop: 2.Perturb randomly b vector of constraints and run shadow-vertex Simplex Method on perturbed polytope to generate certificate of boundedness 3.If it fails in constant number of steps, change the distribution on perturbations and run the algorithm again. TIME: The number of iterations of this loop is polynomial on the bit-length of the input, with high probability (w.h.p.)

6. 1.Reduce the starting LP to a problem of certifying boundedness Boundedness does not depend on RHS vector b, of the constraints The problem of determining whether a set of linear constraints defines an unbounded polyhedron or not. A polyhedron is said to be bounded If the solution set is bounded Maximizecx Subject to:Ax ≤ b

7. 1.Reduce the starting LP to a problem of certifying boundedness Boundedness does not depend on RHS vector b, of the constraints Loop: 2.Perturb randomly b vector of constraints and run shadow-vertex Simplex Method on perturbed polytope to generate certificate of boundedness 3.If it fails in constant number of steps, change the distribution on perturbations and run the algorithm again. TIME: The number of iterations of this loop is polynomial on the bit-length of the input, with high probability (w.h.p.)

8. The shadow of a polytope P onto a 2-dimensions plane S is just the projection of P onto S. The shadow is a polygon Create shadow: Every vertex/edge of the shadow’s polygon is the image of a vertex/edge of the polytope P

9. The shadow of a polytope P onto a 2-dimensions plane S is just the projection of P onto S. The shadow is a polygon Idea: Find S such that the set of vertices of P that projects onto the boundary of the shadow polygon, are exactly the vertices of P that optimize the objective function in S.

10. 1.Input: a starting vertex u 0 of starting polytope P. 2.Choose some random objective function f optimized at u 0. 3.Set S = span(c,f). S is the 2-dimentional subspace which has c and f as bases. 4.Find the shadow of P onto S. Selection of 2-dimensions plane S S = span(f,c) f (opt. by u0) objective function c Image: BU`s prof. Teng presentation

11. 5. Each vertex y on P that projects onto the boundary of the shadow has a unique neighbor on P that projects onto the next vertex of the shadow in clockwise order. 6.Looking all the vertices on P that map to the boundary of the shadow we can move from vertex that optimize f to the vertex that optimize c. u 0 opt. f u opt. c or -c

12. Problem: By using the starting LP and the plane S we proposed, we cannot guarantee that: the number of edges of the shadow will be polynomial in size. [There is the possibility to be exponentially large] Solution: Guarantee that the number of edges of the shadow’s boundary will be polynomially large by perturbing the b-vector

13. The real problem is to bound the number of shadow’s edges 2 cases: Polytope in k-near-isotropic position Polytope in non k-near-isotropic position

14. P is the polytope: Ax≤b k 1

15. Upper bound of total shadow length (Shadow Size). Lower bound expected length of each edge. Perturb polytope P: P := {x | for each i: a i x ≤ b i }Q := {x | for each i: a i x ≤ b i + r i } r i : independent exponentially distributed random variable with expectation λ ( Pr[ri≥t] = e -t/l, t≥0 ) Proof of Poly Size of the shadow Number of edges of the shadow is poly large w.h.p.

16. 1.Starting LP: 2.Perturb b vector: guarantee that the number of edges of the shadow of the new polytope will be polynomially large w.h.p. 3.From the shadow find the vertex that optimizes the objective function c in polynomial number of steps, w.h.p. Maximizecx Subject to:Ax ≤ b

17. P is the polytope: Ax≤b Starting LP Maximizec x Subject to:Ax ≤ b Boundedness Certification Problem B w ≤ b’ B -> A, c, b Reduction Dual LP Minimizeb y Subject to:A T y = c y≥0 Combine Primal & Dual Need feasible x and y: Ax ≤ b A T y = c y≥0 c x = b y Dual

18. Boundedness independent of b-vector Boundedness Certification Problem B w ≤ b Find certificate of boundedness by minimizing and maximizing the objective function (c)

19. 1.Reduce the starting LP to a problem of certifying boundedness Boundedness does not depend on RHS vector b, of the constraints 2.We choose the correct 2-dimensions plane S: care only about the boundary of the shadow 3.Perturb randomly b vector: guarantee that the number of edges of the boundary of the shadow is polynomially large, w.h.p. 4.We can move from a starting vertex of the shadow to this that optimizes c or -c objective function in polynomial number of steps, by doing fixed number of steps. 5.If it fails in constant number of steps, change the distribution on perturbations and run the algorithm again.

20. (From the movie: π) I have uploaded the presentation on my web site: http://www.ccs.neu.edu/home/dkanou/ 11:15, restate my assumptions: 1.Mathematics is the language of nature. 2.Everything around us can be represented and understood through numbers. 3.If you graph these numbers, patterns emerge. Therefore: There are patterns everywhere in nature.