1. Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 1 Linear Programming Overview Formulation of the problem and example Incremental, deterministic algorithm Randomized algorithm Unbounded linear programs Linear programming in higher dimensions

2. Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 2 Algorithm 2D-LP Input:A 2-Dimensional Linear Program (H, c ) Output: Either one optimal vertex or  or a ray along which (H, c ) is unbounded. if UnboundedLP(H, c ) reports (H, c ) is infeasible then return UnboundedLP(H, c ) else h 1 := h; h 2 := h´ ; v 2 := l 1  l 2 h 3,...,h n := remaining half-planes of H for i:= 3 to n do if v i-1  h i then v i :=v i-1 else S i-1 := H i-1  * l i v i := 1-dim-LP(S i-1, c ) if v i not exists then return  return v n compute a random permutation h 3,..., h n

3. Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 3 Unbounded Linear Programs hihi ii :=The outward normal of h i  i := The smaller angle that  makes with c I min, an index with  i min = min  j, 1  j  n   c

4. Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 4 H min := { h j  H | = min } H par := { h j  H | = - min } H par { H min { H par { H min { h i* cc

5. Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 5 Lemma Let H = {h 1,h 2,...,h n } be a set of half-planes. Assuming that  (H min  H par ) is not empty. 1.If l i*  h j* is unbounded in the direction c for every half-plane h j in the set H \ (H min  H par ), then (H, c ) is unbounded along a ray contained in l i*. 2.If l i*  h j* is bounded in the direction c for some h j* in H \ (H min  H par ), then the linear program ({h i*, h j* }, c ) is bounded.

6. Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 6 Algorithm UnboundedLP Input:A 2-Dimensional Linear Program (H, c ) Output: Either one optimal vertex or  or a ray along which (H, c) is unbounded. 1.For each half plane h i  H compute  j 2.Let hi be half plane with  j = min  j, 1  j  n 3.H min := { h j  H | = min } 4.H par := { h j  H | = - min } 5. = H \ (H min  H par ), compute intersection in H min  H par 6.If the intersection is empty then report (H, c ) is feasible else Let h i  H min be the half-plane whose line bounds the intersection if there is half plane h j*  sucht hat l i*  h j* bounded in c then report ({h i*, h j* }, c ) is bounded else report is bounded along l i*  (  )

7. Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 7 Higher Dimensions Let h 1,...,h d H be the d certificate half-spaces that UnboundedLP returns. C i := h 1  h 2 ...  h i Lemma: Let d < i  n, and let C i be defined as above. 1. If v i-1  h i, then v i = v i-1 2. If v i-1  h i, then either C i =  or v i  g i, where g i is the hyperplane that bounds h i.

8. Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 8 Algorithm RandomozedLP Input : A linear program (H, c ). Output : Either one optimal vertex or  or a ray along which (H, c ) is unbounded. if UnboundedLP(H, c ) reports (H, c ) is unbounded then Report a ray along which (H, c ) unbounded. else Let h 1,...,h d  H be the certificate halfplanes returned by UnboundedLP, and let v d be their vertex of intersection Compute a random permutation h d+1,...,h n for i = d+1 to n do if v i-1  h i then v i = v i-1 else S i-1 := H i-1  * bd(h i v i := d-1-dim-LP(S i-1, c ) if v i not exists then return  Return v n

9. Lecture 4: Linear Programming Computational Geometry Prof. Dr. Th. Ottmann 9 Theorem The d-dimensional linear programming problem with n constraints can be solved in O(d!n) expected time using linear storage.