 
Summary: Algorithmic Methods Fall Semester, 2010/11
Exercise 1: November 8, 2010
Lecturer: Prof. Yossi Azar
Write short but full and accurate answers. Each question should start on a new separate page and each
of its parts should not exceed a page.
1. (a) Show that there can exist a degenerate vertex whose corresponding basis is unique.
(b) Prove or disprove: every vertex of an LP is nondegenerate implies that the optimal solution
is unique.
2. Complete the proof of the strong part of the duality theorem for the general form. Note that in
class we assumed the primal is LPS (standard form LP).
3. (a) Prove that if an LPS (standard form LP) has a nondegenerate vertex which is an optimal
solution then the dual problem has a unique optimal solution.
(b) Does the above remain true if the LPS has both a nondegenerate vertex and a degenerate
vertex which are optimal ?
(c) Does the above remain true if the LPS has an optimal solution (not necessarily a vertex) with
m variables of nonzero values (the LPS has m equations ) ?
4. Prove that a variable which has just left the basis in the simplex algorithm cannot reenter on the
very next pivot step.
5. We are given a set of n points in R2, (x1, y1), . . . , (xn, yn). Our goal to find a function f(x) = ax+b
such that
