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From a master of philosophy (m.phil.) in statistical science course focusing on operational research. Topics covered include integer linear programming (ilp), gomory's cutting plane method, extreme points, ellipsoidal algorithm, network flow, coalitional games, and the traveling salesman problem (tsp). Students are expected to understand concepts such as tableaus, optimal solutions, dual simplex algorithm, cramer's rule, and the role of extreme points in optimization.
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Thursday 30 May 2002 1.30 to 4.
Attempt FOUR questions There are six questions in total
The questions carry equal weight
1 Consider the ILP
minimize 3 x 1 + 4x 2 subject to 3 x 1 + x 2 > 4 x 1 + 2x 2 > 4 x 1 , x 2 > 0 x 1 , x 2 integer
Ignoring the integer constraints, the following tableau gives the optimal solution.
x 1 x 2 z 1 z 2
1 0 − 25 − (^1545)
0 1 15 − (^3585)
0 0 − 25 − (^95445)
Explain Gomory’s cutting plane method, illustrating it by showing that from the above tableau one can deduce that the optimal integer solution must satisfy the additional constraint 1 5 z^1 +^
2 5 z^2 >^
3
Use this constraint and the dual simplex algorithm to find the optimal solution to the ILP.
2 Let A be a m × n matrix of integers and let b be a vector in Rm. Let U be the largest absolute value of the entries of A and b. By using Cramer’s rule or otherwise, prove that every extreme point of the polyhedron P = {x ∈ Rn^ : Ax > b} satisfies
−n!U n^6 xj 6 n!U n, j = 1,... , n.
Give an account of the ellipsoidal algorithm for the problem of deciding whether or not P is empty. Describe the inputs to the algorithm and its main steps. You need not derive any detailed formulae, but you should explain enough so that the role of the above result is clear.
5 An instance of the ∆TSP decision problem is an undirected graph (with all possible edges present), a nonnegative integer cost cij = cji for each edge {i, j} and a nonnegative integer L. Edge costs are required to satisfy the triangle inequality. The question is whether there is a tour whose cost is no greater than L. Show that this problem is in the complexity class N P.
The ∆TSP evaluation problem is defined on the same instances (but omitting L), and the problem is to find the length of the shortest tour. Show that there exists a polynomial time algorithm for this problem if and only if there exists a polynomial time algorithm for the decision problem.
An instance of HCP is a graph G (with only some of the possible edges present). The question is whether there exists a tour that visits each vertex exactly once (a Hamiltonian circuit). Show that if HCP is N P-complete then the ∆TSP decision problem is also N P-complete.
Given the same data as a ∆TSP evaluation problem, the ∆TSP optimization problem is to find a minimum length tour; the MST optimization problem is to find a minimum spanning tree. Show that if there exists a polynomial time algorithm for the MST optimization problem then there also exists a polynomial time 1-approximation algorithm for the ∆TSP optimization problem, such that it produces a tour no more than twice the length of the minimal length tour.
6 Define what is meant by an equilibrium pair for a non-zero-sum two-person game.
State conditions under which at least one equilibrium pair is guaranteed to exist.
Two friends have different preferences for composers. Without consulting one another, they must each book for one of three possible concerts. They are pleased if they happen to book for the same concert. This is modelled by a game with the following payoff matrix.
Bach Mozart Schubert Bach
Mozart
Schubert
Find all the equilibrium pairs.
