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Chapter 7: Q6E (page 240)

Give an example of a linear program in two variables whose feasible region is infinite, but such that there is an optimum solution of bounded cost.

Short Answer

Expert verified

Example 1:

x≥0y≥0

Example 2:

x≥0y≥0x−y≤1

Step by step solution

01

Explain Linear Program

Linear program is used for optimization tasks that has constraints and the optimization criterion as linear functions. A linear program has the set of variables that needs to be assign with the real values to satisfy the linear inequalities and to minimize or maximize a given linear objective function.

02

Give an example of linear program

Example 1:

Consider two variablexandy. The linear program is as follows:

x≥0y≥0

Operation or requirement for the constraints :

minx,y(x+y)

The above solution is possible in bounded cost.

Example 2:

Consider the constraints as follows,

x≥0y≥0x−y≤1

Operation required:

maxi,j(x−2y)

Therefore, an example for the linear program of two variables were obtained.

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Most popular questions from this chapter

Question: A linear program for shortest path. Suppose we want to compute the shortest path from node s to node t in a directed graph with edge lengths le>0.

a) Show that this is equivalent to finding an s - tflow fthat minimizes ∑elefesubject to size (f) = 1. There are no capacity constraints.

b) Write the shortest path problem as a linear program.

c) Show that the dual LP can be written as

role="math" localid="1659250472483" maxxs-xtxu-xv≤luvforall(u,v)∈E

d) An interpretation for the dual is given in the box on page 223. Why isn’t our dual LP identical to the one on that page?

There are many common variations of the maximum flow problem. Here are four of them.

(a) There are many sources and many sinks, and we wish to maximize the total flow from all sources to all sinks.

(b) Each vertex also has a capacity on the maximum flow that can enter it.

(c) Each edge has not only a capacity, but also a lower bound on the flow it must carry.

(d) The outgoing flow from each node u is not the same as the incoming flow, but is smaller by a factor of (1-∈U), whererole="math" localid="1659789093525" ∈u is a loss coefficient associated with node u.

Each of these can be solved efficiently. Show this by reducing (a) and (b) to the original max-flow problem, and reducing (c) and (d) to linear programming.

The pizza business in Little Town is split between two rivals, Tony and Joey. They are each investigating strategies to steal business away from the other. Joey is considering either lowering prices or cutting bigger slices. Tony is looking into starting up a line of gourmet pizzas, or offering outdoor seating, or giving free sodas at lunchtime. The effects of these various strategies are summarized in the following payoff matrix (entries are dozens of pizzas, Joey’s gain and Tony’s loss).




TONY




Gourmet

Seating

Freesoda

JOEY

Lower price

+2

0

-3


BiggerSlices

_1

-2

+1

For instance, if Joey reduces prices and Tony goes with the gourmet option, then Tony will lose 2 dozen pizzas worth of nosiness to Joey.

What is the value of this game, and what are the optimal strategies for Tony and Joey?

In a particular network G = (V, E) whose edges have integer capacities ce, we have already found the maximum flow f from node to node t. However, we now find out that one of the capacity values we used was wrong: for edge (u, v) we used cuv whereas it should have been cuv. -1 This is unfortunate because the flow f uses that particular edge at full capacity: f = c.

We could redo the flow computation from scratch, but there’s a faster way. Show how a new optimal flow can be computed inO(|V|+|E|) time.

Consider the following generalization of the maximum flow problem.

You are given a directed network G=(V,E)with edge capacities {ce}. Instead of a single (s,t)pair, you are given multiple pairs (s1,t1),(s2,t2),…,(sk,tk), where the siare sources of Gand tithe are sinks of G. You are also given kdemands d1,…,dk. The goal is to find kflows f(1),…,f(k)with the following properties:

  • f(i)is a valid flow fromSi toti .
  • For each edge e, the total flowfe(1)+fe(2)+…+fe(k) does not exceed the capacityce .
  • The size of each flowf(i) is at least the demand di.
  • The size of the total flow (the sum of the flows) is as large as possible.

How would you solve this problem?
See all solutions

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