Lec04 min cost linear problems

Lecture 4: Min-Cost Linear Problems

Wai-Shing Luk (陆伟成)

Fudan University

2012 年 8 月 13 日

W.-S. Luk (Fudan Univ.) Lecture 4: Min-Cost Linear Problems 2012 年 8 月 13 日 1 / 18

Elementary Optimal Problems

Elementary Flow Problem: Elementary Potential Problem:

min dT x + p max b T u − (c T y + q)
s. t. c ≤ x, s. t. y ≤ d,
AT x = b, b(V ) = 0 Au = y

Theorem
The problems are dual to each other if
p + q = −c T d, (x − c)T (d − y ) = 0, c ≤ x, y ≤ d

Proof.
Since b T u = (AT x)T u = x T Au = x T y , [min] − [max] = (d T x + p) −
(b T u − [c T y + q]) = d T x + c T y − x T y + p + q = (x − c)T (d − y ) ≥ 0
[min] - [max] when equality holds.

Remark 1

We can formulate a linear problem either in its primal form or in its
dual form, depending on which one is more appropriate, for example,
whether design variables are in integral domain:
max-ﬂow problem (i.e. d T = [−1, −1, · · · , −1]T ) may be better to be
solved by a dual method.


Linear Optimal Problems

Optimal Flow Problem: Optimal Potential Problem:

min dT x + p max b T u − (c T y + q)
s. t. c − ≤ x ≤ c + , s. t. d − ≤ y ≤ d + ,
AT x = b, b(V ) = 0 Au = y

By modifying the network:
This problem can be reduced to the elementary case [?, pp.275–276]
Piecewise linear convex cost can be reduced to this linear problem [?,
p.239,p.260]

The problem has been studied extensively with a lot of applications.


Remark

We can transform the cost function to be non-negative by reversing
the orientation of the negative cost edges.
Then reduce the problem to the elementary case.
Most software packages only solve the problems in primal forms,
usually with c − = 0.


Algorithms for Optimal Flow Problems

Successive shortest path algorithm
Cycle cancellation method
iteratively insert additional minimal ﬂows according to a negative cycle
of the residual network, until no negative cycle is found.


For Special Cases

Max flow problem (d = −[1, · · · , 1])
Ford-Fulkerson algorithm: iteratively insert additional minimal flows
according to an argument path of the residual network, until no
argument path of the residual network is found.
Preflow Push-Relabel algorithm (dual method???)
Matching problems ([c − , c + ] = [0, 1])
Edmond’s blossom algorithm


Min-Cost Flow Problem (MCFP1)

Consider:
min dT x
s. t. 0 ≤ x ≤ c,
AT x = b, b(V ) = 0
where d is assumed to be non-negative.

Note: Optimal ﬂow problem can be transformed into this problem by
letting x = x − c − .

Algorithm idea: descent method: given a feasible x0 , ﬁnd a better
solution x1 = x0 + αp, where α is positive.


Review of General Descent Method

Input: f (x), initial x
Output: x ∗
while not converged do
Choose descent direction p;
Choose the step size α;
x := x + αp;
end


Some Common Descent Directions

For convex problems, the search direction must satisfy f (x) · p < 0
Gradient descent: p = − f (x)T
Steepest descent: (complicated)
Newton’s method: p = − 2 f (x)−1 f (x)

Note: Here, we have a natural way to choose p!


Min-Cost Flow Problem (MCFP1)

Let x1 = x0 + αp, then we have:

min d T x0 + αd T p ⇒ dT < 0
s. t. −x0 ≤ αp ≤ c − x0 ⇒ residual graph
AT p = 0 ⇒ p is a cycle!

In other words, choose p to be a negative cycle!
Simple negative cycle, or
Minimum mean cycle (c.f. Lecture 3)


Primal Method for MCFP1

Input: G (V , E ), c, d
Output: optimal opt x ∗
Initialize a feasible x and certain data structure;
while ( a negative cycle p found in G (x)) do
Choose a step size α;
If α is unbound, return UNBOUNDED;
If α = 0, break;
x := x + αp;
Update corresponding data structures;
end
return OPTIMAL;


Remarks

G (x) denotes the residual graph.
Negative cycle can be found using Bellman-Ford-like algorithms (c.f.
Lecture 2).
A precede graph or other data structures are used for ﬁnding negative
cycles eﬃciently.
Usually α is chosen such that one constraint is tight.


Dual Problem of MCFP1

It is well known that the dual problem of MCFP1 is given by

min cT I
s. t. y ≤ d + I ,
Au = y
I ≥0

where I is a slack variable.
Note: Delay padding problem can be formulated as this problem
where I represents the buﬀer delay.


Dual Problem of MCFP1 (cont’d)

Note: The problem can be transformed into the elementary potential
problem by splitting an edge between u(i) and u(j):

u(i) − u(h) = I (i, j) (1)
u(h) − u(j) ≤ d(i, j) (2)
u(i) − u(h) ≥ 0 (3)


Min-Cost Potential Problem (MCPP)

Consider:
min cT y
s. t. y ≤ d,
Au = y
where c is assumed to be non-negative.
Algorithm idea: given an initial feasible u0 , ﬁnd a better solution
u1 = u0 + βq, where β is positive:

min c T y0 + c T y ⇒ cT y < 0
s. t. y ≤ d − Au0 ⇒ residual graph
βAq = y ⇒ q is a “cut”!


Method for MCPP

Input: G (V , E ), c, d
Output: optimal value OPT, u ∗
Initialize a feasible u and certain data structure;
while ( a negative cut q found in G (u)) do
Choose a step size β;
If β is unbounded,return UNBOUNDED;
If β = 0, break;
u := u + βq;
Update corresponding data structures;
end
return OPTIMAL;


Remarks

Usually β is chosen such that one constraint is tight.
Min-cost potential problem is a dual of min-cost ﬂow problem, hence
algorithms can solve both problems.


Lec04 min cost linear problems

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