1. Topics in Algorithms
Lap Chi Lau

2. TU-simplex-ellipsoid

3. Totally Unimodular Matrices
Simplex Algorithm
Elliposid Algorithm
Lecture 11: Feb 23

4. Integer Linear programs
Method: model a combinatorial problem as a linear program.
Goal: To prove that the linear program has an integer optimum solution for every objective function.
Goal: Every vertex (basic) solution is an integer vector.
Consequences:
The combinatorial problem (even the weighted case) is polynomial time solvable.
Min-max theorems for the combinatorial problem through the LP-duality theorem.

5. Method 1: Convex Combination
A point y in Rn is a convex combination of
if y is in the convex hull of
Fact: A vertex solution is not a convex combination of some other points.
Method 1: A non-integer solution must be a convex combination of some other points.
Examples: bipartite matching polytope, stable matching polytope.

6. Method 2: Linear Independence
Tight inequalities: inequalities achieved as equalities
Vertex solution: unique solution of n linearly independent tight inequalities
Think of 3D.
Method 2: A set of n linearly independent tight inequalities must have some
tight inequalities of the form x(e)=0 or x(e)=1. Proceed by induction.
Examples: bipartite matching polytope, general matching polytope.

7. Method 3: Totally Unimodular Matrices
m constraints, n variables
Vertex solution: unique solution of n linearly independent tight inequalities
Can be rewritten as:
That is:

8. Method 3: Totally Unimodular Matrices
Assuming all entries of A and b are integral
When does has an integral solution x?
By Cramer’s rule
here Ai is the matrix where each column is equal to the
corresponding column in A except the i-th column is equal to b.
x would be integral if det(A) was equal to +1 or -1.

9. Method 3: Totally Unimodular Matrices
A matrix is totally unimodular if the determinant
of each square submatrix of is 0, -1, or +1.
Theorem 1: If A is totally unimodular,
then every vertex solution of is integral.
Proof (follows from previous slides):
a vertex solution is defined by a set of n linearly independent tight inequalities.
Let A’ denote the (square) submatrix of A which corresponds to those inequalities.
Then A’x = b’, where b’ consists of the corresponding entries in b.
Since A is totally unimodular, det(A) = 1 or -1.
By Cramer’s rule, x is integral.

10. Example of Totally Unimodular Matrices
A totally unimodular matrix must have every entry equals to +1,0,-1.
Gaussian elimination
And so we see that x must be an integral solution.

11. Example of Totally Unimodular Matrices
is not a totally unimodular matrix,
as its determinant is equal to 2.
x is not necessarily an integral solution.

12. Method 3: Totally Unimodular Matrices
Primal
Dual
Transpose of A
Theorem 2: If A is totally unimodular, and primal and dual are feasible,
then both have integer optimal solutions.
Proof: if A is totally unimodular, then so is its transpose.

13. Application 1: Bipartite Graphs
Let A be the incidence matrix of a bipartite graph.
Each row i represents a vertex v(i),
and each column j represents an edge e(j).
A(ij) = 1 if and only if edge e(j) is incident to v(i).
edges
vertices

14. Application 1: Bipartite Graphs
We’ll prove that the incidence matrix A of a bipartite graph is totally unimodular.
Consider an arbitrary square submatrix A’ of A.
Our goal is to show that A’ has determinant -1,0, or +1.
Case 1: A’ has a column with only 0.
Then det(A’)=0.
Case 2: A’ has a column with only one 1.
By induction, A’’ has determinant -1,0, or +1.
And so does A’.

15. Application 1: Bipartite Graphs
Case 3: Each column of A’ has exactly two 1’s. +1
We can write -1
Since the graph is bipartite, each column has one 1 in Aup and one -1 in Adown
So, by multiplying by +1 the rows in Aup and by -1 those in Adown,
we get that the rows are linearly dependent, and thus det(A’)=0, and we’re done. +1 -1

16. Application 1: Bipartite Graphs
Maximum bipartite matching
Incidence matrix of a bipartite graph,
hence totally unimodular,
And another proof that this LP has integral OS.

17. Application 1: Bipartite Graphs
Maximum general matching
The linear program for general matching does not
come from a totally unimodular matrix, and this is why Edmonds’ result is regarded as a major breakthrough.

18. Application 1: Bipartite Graphs
Theorem 2: If A is totally unimodular,
then both the primal and dual programs are integer programs.
Maximum matching <= maximum fractional matching <=
minimum fractional vertex cover <= minimum vertex cover
Theorem 2 show that the first and the last inequalities are equalites.
The LP-duality theorem shows that the second inequality is an equality.
And so we have maximum matching = minimum vertex cover.

19. Application 2: Directed Graphs
Let A be the incidence matrix of a directed graph.
Each row i represents a vertex v(i),
and each column j represents an edge e(j).
A(ij) = +1 if vertex v(i) is the tail of edge e(j).
A(ij) = -1 if vertex v(i) is the head of edge e(j).
A(ij) = 0 otherwise.
The incidence matrix A of a directed graph is totally unimodular.
Consequences:
The max-flow problem (even min-cost flow) is polynomial time solvable.
Max-flow-min-cut theorem follows from the LP-duality theorem.

20. LP-solver Black Box Problem Solution Polynomial time LP-formulation
Vertex solution
LP-solver
integral

21. Simplex Method Simplex method: A simple and effective approach to
solve linear programs in practice.
It has a nice geometric interpretation.
Idea: Focus only on vertex solutions,
since no matter what is the objective function,
there is always a vertex which attains optimality.

22. Moving along this direction
Simplex Method
Simplex Algorithm:
Start from an arbitrary vertex.
Move to one of its neighbours
which improves the cost. Iterate.
Key: local minimum = global minimum
Global minimum
Moving along this direction
improves the cost.
There is always one neighbor
which improves the cost.
We are here

23. Simplex Method Simplex Algorithm: Start from an arbitrary vertex.
Move to one of its neighbours which improves the cost. Iterate.
Which one?
There are many different rules to choose a neighbor,
but so far every rule has a counterexample so that
it takes exponential time to reach an optimum vertex.
MAJOR OPEN PROBLEM: Is there a polynomial time simplex algorithm?

24. Simplex Method
For combinatorial problems, we know that vertex solutions correspond
to combinatorial objects like matchings, stable matchings, flows, etc.
So, the simplex algorithm actually defines a combinatorial algorithm for these problems.
For example, if you consider the bipartite matching polytope and
run the simplex algorithm, you get the augmenting path algorithm.
The key is to show that two adjacent vertices differ by an augmenting path.
Recall that a vertex solution is the unique solution of n linearly independent tight inequalities. So, moving along an edge in the polytope means to replace one tight inequality by another one. There is one degree of freedom and this corresponds to moving along an edge.

25. Ellipsoid Method Goal: Given a bounded convex set P,
find a point x in P.
Key: show that the volume
decreases fast enough
Ellipsoid Algorithm:
Start with a big ellipsoid which contains P.
Test if the center c is inside P.
If not, there is a linear inequality ax <=b for which c is violated.
Find a minimum ellipsoid which contains the intersection of
the previous ellipsoid and ax <= b.
Continue the process with the new (smaller) ellipsoid.

26. Ellipsoid Method
Goal: Given a bounded convex set P, find a point x in P.
Why it is enough to test if P contains a point???
Because optimization problem can be reduced to this testing problem.
Any point which satisfies this new system is an optimal solution of the original system.

27. Ellipsoid Method Important property: We just need to know the previous
ellipsoid and a violated inequality.
This can help to solve some exponential size LP if we have a separation oracle.
Separation orcale: given a point x, decide in polynomial time
whether x is in P or output a violating inequality.

28. Application of the Ellipsoid Method
Maximum matching
To solve this linear program,
we need a separation oracle.
Given a fractional solution x, the separation oracle needs to determine if x is a feasible solution, or else output a violated inequality.
For this problem, it turns out that we can design a polynomial time separation oracle by using the minimum cut algorithm!
For each odd set S, exponentially many!

29. What We Have Learnt (or Heard)
Stable matchings
Bipartite matchings
Minimum spanning trees
General matchings
Maximum flows
Shortest paths
Minimum Cost Flows
Submodular Flows
Linear programming

30. What We Have Learnt (or Heard)
How to model a combinatorial problem as a linear program.
See the geometric interpretation of linear programming.
How to prove a linear program gives integer optimal solutions?
Prove that every vertex solution is integral.
By convex combination method.
By linear independency of tight inequalities.
By totally unimodular matrices.
By iterative rounding (to be discussed).
By randomized rounding (to be discussed).

31. What We Have Learnt (or Heard)
How to obtain min-max theorems of combinatorial problems?
LP-duality theorem, e.g. max-flow-min-cut, max-matching-min-vertex-cover.
See combinatorial algorithms from the simplex algorithm,
and even give an explanation for the combinatorial algorithms
(local minimum = global minimum).
We’ve seen how results from combinatorial approach follow from results in linear programming. Later we’ll see many results where linear programming is the only approach we know of!