Semidefinite Programming

Lecture 22: Apr 2

Semidefinite Programming

What is semidefinite programming?• A relaxation of quadratic programming.• A special case of convex programing.• A generalization of linear programming.• Can be optimized in polynomial time.

What is it good for?

• Shannon capacity

• Perfect graphs

•Approximation algorithms

• Image segmentation and

clustering

• Constraint satisfaction problems

• Number theory, quantum computation,

etc…

Maximum Cut

(Maximum Cut) Given an undirected graph, with an edge weight w(e) on each edge e, find a partition (S,V-S) of V so as to maximum the total weight of edges in this cut, i.e. edges that have one endpoint in S and one endpoint in V-S.

Maximum Cut

When is computing maximum cut easy?

When we are given a bipartite graph.

The maximum cut problem can also be interpreted asthe problem of finding a maximum bipartite subgraph.

There is a simple greedy algorithm with approximation ratio ½.

Similar to vertex cover.

Quadratic Program for MaxCut

The two sides of the partition.

if they are on opposite sides.

if they are on the same side.

Quadratic Program for MaxCut

This is unlikely to be solved in polynomial time, otherwise P=NP.

This quadratic program is called strict quadratic program,

because every term is of degree 0 or degree 2.

Vector Program for MaxCut

Vector Program for MaxCut

This is a relaxation of the strict quadratic program (why?)

Vector program: linear inequalities over inner products.

Vector program = semidefinite program.

Can be “solved” in polynomial time (ellipsoid, interior point).

Geometric Interpretation

Think of as an n-dimensional vector.

Contribute more to the objectiveif the angle is bigger.

Demonstration

Rubber band method.

László Lovász

Algorithm

(Max-Cut Algorithm)

1. Solve the vector program. Let be an optimal solution.

2. Pick r to be a uniformly distributed vector on the unit sphere .

3. Let

Analysis

Claim:

Analysis

Suppose and has an edge.

Contribution to semidefinite program:

Contribution to the solution:

Approximation Ratio:

Let W be the random variable denoting the weight of edges in the cut.

Analysis

Proof: Linearity of expection.

Claim:

(Max-Cut Algorithm)

1. Solve the vector program. Let be an optimal solution.

2. Pick r to be a uniformly distributed vector on the unit sphere .

3. Let

Algorithm

Repeat a few times to get a good approximation with high probability.

This algorithm performs extremely well in practice.

Try to find a tight example.

Remarks

Hard to imagine a combinatorial algorithm with the same performance.

Assuming the “unique games conjecture”,

this algorithm is the best possible!

That is, it is NP-hard to find a better approximation algorithm!

Constraint Satisfaction Problems

(Max-2-SAT)

Given a formula in which each clause contains two literals,

find a truth assignment that satisfies the maximum number of clauses.

e.g.

An easy algorithm with approximation ratio ½.

An LP-based algorithm with approximation ratio ¾.

An SDP-based algorithm with approximation ratio 0.87856.

Vector Program for MAX-2-SAT

(Max-2-SAT)

Given a formula in which each clause contains two literals,

find a truth assignment that satisfies the maximum number of clauses.

Additional variable (trick):

A variable is set to be true if:

A variable is set to be false if:

Vector Program for MAX-2-SAT

Denote v(C) to be the value of a clause C, which is defined as follows.

Consider a clause containing 2 literals, e.g. . Its value is:

Vector Program for MAX-2-SAT

Objective:

where a(ij) and b(ij) is the sum of coefficients.

(MAX-2-SAT Algorithm)

1. Solve the vector program. Let be an optimal solution.

2. Pick r to be a uniformly distributed vector on the unit sphere .

3. Let be the “true” variables.

Algorithm

Analysis

Term-by-term analysis.

Contribution to semidefinite program:

Contribution to the solution:

Approximation Ratio:

First consider the second term.

Analysis

Term-by-term analysis.

Contribution to semidefinite program:

Contribution to the solution:

Approximation Ratio:

Consider the first term.

(MAX-2-SAT Algorithm)

1. Solve the vector program. Let be an optimal solution.

2. Pick r to be a uniformly distributed vector on the unit sphere .

3. Let be the “true” variables.

Algorithm

This is a 0.87856-approximation algorithm for MAX-2-SAT.

Can be improved to 0.931!

More SDP in approximation algorithms

• Sparsest cut: O(√log n)• Constraint satisfaction problems• Correlation clustering• Graph colouring

A very powerful tool in the design of approximation

algorithms.

Useful to know geometry and algebra.

Next two lectures: SDP and perfect graphs.

Summary: Topics

1. Classical problem: TSP, Steiner trees

2. Covering problem: vertex cover, set cover

3. Packing problem: knapsack, bin packing

4. Graph partitioning problem: multiway cut, multi-cut

5. Job scheduling: makespan, general assignment

6. Network design: Steiner network, degree constrained spanning trees

7. Constraint satisfaction: maximum cut, max 2-SAT

Summary: Techniques

1. Combinatorial arguments: TSP, Steiner trees, multiway cut

2. Greedy algorithm and Randomized rounding: set cover

3. Dynamic programming: knapsack, bin packing

4. Region Growing: multi-cut

5. Iterative relaxation: scheduling, network design problems

6. Primal-dual method: vertex cover, multi-cut in tree

7. Semidefinite programming: maximum cut, constraint satisfaction

Concluding Remarks

Learn to design better heuristics.

e.g. iterative rounding, SDP performs extremely well in practice.

Use LP or SDP as a good way to estimate the optimal value.

e.g. help to test the performance of a heuristic.

Relax the search space to a convex set.

Remaining Schedule

Apr 10-11: NO CLASSES

Apr 17-18: Last week of lecture (perfect graphs, SDP)

Apr 23-24: Project presentation (15 minutes)

May 1: Project report deadline

May 8: Homework 3 deadline