Approximation%20Algoirthms:%20Semidefinite%20Programming

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Approximation Algoirthms Semidefinite Programming

• Lecture 19 Mar 22

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Semidefinite Programming

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

• What is it good for?
• Shannon capacity
• Perfect graphs
• Number theory
• Quantum computation
• Approximation algorithms
• Image segmentation and clustering
• Constraint satisfaction problems
• etc

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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.
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Maximum Cut
When is computing maximum cut easy?
When we are given a bipartite graph.
The maximum cut problem can also be interpreted
as the problem of finding a maximum bipartite
subgraph.
There is a simple greedy algorithm with
approximation ratio ½.
Similar to vertex cover.
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Quadratic Program for MaxCut
The two sides of the partition.
if they are on opposite sides.
if they are on the same side.
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Quadratic Program for MaxCut
This quadratic program is called strict quadratic
program, because every term is of degree 0 or
degree 2.
Of course this is unlikely to be solved in
polynomial time, otherwise PNP.
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Vector Program for MaxCut
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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).
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Geometric Interpretation
Think of as an n-dimensional vector.
Contribute more to the objective if they are more
far apart.
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Demonstration
Rubber band method.
László Lovász
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Algorithm

• (Max-Cut Algorithm)
• Solve the vector program. Let
  be an optimal solution.
• Pick r to be a uniformly distributed vector on
  the unit sphere .
• Let

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Analysis
Claim
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Analysis
Suppose and has an edge.
Contribution to semidefinite program
Contribution to the solution
Approximation Ratio
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Analysis
Proof Linearity of expection.
Let W be the random variable denoting the weight
of edges in the cut.
Claim
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Algorithm

• (Max-Cut Algorithm)
• Solve the vector program. Let
  be an optimal solution.
• Pick r to be a uniformly distributed vector on
  the unit sphere .
• Let

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.
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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!
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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.
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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
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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
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Vector Program for MAX-2-SAT
Objective
where a(ij) and b(ij) is the sum of coefficients.
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Algorithm

• (MAX-2-SAT Algorithm)
• Solve the vector program. Let
  be an optimal solution.
• Pick r to be a uniformly distributed vector on
  the unit sphere .
• Let be
  the true variables.

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Analysis
Term-by-term analysis.
First consider the second term.
Contribution to semidefinite program
Contribution to the solution
Approximation Ratio
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Analysis
Term-by-term analysis.
Consider the first term.
Contribution to semidefinite program
Contribution to the solution
Approximation Ratio
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Algorithm

• (MAX-2-SAT Algorithm)
• Solve the vector program. Let
  be an optimal solution.
• Pick r to be a uniformly distributed vector on
  the unit sphere .
• Let be
  the true variables.

This is a 0.87856-approximation algorithm for
MAX-2-SAT.
Can be improved to 0.931!
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More SDP in approximation algorithms

• Sparsest cut O(vlog n)
• Graph colouring
• Constraint satisfaction problems
• Correlation clustering

Perhaps the most powerful weapon in the design of
approximation algorithms.
Useful to know geometry and algebra.
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Semidefinite Programming

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

• What is it good for?
• Shannon capacity
• Perfect graphs
• Number theory
• Quantum computation
• Approximation algorithms
• Image segmentation and clustering
• Constraint satisfaction problems
• etc

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NP-completeness
!
Do your best then.
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Topics Covered?
A diverse set of techniques.

1. Combinatorial arguments TSP, Steiner trees,
   multiway cut
2. Dynamic programming knapsack
3. Iterative rounding network design problems
4. Randomized rounding congestion minimization, set
   cover
5. Semidefinite programming maximum cut
6. Primal-dual algorithms? Greedy algorithms?

A popular topic graph partitioning problems.
Multicut, sparsest cut, metric labeling,
correlation clustering, etc.
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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.
What is approximable?
Why LP/SDP?