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 as
the 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 objective
if the angle is bigger.
Demonstration Rubber band method. László Lovász
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
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)
Solve the vector program. Let be an optimal solution.
Pick r to be a uniformly distributed vector on the unit sphere .
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.
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